Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices

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Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices

by Alexander Pankov

The Imperial College Press, London, 2005

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Preface

In the past two decades, there has been an explosion of interest to the study of wave propagation in spatially discrete nonlinear systems. Probably, the most prominent example of such a system is the famous Fermi-Pasta-Ulam (FPU) lattice introduced in the pioneering work [Fermi et. al (1955)]. E. Fermi, J. Pasta and S. Ulam studied numerically the lattices of identical particles, i. e. monoatomic lattices, with cubic and quartic interaction potentials. These lattices are known today as α- and β-models, respectively. The aim of E. Fermi, J. Pasta and S. Ulam was to show the relaxation to equipartition of the distribution of energy among modes. Surprisingly enough, their numerical simulation yielded the opposite result. They observed that, at least at low energy, the energy of the system remained confined among the initial modes, instead of spreading towards all modes. This work motivated a great number of further numerical and analytical investigations (for a relatively recent survey of the subject see [Poggi and Ruffo (1997)]). We mention here the so-called Toda lattice which is a completely integrable system. Due to the integrability, the dynamics of Toda lattice is well-understood (see [Toda (1989); Teschl (2000)]). Unfortunately, the Toda lattice is the only known completely integrable lattice of FPU type. Overwhelming majority of existing results concern either particular explicit solutions, both exact and approximate, or numerical simulation. Moreover, almost exclusively spatially homogeneous, i. e. monoatomic, lattices are under consideration, although inhomogeneous lattices (multiatomic lattices, lattices with impurities, etc.) are of great interest. One of the first rigorous results about general FPU type lattices was

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Travelling Waves and Periodic Oscillations in FPU Lattices

obtained in [Friesecke and Wattis (1994)]. G. Friesecke and J. Wattis have proved the existence of solitary travelling waves in monoatomic FPU lattices under some general assumptions on the potential of interparticle interaction. The class of potentials includes the α- and β- models, the Toda potential, the Lennard-Jones potential, and others. The approach of G. Friesecke and J. Wattis is based on an appropriate constrained minimization procedure and the concentration compactness principle of P.-L. Lions [Lions (1984)]. In this approach the wave speed is unknown and is determined a posteriori through the corresponding Lagrange multiplier. Later on D. Smets and M. Willem [Smets and Willem (1997)] considered the travelling wave problem as a problem with prescribed speed. Under another set of assumptions they have proved the existence of travelling waves for every prescribed speed beyond the speed of sound (naturally defined). The proof relies upon an appropriate version of the mountain pass theorem without Palais-Smale condition. In [Pankov and Pfl¨ uger (2000b)], K. Pfl¨ uger and the author revised the last approach considerably choosing periodic travelling waves as a starting point. The existence of periodic waves is obtained by means of the standard mountain pass theorem. Then one gets solitary waves in the limit as the wave lengths goes to infinity. This approach applies to many other problems (see, e. g. [Pankov and Pfl¨ uger (1999); Pankov and Pfl¨ uger (2000a)]). We mention also the series of papers [Friesecke and Pego (1999); Friesecke and Pego (2002); Friesecke and Pego (2004a); Friesecke and Pego (2004b)], where near sonic solitary waves are studied. Under some generic assumptions on the potential of interaction near the origin the existence of such waves is obtained by means of perturbation from the standard Korteweg-de Vries (KdV) soliton. Many properties of near sonic waves are discussed including their dynamical stability. Another line of development was originated by B. Ruf and P. Srikanth [Ruf and Srikanth (1994)] who considered time periodic motions of finite FPU type lattices not necessary consisting of identical particles. Similar problem for infinite lattices, still inhomogeneous, was studied in [Arioli and Chabrowski (1997); Arioli and Gazzola (1995); Arioli and Gazzola (1996); Arioli et. al (1996); Arioli and Szulkin (1997)] under more restrictive assumptions on the potential. Another class of discrete media consists of chains of coupled nonlinear oscillators. One of the most known models of such kind is the so-called

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Preface

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Frenkel-Kontorova model introduced by Ya. Frenkel and T. Kontorova in 1938. As we have learned from [Braun and Kivshar (2004)], the same model have been appeared even before, in works by L. Prandtl and U. Dehlinger (1928–29). For physical applications of the Frenkel-Kontorova and related models we refer to [Braun and Kivshar (1998)] and [Braun and Kivshar (2004)]. Also chains of oscillators as systems that support breathers, i. e. spatially localized time periodic solutions, were studied in many works (see [Aubry (1997); James (2003); Livi et. al (1997); MacKay and Aubry (1994); Morgante et. al (2002)] and references therein). Some other mathematical results that concern time periodic solutions and travelling waves in such systems can be found in [Bak (2004); Bak and Pankov (2004); Bak and Pankov (to appear); Iooss and Kirschgässner (2000)]. Finally, we mention the third class of discrete systems of common interest – discrete nonlinear Schrödinger equations. Such equations are not considered here (see [Flach and Willis (1998); Hennig and Tsironis (1999); Kevrekidis and Weinstein (2003); Pankov and Zakharchenko (2001); Weinstein (1999)] and references therein). Contents The main aim of this book is to present rigorous results on time periodic oscillations and travelling waves in FPU lattices. Also we consider briefly similar results for chains of oscillators. Actually, we confine ourself in the circle of the results obtained by variational methods. Therefore, other approaches, like bifurcation theory and perturbation analysis, are not presented here. As we mentioned before, discrete nonlinear Schrödinger equations are outside the scope of the book. In Chapter 1 we discuss general properties of equations that govern the dynamics of FPU lattices and chains of oscillators, with special attention paid to the well-posedness of the Cauchy problem. Also we remind here basic facts from the spectral theory of linear difference operators that are relevant to linear FPU lattices. Chapter 2 deals with the existence of time periodic solutions in the latices of FPU type. Since we employ global variational techniques, it is not natural to restrict the analysis to the case of spatially homogeneous, i. e. monoatomic, lattices. Instead, we allow periodic spatial inhomogeneities that means that we consider regular multiatomic lattices. We give complete proofs of all principal results. At the same time, for the results that

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require more technicalities we outline basic ideas and skip details. Also, skipping technical details, we present couple of results on the existence of time periodic solutions in some chains of nonlinear oscillators. In Chapters 3 and 4 we study travelling waves in monoatomic FPU lattices. The first of them is devoted to waves with prescribed speed. This statement of problem seems to be most natural. Here we consider two types of travelling waves, periodic and solitary. In fact, we treat solitary waves as a limit case of periodic waves when the wavelength goes to infinity. In Chapter 4 we give some additional results. First of all, we present in details the approach of G. Friesecke and J. Wattis. This approach is technically more involved and, therefore, is postponed to the last chapter. Also we discuss here several other results, including exponential decay of solitary waves, as well as travelling waves in chains of oscillators. Each chapter, except Chapter 3, ends with a special section devoted to various comments and open problems. Comments and open problems that concern travelling waves are put on the end of Chapter 4. Open problems we offer reflect author’s point of view on what should be done next. Some of them are accessible by existing methods, while others are probably hard enough. For reader’s convenience we include four appendices. Their aim is to remind basic facts about functional spaces, concentration compactness, critical points and finite differences, and make the presentation more or less self-contained. Audience As audience we have researchers in mind. Although the book is formally self-contained, some acquaintance with variational methods and nonlinear analysis is recommended. Appropriate references are [Mawhin and Willem (1989); Rabinowitz (1986); Struwe (2000); Willem (1996)] (variational methods) and [Zeidler (1995a); Zeidler (1995b)] (nonlinear analysis). At the same time the present book is accessible to graduate students as well, especially in combinations with the books on variational methods listed above. Acknowledgements The present book was prepared during author’s staying at Texas A&M University and the College of William and Mary as visiting professor. The

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work was also supported in part by NATO, grant 970179. The author thanks T. Bartsch, G. Papanicolaou, P. H. Rabinowitz, A. Szulkin, Z.-Q. Wang and M. Willem for many interesting discussions and valuable information. Last but not least, I am deeply grateful to my wife Tanya for her generous support. A. Pankov

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Contents

Preface 1.

Infinite Lattice Systems 1.1 1.2 1.3 1.4 1.5

2.

vii 1

Equations of motion . . . . . . . . . . The Cauchy problem . . . . . . . . . . Harmonic lattices . . . . . . . . . . . . Chains of coupled nonlinear oscillators Comments and open problems . . . . .

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Time Periodic Oscillations

1 7 10 16 24 27

2.1 Setting of problem . . . . . . . . . . . . . 2.2 Positive definite case . . . . . . . . . . . . 2.3 Indefinite case . . . . . . . . . . . . . . . . 2.3.1 Main result . . . . . . . . . . . . . 2.3.2 Periodic approximations . . . . . . 2.3.3 Proof of main result . . . . . . . . 2.4 Additional results . . . . . . . . . . . . . . 2.4.1 Degenerate case . . . . . . . . . . . 2.4.2 Constrained minimization . . . . . 2.4.3 Multibumps . . . . . . . . . . . . . 2.4.4 Lattices without spatial periodicity 2.4.5 Finite lattices . . . . . . . . . . . . 2.5 Chains of oscillators . . . . . . . . . . . . 2.6 Comments and open problems . . . . . . . 3.

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Travelling Waves: Waves with Prescribed Speed xiii

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27 34 42 42 46 54 56 56 58 59 61 62 64 70 75

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3.1 Statement of problem . . . . . . . . . . . . 3.2 Periodic waves . . . . . . . . . . . . . . . . 3.2.1 Variational setting . . . . . . . . . . 3.2.2 Monotone waves . . . . . . . . . . . 3.2.3 Nonmonotone and subsonic waves . . 3.3 Solitary waves . . . . . . . . . . . . . . . . . 3.3.1 Variational statement of the problem 3.3.2 From periodic waves to solitary ones 3.3.3 Global structure of periodic waves . 3.3.4 Examples . . . . . . . . . . . . . . . 3.4 Ground waves: existence and convergence . 3.4.1 Ground waves: periodic case . . . . . 3.4.2 Solitary ground waves . . . . . . . . 3.4.3 Monotonicity . . . . . . . . . . . . . 3.5 Near sonic waves . . . . . . . . . . . . . . . 3.5.1 Amplitude estimate . . . . . . . . . . 3.5.2 Nonglobally defined potentials . . . . 4.

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Travelling Waves: Further Results 4.1 Solitary waves and constrained minimization . . . . . 4.1.1 Statement of problem . . . . . . . . . . . . . . . 4.1.2 The minimization problem: technical results . . 4.1.3 The minimization problem: existence . . . . . . 4.1.4 Proof of main result . . . . . . . . . . . . . . . 4.1.5 Lennard-Jones type potentials . . . . . . . . . . 4.2 Other types of travelling waves . . . . . . . . . . . . . 4.2.1 Waves with periodic profile functions . . . . . . 4.2.2 Solitary waves whose profiles vanish at infinity . 4.3 Yet another constrained minimization problem . . . . 4.4 Remark on FPU β-model . . . . . . . . . . . . . . . . 4.5 Exponential decay . . . . . . . . . . . . . . . . . . . . 4.6 Travelling waves in chains of oscillators . . . . . . . . . 4.7 Comments and open problems . . . . . . . . . . . . . .

Appendix A

Functional Spaces

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A.1 Spaces of sequences . . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Spaces of functions on real line . . . . . . . . . . . . . . . . 168 Appendix B Concentration Compactness

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Contents

Appendix C

Critical Point Theory

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C.1 Differentiable functionals . . . . . . . . . . . . . . . . . . . . 177 C.2 Mountain pass theorem . . . . . . . . . . . . . . . . . . . . 178 C.3 Linking theorems . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix D

Difference Calculus

183

Bibliography

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Index

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Chapter 1

Infinite Lattice Systems

1.1

Equations of motion

We consider a one dimensional chain of particles with nearest neighbor interaction. Equations of motion of the system read 0 m(n) q¨(n) = Un+1 q(n + 1) − q(n) − Un0 q(n) − q(n − 1) ,

n ∈ Z. (1.1)

Here q(n) = q(t, n) is the coordinate of n-th particle at time t, m(n) is the mass of that particle, and Un is the potential of interaction between n-th and (n − 1)-th particles. We always assume that there are positive constants m0 and M0 such that m0 ≤ m(n) ≤ M0 for every n ∈ Z. Equations (1.1) form an infinite system of ordinary differential equations which is a Hamiltonian system with the Hamiltonian H=

∞ 2 X p (n) + Un q(n + 1) − q(n) , 2m(n) n=−∞

(1.2)

where p(n) = m(n) q(n) ˙ is the momentum of n-th particle. Formally this statement is readily verified. However, to make it precise first one has to specify the phase space. The simplest, but not so natural from the point of view of physics, choice of the configuration space is the space l2 of two-sided sequences1 1 For

the definitions and notations of spaces of sequences see Appendix A.1. 1

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q = {q(n)}n∈Z . This corresponds to the boundary condition lim q(n) = 0

n→±∞

(1.3)

at infinity. In this case the phase space is l2 × l2 and Eq. (1.1) can be written as the first-order system u˙ = J ∇H(u), where u=

q , p

J=

0 I −I 0

: l2 × l2 → l2 × l2 ,

I is the identity operator and ∇H the functional gradient of H 0 0 q(n + 1) − q(n) Un q(n) − q(n − 1) − Un+1 . ∇H(u)(n) = p(n)/m(n) Denote by G the nonlinear operator defined by G(q)(n) = Un0 q(n) , n ∈ Z,

(1.4)

where q = {q(n)}, and consider operators of right and left differences (∂ + q)(n) := q(n + 1) − q(n) and (∂ − q)(n) := q(n) − q(n − 1), respectively. We suppose that G is a “good” nonlinear operator in l2 . Then −∂ + G(∂ − q) ∇H(u) = , (1.5) p/m while Eq. (1.1) becomes a “divergence form” equation m¨ q = ∂ + G(∂ − q).

(1.6)

Note that ∂ + and ∂ − are bounded linear operators in l2 and (∂ + )∗ = −∂ − . Another form of Eq. (1.6) is m¨ q = ∂ − G+ (∂ + q),

(1.7)

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where 0 G+ (q)(n) = Un+1 q(n) .

(1.8)

However, more natural and most important choice of configuration space is the space X = e l2 that consists of two-sided sequences q = {q(n)}n∈Z such + 2 that ∂ q ∈ l . Endowed with the norm 1/2 1/2 kqkX = k∂ + qk2l2 + |q(0)|2 = k∂ − qk2l2 + |q(0)|2 , X is a Hilbert space. Obviously, k∂ − qkl2 = k∂ + qkl2 . Operators ∂ + and ∂ − are linear bounded operators from the space X onto l2 and have one dimensional kernel that consists of constant sequences. Equation (1.1) (equivalently, (1.6)) is a Hamiltonian system on the phase space e l2 × l2 . In this case the corresponding symplectic form [Marsden and Ratiu (1994)] is degenerate. Nevertheless, the Hamiltonian H defined by (1.2) is a conserved quantity provided H(q, p) is C 1 on e l2 × l2 . This can be verified by a direct calculation. Now we introduce a reformulation of Eq. (1.6) in e l2 as an equation in l . Denote by 2

r(n) := q(n + 1) − q(n), i. e. r = ∂ + q, the relative displacements of adjacent lattice sites and set b(n) := a(n − 1) = m(n)−1/2 . Then Eq. (1.1) gives immediately h i 0 r¨(n) = a2 (n) Un+1 r(n + 1) − Un0 r(n) h i 0 − a2 (n − 1) Un0 r(n) − Un−1 r(n − 1) .

(1.9)

Note that r ∈ l2 whenever q ∈ e l2 . In operator form, Eq.(1.9) reads r¨ = ∂ − a2 ∂ + G(r) . Also it can be written as (see Appendix D, Eq. (D.5)) r¨ = ∂ + b2 ∂ − G+ (r) ,

(1.10)

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where 0 G+ (r)(n) = Un+1 r(n) . Equation (1.10) is equivalent to the following first-order system r ∂ + (bs) u˙ = F (u), u = , F (u) = . (1.11) s b ∂ − G+ (r) This is a Hamiltonian system u˙ = J∇H(u),

(1.12)

where J=

0 ∂+b − −b ∂ 0

(1.13)

and H(r, s) =

∞ X s(n)2 + Un+1 r(n) . 2 n=−∞

(1.14)

In fact, here s = bp = p/m1/2 . The phase space of this system is l2 × l2 . It is readily verified that (∂ + b)∗ = −b∂ − . Certainly, H(r, s) defined by (1.14) is a conserved quantity if H is C 1 on l2 × l2 . Now let us discuss the relation between solutions of Eq. (1.6) and Eq. (1.10) (or (1.11)). Consider a solution q = q(t, n) of Eq. (1.6) such that q is a C 1 function of t with values in X = e l2 and q˙ is a C 1 function with values in l2 . Then r = ∂ + q and s = ap are C 1 functions with values in l2 , and u = (r, s) obviously solves (1.11). Moreover, the well-posedness, local or global in time, of the Cauchy problem for Eq. (1.6) in e l2 ×l2 implies 2 the same property for Eq. (1.11) (and (1.10)) in the space l × l2 . Conversely, consider the Cauchy problem for (1.6), with q|t=0 = q (0) ∈ e l2 ,

q| ˙ t=0 = q (1) ∈ l2 .

Set r(0) = ∂ + q (0) ,

s(0) = m1/2 q (1) .

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Let u = (r, s) be the solution of Cauchy problem for Eq. (1.11), with u|t=0 = r(0) , s(0) ∈ l2 × l2 . Define q(t) = q (0) +

Z

t

bs(τ ) dτ. 0

Equation (1.11) implies that ∂ + q = r. We have q˙ = bs and, by Eq. (1.11), q¨ = bs˙ = b2 ∂ − G+ (r) = b2 ∂ − G+ (∂ + q). Hence, m¨ q = ∂ − G+ (∂ + q) which is another form (1.7) of Eq. (1.6). Obviously q| ˙ t=0 = q (1) . Thus, the two statements of problem, (1.6) and (1.11), are equivalent. Now let us consider several examples. In these examples we always have m(n) ≡ 1, Un (r) ≡ U (r). Example 1.1

Let U (r) =

c0 2 r , 2

c0 > 0.

Then we obtain the discrete wave equation q¨ = c0 ∆q,

(1.15)

where ∆ = ∂ + ∂ − is the discrete one-dimensional Laplacian. Example 1.2

If U (r) =

c0 2 c1 3 r + r , 2 3

c0 > 0,

c1 6= 0,

(cubic interaction), we obtain the so-called Fermi-Pasta-Ulam α-model. Example 1.3

The Fermi-Pasta-Ulam β-model is defined by U (r) =

(quartic interaction).

c0 2 c2 4 r + r , 2 4

c0 > 0,

c2 > 0

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Example 1.4 The well-known Toda lattice [Toda (1989)] has the potential of interaction U (r) = ab−1 (e−br + br − 1). This is an example of completely integrable Hamiltonian system [Toda (1989); Teschl (2000)]. Example 1.5 The Lennard-Jones potential [Friesecke and Wattis (1994)] is a singular potential defined by 2 U (r) = a (d+r)−12 −2d−6 (d+r)−6 +d−12 = a (d+r)−6 −d−6 ,

r > −d.

More generally, one considers the potentials 2 U (r) = a (d + r)−k − d−k , with a > 0, d > 0 and positive integer k [Friesecke and Matthies (2002)]. In classical FPU lattices m(n) ≡ m0 and Un does not depend on n ∈ Z. Such lattices are often called monoatomic lattices. More general class of FPU type lattices consists of systems with periodic dependence of m(n) and Un on n ∈ Z, i. e. m(n + N ) = m(n), Un+N = Un . This is the class of multiatomic, or N-atomic lattices. If N = 2, such systems are called diatomic lattices. Another interesting class of FPU type systems, lattices with impurities, appears when we assume that m(n) and Un are close to periodic for large (0) values of |n|, i. e. m = m + m(0) and Un (r) = U n (r) + Un (r), where m(n) and U n are N -periodic in n, while lim m(0) (n) = 0,

|n|→∞

lim Un(0) (r) = 0.

|n|→∞

(0)

In particular, if m(0) (n) and Un vanish for all, but finite, values of n, such a system can be viewed as a multiatomic lattice perturbed by replacing a finite number of original particles by particles of another sort, i. e. introducing an impurity.

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1.2

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The Cauchy problem

Consider the Cauchy problem for Eq. (1.12) or, equivalently, for Eq. (1.13) with the data r|t=0 = r(0) ∈ l2 ,

r| ˙ t=0 = r(1) ∈ l2 ,

(1.16)

u|t=0 = u(0) = r(0) , s(0) ∈ l2 × l2 .

(1.17)

or, respectively,

We impose the following assumptions. (i) There exist M0 ≥ m0 ≥ 0 such that m0 ≤ m(n) ≤ M0 ,

n ∈ Z.

(ii) The potential Un (r) is a C 1 function on R, Un (0) = Un0 (0) = 0 and for every R there exists C(R) > 0 such that for all n ∈ Z 0 Un (r1 ) − Un0 (r2 ) ≤ C(R) |r1 − r2 |, |r1 |, |r2 | ≤ R. (1.18) Lemma 1.1 Under assumption (ii) the operators G and G+ (see Eqs. (1.4) and (1.8)) act in l2 and are bounded locally Lipschitz continuous operators, i. e.

G q (1) − G q (2) ≤ C(R) q (1) − q (2) l2 , q (1) l2 , q (2) l2 ≤ R, l2

and similar inequalities for G+ . Proof . Let q ∈ l2 and kqkl2 ≤ R. Then kqkl∞ ≤ R and (1.18) implies that kG(q)kl2 ≤ C kqkl∞ kqkl2 ≤ C(R) kqkl2 . Hence, G acts from l2 into itself. The remaining parts of the lemma are similar. Since b = m−1/2 is a bounded sequence and the operators ∂ + and ∂ − are bounded in l2 , the operator F in (1.11) acts from l2 ×l2 into itself and is a bounded locally Lipschitz continuous operator in that space. Therefore, the standard infinite dimensional version of the classical Picard theorem (see, e. g., [Dalec’kii and Krein (1974); Zeidler (1986); Zeidler (1995b)]) implies the following local well-posedness result

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Theorem 1.1

Under assumptions (i) and (ii), for every u(0) = r(0) , s(0) ∈ l2 × l2

there exists a unique solution u = (r, s) ∈ C 1 (−a, a), l2 × l2

of problem (1.13), (1.17). For any ball B ⊂ l2 × l2 of initial data u(0) the 0 number a ∈ R can be chosen independent on u(0) and for every a ∈ (0, a) 1 0 0 2 2 the solution u as an element of C [−a , a ], l × l depends continuously on the initial data u(0) ∈ B. The problem of global existence is more delicate. A simple result of such kind is Theorem 1.2 Suppose that assumptions (i) and (ii), with (1.18) replaced by stronger inequality 0 Un (r1 ) − Un0 (r2 ) ≤ C |r1 − r2 |, r1 , r2 ∈ R, (1.19) are satisfied. Then for every u(0) ∈ l2 × l2 problem (1.13), (1.17) has a unique solution defined on R. This theorem is a particular case of a well-known result (see, e. g., [Dalec’kii and Krein (1974)], Theorem 1.2 of Chapter 8). However, it has a limited range of applications, since Eq. (1.19) means in particular that the potential Un has at most quadratic growth at infinity. Theorem 1.3 either

Suppose that assumptions (i) and (ii) are satisfied and

(a) Un (r) ≥ 0 for all n ∈ Z and r ∈ R, or (b) there exists a nondecreasing continuous function h(r), r ≥ 0, such that lim h(r) = +∞ and for every n ∈ Z r→∞

Un (r) ≥ h |r| , r ∈ R. Then for every u(0) = r(0) , s(0) ∈ l2 × l2 Cauchy problem (1.13), (1.17) has a unique global solution, i. e. solution defined for all t ∈ R.

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Proof . It is readily verified that the Hamiltonian H(r, s) defined by Eq. (1.14) is a C 1 functional on l2 × l2 and, hence, a conserved quantity. Thus, for any solution u(t) = r(t), s(t) of problem (1.13), (1.17) H r(t), s(t) = H r(0) , s(0) . In case (a) this implies that 1 ks(t)k2 ≤ H r(0) , s(0) . 2 Since r˙ = ∂ + (bs), we obtain by integration that kr(t)k remains bounded on any finite interval of existence of solution. This is enough to conclude that the solution is defined everywhere on R (see, e. g., [Reed and Simon (1975)], Theorem X.74). Now we consider case (b). Fix H0 ≥ 0 such that H r(0) , s(0) ≤ H0 . Conservation of the Hamiltonian implies that 1 s(t, n)2 + h |r(t, n)| ≤ H0 . 2 Let r > 0 be a solution of the equation h(r) = H0 . Obviously, such a solution exists. Then |r(t, n)| ≤ r. Now we introduce a modified potential defined as follows. Let ψ(r) be an even function such that  if 0 ≤ r ≤ r, 1 ψ(r) = −r + r + 1 if r ≤ r ≤ r + 1,  0 if r ≥ r. en (r) by the formula Define U Z en (r) = U

r

h

i ψ(%) Un0 (%) + 1 − ψ(%) % d%.

0

en satisfies assumption (1.19). Hence, It is a simple exercise to verify that U the modified Hamiltonian satisfies the assumptions of Theorem 1.2. On the e coincides with H. Therefore, u(t) extends solution u(t) = r(t), s(t) , H to a global solution of the modified system. An elementary, but somewhat

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en (r) ≥ e long, calculation shows that U h |r| , where  h(r),   3 0 ≤ r ≤ r, Z r  3 2  r r r   (r + 1 − r) h(r) + h(%) d% + −r + , r ≤ r ≤ r + 1, e 3 2 6 h(r) = r  Z r+1   (r + 1)2 r3    r2 + h(%) d% + + , r ≥ r + 1. 3 6 r Since e h(r) ≥ e h(r) = h(r) = H0

for r ≥ r,

the argument from the beginning of the proof of case (b) shows that the extended solution satisfies r(t) ≤ r and, therefore, is a solution of the original problem. Remark 1.1 Theorems 1.1–1.3 imply (and are equivalent to) corresponding results on well-posedness of the Cauchy problem for Eq. (1.1) on e l2 × l2 . Certainly, these parallel statements can be obtained directly by similar arguments. Local Theorem 1.1 applies to all examples considered in Section 1.1 except the singular Lennard-Jones potential. Global existence for the βmodel (Example 1.3), as well as for the Toda lattice (Example 1.4), follows immediately from Theorem 1.3. For the α-model (Example 1.2) we expect that global in time solutions do exist for some set of initial data, while for all other initial data the solutions blow up at finite time. For the lattice with Lennard-Jones potential the existence of global so lution holds true for all initial data u(0) = r(0) , s(0) ∈ l2 × l2 such that r(0) (n) > −d for all n ∈ Z. In this case one can follow the proof of Theorem 1.3, case (b), modifying the potential near the singularity and behind it to reduce the problem to a nonsingular one. 1.3

Harmonic lattices

Here we consider harmonic lattices, i. e. lattices with quadratic interaction potential Un (r) =

c(n) 2 r . 2

Throughout this section we assume that

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(h) there exist positive constants m0 , M0 and C such that 0 < m0 ≤ m(n) ≤ M0 and |c(n)| ≤ C. The dynamics of the harmonic lattice is governed by the equation m¨ q = ∂ + c ∂ − q,

(1.20)

with the phase space e l2 × l2 (see Eq. (1.9)). In other words, we consider solutions such that q is a C 1 function with values in e l2 , while q˙ is a C 1 function with values in l2 . Making use the change of unknown x = m1/2 q, we see that Eq. (1.20) is equivalent to the equation x ¨ = m−1/2 ∂ + c ∂ − m−1/2 x

(1.21)

considered in the phase space e l2 × l2 . Let us introduce the operator A by the formula −Ax = m−1/2 ∂ + c ∂ − m−1/2 x.

(1.22)

Notice that the operators −∂ + and ∂ − are formally adjoint and the same holds true for the operators −m−1/2 ∂ + and ∂ − m−1/2 . Now Eq. (1.21) becomes x ¨ = −Ax.

(1.23)

The operator A defined by (1.22) is a bounded linear operator in the space e l2 and, actually, the range of A lies in l2 . The last follows from the fact that the operator ∂ − maps e l2 into l2 , while the multiplication by any bounded sequence, as well as the operator ∂ + , leave l2 invariant. Moreover, being restricted to the space l2 , the operator A is self-adjoint. In what follows the restriction of A to l2 is still denoted by A. The Cauchy problem for Eq.(1.20), with q|t=0 = q (0) ∈ e l2 ,

q| ˙ t=0 = q (1) ∈ l2 ,

is equivalent to the Cauchy problem for Eq. (1.23), with x|t=0 = x(0) := m1/2 q (0) ∈ e l2 ,

x| ˙ t=0 = x(1) := m1/2 q (1) ∈ l2 .

(1.24)

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It is well-known (see, e. g., [Dalec’kii and Krein (1974); Deimling (1977)]) that the solution of (1.23), (1.24) is given by x = cos tA1/2 x(0) + A−1/2 sin tA1/2 x(1) , (1.25) where the operator functions cos tA1/2 and A−1/2 sin tA1/2 are defined by means of power series expansions ∞ X (−1)k t2k k cos tA1/2 := A , (2k)!

A−1/2 sin tA

1/2

:=

k=0 ∞ X

k=0

(−1)k t2k+1 k A . (2k + 1)!

Being considered in the space of all bounded linear operators in e l2 , these 2 e expansions are norm convergent. The same holds true with l replaced by l2 . Note that in the notations of these functions A1/2 has no independent meaning. Since x(1) ∈ l2 , the function A−1/2 sin tA1/2 in the second term on the right hand part of Eq. (1.25) can be considered as a function of self-adjoint operator A in l2 . A direct calculation shows that "∞ # X (−1)k+1 t2k+2 1/2 cos tA −I = A (2k + 2)! k=0 Z t =− A−1/2 sin τ A1/2 dτ A. 0

Therefore, formula (1.25) becomes Z t (0) −1/2 1/2 A sin τ A dτ Ax(0) +A−1/2 sin tA1/2 x(1) . (1.26) x=x − 0

Another way to derive (1.26) is to look for the solution of (1.23), (1.24) in the form x = x(0) + u, where u is a function with values in l2 , and reduce the problem to the nonhomogeneous Cauchy problem u ¨ = −Au − Ax(0) ,

u|t=0 = 0,

u| ˙ t=0 = x(1) ,

in the space l2 , with the operator A being self-adjoint. 2 The advantage of Eq. (1.26) is that, since x(1) and Ax(0) are both in l , −1/2 1/2 of the the solution is expressed in terms of the function A sin tA 2 self-adjoint operator A in l .

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Actually, A is a second order difference operator of the form (Ax)(n) = a(n) x(n + 1) + a(n − 1) x(n − 1) + b(n) x(n)

(1.27)

(Jacobi operator), where c(n)

a(n) = − p

m(n) m(n + 1)

(1.28)

and b(n) =

c(n) + c(n − 1) . m(n)

(1.29)

This follows immediately from Eq. (1.22). Looking for solutions to Eq. (1.23) in the form2 x = exp(iωt) u(n), we arrive at the spectral problem Au − λu = 0,

(1.30)

where λ = ω 2 . The spectral theory of Jacobi operators is well-understood (see, e. g., [Teschl (2000)] and references therein). Since, due to assumption (h), the operator A is a bounded self-adjoint operator in l2 , its spectrum σ(H) is a bounded closed subset of R. More precisely, one has (see, e. g., [Teschl (2000)], Lemma 1.8 and (1.151)) Proposition 1.1 (i) Let h i a+ = sup b(n) + a(n) + a(n − 1) , n∈Z h i a− = inf b(n) − a(n) − a(n − 1) . n∈Z

Then σ(A) ⊂ [a− , a+ ]. (ii) If c(n) > 0 for all n ∈ Z, then the operator A is nonnegative, σ(A) ⊂ 0, 4 kckl∞ km−1 kl∞ , and 0 is in the essential spectrum σess (A). 2 These

are standing waves if ω ∈ R.

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Now let us consider the case of harmonic N -atomic lattices, i. e. we assume that m(n + N ) = m(n) and c(n + N ) = c(n) for all n ∈ Z. First, we introduce (a discrete version of) the Floquet transform as X f (n − kN ) eixθ . (1.31) (Uf )(n, θ) = k∈Z

Sometimes this transform is called Gelfand transform. (For the theory of continuous Floquet transform and its applications we refer to [Kuchment (1993)]). The parameter θ is called quasimomentum. Observe that (Uf )(n + N, θ) = eiθ (Uf )(n, θ).

(1.32)

This is the so-called Floquet condition. In physics literature this condition is called the Bloch condition. Also we have (Uf )(n, θ + 2π) = (Uf )(n, θ),

(1.33)

i. e. the Floquet transform is 2π-periodic with respect to quasimomentum. Thus, the quasimomentum θ can be considered as the angle coordinate on the unit circle S1 . The periodicity assumption implies that the operator A commutes with the Floquet transform (UAf )(n, θ) = A(Uf )(n, θ).

(1.34)

The operator A on the right hand side of Eq. (1.34) acts on the function of the variable n satisfying Floquet condition (1.32), with θ being a parameter. Let E denote the space of all complex valued functions on the set IN = {0, 1, . . . , N − 1} and Eθ the space of all functions u on Z satisfying the Floquet condition u(n + N ) = eiθ u(n),

n ∈ Z.

(1.35)

Being endowed with the standard inner product, E becomes an N dimensional Euclidian space. The restriction u|IN defines an isomorphism between Eθ and E. The Floquet transform is a unitary operator U : l2 → L2 (S1 ; E). The inverse operator U −1 is defined by the formula Z dθ U −1 v (n) = v(n, θ) , 2π 1 S where v(n, θ) is extended from IN to Z according to (1.35).

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The operator A is well-defined on Eθ and is, in fact, a self-adjoint operator. Denote this operator by A(θ). What we have described before, can be summarized in terms of direct integrals as Theorem 1.4 The Floquet transform provides a unitary equivalence between the operator A and the direct integral expansion Z ⊕ dθ A(θ) . 2π 1 S For the discussion of the notion of direct integral see, e. g., [Reed and Simon (1978)]. Due to the isomorphism between Eθ and E, the operators A(θ) can be considered as acting in E. Actually, A(θ) form a real analytic family of self-adjoint operators in the space E. Let λ1 (θ) ≤ λ2 (θ) ≤ · · · ≤ λN (θ) be the eigenvalues of A(θ). The functions λj (θ), j = 1, 2, . . . , N , are continuous (actually, piecewise analytic) on S1 . They are called band functions or dispersion relations. Let λ− λj (θ), j = min 1 θ∈S

Theorem 1.5

λ+ λj (θ). j = max 1 θ∈S

The spectrum σ(A) is absolutely continuous and N [ − + σ(A) = λj , λ j .

(1.36)

j=1

The intervals in (1.36) are called spectral bands, while the intervals − λ+ , λ are called spectral gaps. Some of the gaps may be empty (closed). j j+1 In general, at most (N − 1) gaps open up. The eigenfunctions of A(θ) are the generalized eigenfunctions of A called Bloch eigenfunctions. Corresponding solutions of Eq. (1.23) x(t, n) = exp(iωt) u(n),

ω 2 = λj (θ),

– so-called Bloch waves – have infinite energy. If c ≥ 0, then A ≥ 0 and σ(A) is nonnegative. Therefore, all the eigenfrequencies are real and the Bloch waves are bounded. In the N -atomic case the Bloch waves are proper analoges of plane waves that occur in the monoatomic case.

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For more details about spectral theory of periodic Jacobi operators, including generalized eigenfunction expansions, we refer to [Teschl (2000)]. Now we formulate a spectral result on harmonic lattices with impurities. Let m = m + m(0) and c = c + c(0) , where m and c are N -periodic, while lim m(0) (n) = lim c(0) (n) = 0.

|n|→∞

|n|→∞

Denote by A the operator (1.22) with the coefficients m and c. Theorem 1.6 σess (A) = σ(A). If in addition |n| c(0) (n) ∈ l1 , then the point spectrum of A is finite and is located in R \ σ(A), and the essential spectrum of A is absolutely continuous. See [Teschl (2000)], Theorem 7.11. Finally, let us consider the case of monoatomic harmonic lattice. The problem reduces to the equation x ¨(n) = c20 x(n + 1) + x(n − 1) − 2x(n) , (1.37) where we assume that c0 > 0. The so-called plane wave solutions are given by x(t, n) = exp i(κn ± ωt) = e±iωt eiκn , where the wavelength κ −1 and the frequency ω are related by the dispersion relation √ √ κ ω = ± 2 c0 1 − cos κ = ±c0 sin . 2 Note that plane waves are just Bloch waves in the monoatomic case. For the spectral theory of differential and difference operators we refer to [Atkinson (1964); Berezanskii (1968); Berezin and Shubin (1991); Dunford and Schwartz (1988b); Edmunds and Evans (1987); Glazman (1966); Lanczos (1961); Levitan and Sargsyan (1975); Reed and Simon (1978); Schechter (1981)]. 1.4

Chains of coupled nonlinear oscillators

Here we consider another lattice model – an infinite chain of coupled nonlinear oscillators.

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Let q(n) be a generalized coordinate of n-th oscillator and Un (r) the potential of the oscillator, i. e. the dynamics of n-th oscillator is given by q¨(n) = −Un0 q(n) , provided the interaction between the oscillators is absent. Suppose that every oscillator interacts with two its neighbors via linear forces. Then the master equations of the system are q¨(n) = −Un0 q(n) +a(n−1) q(n−1)−q(n) −a(n) q(n)−q(n+1) , (1.38) where n ∈ Z. Let Un (r) = −

c(n) 2 r + Vn (r), 2

where V (0) = V 0 (0) = 0. This means that r = 0 is a rest point of each oscillator. Set b(n) = c(n) − a(n) − a(n − 1). Then Eq. (1.38) becomes q¨(n) = a(n) q(n+1)+a(n−1) q(n−1)+b(n) q(n)−Vn0 q(n) , n ∈ Z. (1.39) We impose the following boundary condition at infinity lim q(n) = 0.

n→±∞

(1.40)

This condition means that at infinity the system is at rest. In what follows we consider the case when the oscillators are anharmonic, i. e. Vn 6= 0. The case of harmonic chains is described by linear equation (1.23) with m ≡ 1 and, hence, reduces to the spectral theory of difference operators. We impose the following two assumptions (cf. assumptions (i) and (ii) of Section 1.2): (i0 ) the sequences a(n) and c(n) are bounded; (ii0 ) the potential Vn (r) is C 1 on R, Vn (0) = Vn0 (0) = 0 and for every R > 0 there exists C = C(R) > 0 such that 0 Vn (r1 ) − Vn0 (r2 ) ≤ C|r1 − r2 |, |r1 − r2 | ≤ R, (1.41) for all n ∈ Z.

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The natural configuration space that incorporates boundary condition (1.40) is the space l2 of two-sided sequences. The corresponding phase space is l2 × l2 . Let us introduce the linear operator (Aq)(n) = a(n) q(n + 1) + a(n − 1) q(n − 1) + b(n) q(n) and the nonlinear operator B(q)(n) = −Vn0 q(n) . Under assumptions (i0 ) and (ii0 ) A is a bounded self-adjoint operator in l2 , while the operator B is bounded and locally Lipschitz continuous. Equation (1.39) together with boundary condition (1.40) can be naturally written as an operator differential equation q¨ = Aq + B(q)

(1.42)

in the space l2 . Equation (1.42) is a Hamiltonian system on l2 ×l2 , with the Hamiltonian H(q, p) =

+∞ i X 1h kpk2 − (Aq, q) + Vn q(n) , 2 n=−∞

(1.43)

where p = q. ˙ Under assumptions (i0 ) and (ii0 ) H(p, q) is a C 1 functional 2 2 on l × l and, hence, H is a conserved quantity. Consider the Cauchy problem for (1.42) with the initial data q|t=0 = q (0) ∈ l2 ,

q| ˙ t=0 = q (1) ∈ l2 .

(1.44)

Its local well-posedness under assumptions (i0 ) and (ii0 ) is a consequence of standard results (cf. Theorem 1.1). Exactly as in the case of FPU lattices (see Theorems 1.2 and 1.3) we have also the following results. Theorem 1.7 Assume (i0 ) and (ii0 ) with the constant C independent of R. Then for every q (0) ∈ l2 and q (1) ∈ l2 Cauchy problem (1.42), (1.43) has a unique solution defined for all t ∈ R. Theorem 1.8 In addition to assumptions (i0 ) and (ii0 ), assume that the operator A is non-positive, i. e. (Aq, q) ≤ 0 for all q ∈ l2 . Suppose also that one of the following two conditions holds: (a) Vn (r) ≥ 0 for all n ∈ Z and r ∈ R

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or (b) there exists a nondecreasing function h(r), r ≥ 0, such that lim h(r) = +∞ and Vn (r) ≥ h |r| for all n ∈ Z and r ∈ R. r→+∞

Then for every q (0) ∈ l2 and q (1) ∈ l2 problem (1.42), (1.43) has a unique solution defined for all t ∈ R. As it follows from the next proposition, under some additional condition one can skip the assumption of nonpositivity of A in Theorem 1.8. Proposition 1.2 Under assumptions of Theorem 1.8, case (b), without (Aq, q) ≤ 0, suppose that lim

r→+∞

h(r) = +∞. r2

Then problem (1.42), (1.43) has a unique global solution for every q (0) ∈ l2 and q (1) ∈ l2 . Proof . Write the potential Un in the form Un (r) = −

c(n) − 2λ 2 r + Vn (r) − λr2 ), 2

with λ > 0 large enough. Then the new operator A that corresponds to the coefficients a(n) and c(n) − 2λ is non-positive. At the same time ! r2 2 2 . Vn (r) − λr ≥ h |r| − λr = h |r| 1 − λ h |r| This yields Vn (r) − λr2 ≥ k1 h |r| − k2 , with some k1 ∈ (0, 1) and k2 ≥ 0. Now it is enough to apply Theorem 1.8, with h(r) replaced by k1 h(r) − k2 . The proof is complete. Now we consider few examples. In all these examples a(n) and Un (r) do not depend on n ∈ Z. Example 1.6

Taking a(n) ≡ a > 0, c(n) ≡ 0 and Vn (r) = 1 − cos r,

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one obtains the Frenkel-Kontorova model. The master equation of this model reads q¨(n) = a (∆d q)(n) − sin q(n),

(1.45)

where (∆d q)(n) = q(n + 1) + q(n − 1) − 2q(n) is the 1-dimensional discrete Laplacian. Equation (1.45) is a straightforward discretization of the famous sin-Gordon equation qtt − aqxx + sin q = 0. The last equation is a completely integrable system (see, e. g. [Ablowitz and Segur (1981)]). At the same time its discrete counterpart, Eq. (1.45), is not completely integrable. Example 1.7

When a(n) ≡ a > 0, c(n) ≡ −m2 ≤ 0 and Vn (r) = −

λ rk+1 , k+1

k > 0 integer,

we obtain the discrete nonlinear Klein-Gordon (DNKG) equation (m2 > 0) q¨(n) = a (∆d q)(n) − m2 q(n) + λq k (n)

(1.46)

and discrete nonlinear wave (DNW) equation (m2 = 0) q¨(n) = a (∆d )(n) + λq k (n).

(1.47)

The cubic (k = 3) and quadratic (k = 2) cases are of particular interest. Example 1.8

If a(n) ≡ a > 0, c(n) ≡ −m2 ≤ 0 and Vn (r) = −

λ |r|p+1 , p+1

p > 0,

we obtain the modified discrete Klein-Gordon (m2 > 0) and nonlinear wave (m2 = 0) equations q¨ = a ∆d q − m2 q + λ|q|p−1 q and q¨ = a ∆d q + λ|q|p−1 q, respectively.

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Example 1.9

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If a(n) ≡ a, c(n) = m2 > 0 and Vn (r) =

λ 4 r , 4

λ > 0,

then we obtain the so-called discrete ϕ4 -equation q¨ = a∆d q + m2 q − λq 3 = 0.

(1.48)

If λ > 0 and k is odd, then it follows from Theorem 1.8 that the Cauchy problem for equaions from Example 1.7 is globally well-posed. The same holds in Example 1.8 with λ > 0 and p > 1. This is so, because in those cases the operator A = a ∆d − m2 is a non-positive operator defined in l2 . The Cauchy problem for the quadratic DNKG equation with any λ 6= 0 has a unique global solution for small initial data, as it follows from Theorem 1.9 below. Let Vn (r) = −

λ(n) 3 r , 3

where λ(n) is a bounded sequence. Also we suppose that the operator A is negative definite, i. e. there exists α0 > 0 such that (Aq, q) ≤ −α0 kqk2l2

∀q ∈ l2 .

Set 1 1X 1 1 ϕ(q) = − (Aq, q) − λ(n) q 3 (n) =: ϕ0 (q) + ϕ1 (q). 2 3 2 3 n∈Z

Note that ϕ0 (q)1/2 is an equivalent norm on l2 . We have that H(p, q) =

1 kpk2l2 + ϕ(q). 2

Since ϕ1 (q) ≤ c0 kqk33 ≤ c00 kqk32 , l l there exists a constant κ > 0 such that ϕ1 (q) 1/3 ≤ κ ϕ0 (q)1/2 ,

q ∈ l2 .

(1.49)

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Let 2 γ := inf sup ϕ(τ q) : q ∈ l , q 6= 0 .

(1.50)

τ ≥0

We have that γ≥

1 . 6κ6

(1.51)

Indeed, ϕ(τ q) =

τ2 τ3 ϕ0 (q) + ϕ1 (q). 2 3

If ϕ1 (q) ≥ 0, then sup ϕ(τ q) = +∞. τ ≥0

If ϕ1 (q) < 0, then, maximizing the cubic polynomial ϕ(τ q), we obtain that ϕ0 (q) 1 ϕ0 (q)3 sup ϕ(τ q) = ϕ − q = , ϕ1 (q) 6 ϕ1 (q)2 τ ≥0 and the required follows from (1.49). Now let us define W = q ∈ l2 : 0 ≤ ϕ(τ q) < γ

∀τ ∈ [0, 1] .

It is readily verified that the set W is star-shaped with respect to the origin, i. e. if q ∈ W , then θq ∈ W for every θ ∈ [0, 1]. Lemma 1.2

For every % > 0 such that %≤

9 , 4κ

% κ2 3/2 + % < γ, 2 3

the set W contains the open set B = q ∈ l2 : ϕ0 (q) < % . Proof . By (1.49), τ2 τ 3 κ3 τ2 τ 3 κ3 ϕ0 (q) − ϕ0 (q)3/2 ≤ ϕ(τ q) ≤ ϕ0 (q) + ϕ0 (q)3/2 . 2 3 2 3 Thus, ϕ(τ q) ≥ 0 for every τ ∈ [0, 1], provided 1 τ κ3 − ϕ0 (q)1/2 ≥ 0 ∀τ ∈ [0, 1]. 2 3

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Hence, if ϕ0 (q) ≤ 9/(4κ2 ), the second condition for % implies that ϕ(τ q) < γ. Let W∗ = q ∈ l2 : ϕ0 (q) + ϕ1 (q) > 0, ϕ(q) < γ . By the continuity of the functionals ϕ0 , ϕ1 and ϕ, the set W∗ is open. Lemma 1.3

We have that W = W∗ ∪ B.

Proof . Due to Lemma 1.2, it sufficies to show that W = W∗ ∪ {0}. Let q ∈ W , q 6= 0. If ϕ1 (q) ≥ 0, then ϕ0 (q) + ϕ1 (q) > 0 and obviously ϕ(q) < γ. If ϕ1 (q) < 0, then ϕ0 (q) sup ϕ(τ q) = ϕ − q ≥ γ. ϕ1 (q) τ ≥0 Hence, −ϕ0 (q)/ϕ1 (q) > 1 and ϕ(q) < γ. This implies that q ∈ W∗ . Conversely, let q ∈ W∗ . If ϕ1 (q) ≥ 0, then sup ϕ(τ q) = ϕ(q) < γ τ ∈[0,1]

and q ∈ W . If ϕ1 (q) < 0, then the inequality −ϕ0 (q)/ϕ1 (q) > 1 implies that sup ϕ(τ q) = ϕ(q) τ ∈[0,1]

and we conclude.

Since W∗ and B are open set, the set W is open. Lemma 1.4

The set W is bounded.

Proof . If ϕ1 (q) ≥ 0, then ϕ(q) ≥ ϕ0 (q)/2 and ϕ0 (q) > 2γ. If ϕ1 (q) < 0, then Lemma 1.3 shows that ϕ1 (q) > −ϕ0 (q). Hence, ϕ(q) > ϕ0 (q)/6 and ϕ0 (q) < 6γ. Thus, W ⊂ {q ∈ l2 : ϕ0 (q) < 6γ}, i. e. W is contained in a bonded set and we conclude. Now we are ready to prove Theorem 1.9

Suppose that the operator A is negative definite and Vn (r) = −

λ(n) 4 r , 3

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where the sequence λ(n) is bounded. Let q (0) ∈ W and q (1) ∈ l2 satisfy

1

q (1) 22 + ϕ q (0) < γ. l 2 Then the Cauchy problem for (1.42) with the initial data q (0) and q (1) has a unique global solution. Proof . The existence and uniqueness of the local solution q(t) is obvious. To prove that q(t) extends to a solution defined for all t ∈ R it is enough to show that q(t) remains bounded. We show that q(t) ∈ W . Assume not. Let t1 > 0 be the smallest value of t > 0 for which q(t) ∈ / W . Then q(t1 ) ∈ ∂W , the boundary of W . Since W is a star-shaped set, then θq(t1 ) ∈ W for all θ ∈ [0, 1]. Hence, ϕ θq(t1 ) < γ. Passing to the limit as θ → 1, we obtain that ϕ q(t1 ) ≤ γ. If ϕ q(t1 ) < γ, then, by definition of W and the fact that ϕ θ q(t1 ) < γ ∀θ ∈ [0, 1), we conclude that q(t1 ) ∈ W , a contradiction. Hence ϕ q(t1 ) = γ. Since the Hamiltonian H is a conserved quantity, we have that

2 1 γ = ϕ q(t1 ) ≤ H p(t1 ), q1 (t1 ) = q(t ˙ 1 ) l2 + ϕ q(t1 ) 2

2 1 = H q (0) , q (1) = q (1) l2 + ϕ q (0) 2 < γ. This contradiction shows that q(t) ∈ W and, hence, q(t) remains bounded for all t > 0. Since Eq. (1.42) is invariant with respect to the inversion of time, the same holds for t < 0 and the proof is complete. (0) (1) Note

(0)

that

the assumptions on q and q are satisfied when the norms

q and q (1) are small enough.

1.5

Comments and open problems

In Sections 1.1–1.2 we have presented, in an appropriate form, more or less commonly known results. Theorem 1.3 is an extension of the global existence result given in [Friesecke and Pego (1999)].

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For many nonlinear wave type equations global solution do exist for small, in an appropriate sense, initial data (see, e. g., [Reed and Simon (1979); Reed (1976)]). The following problem remains open. Problem 1.1 Is it possible to find a norm in the phase space so that smallness of initial data would imply global existence for the α-model? The study of FPU lattices was initiated by E. Fermi, J. Pasta and S. Ulam in [Fermi et. al (1955)]. Since that time it is appeared an extensive physics literature on FPU type chains, as well as articles devoted to numerical simulation of lattice systems. See [Braun and Kivshar (1998); Braun and Kivshar (2004); Eilbeck (1991); Eilbeck and Flesch (1990); Flach and Willis (1998); Flytzanis et. al (1989); Peyrard et. al (1986); Rosenau (1989); Wattis (1993a); Wattis (1993b); Wattis (1996)], to mention a few. Applications of the Frenkel-Kontorova model in physics are discussed in [Braun and Kivshar (1998); Braun and Kivshar (2004)]. The contents of Section 1.4 is borrowed from [Bak and Pankov (to appear)]. In fact, Theorem 1.9 is a discrete counterpart of a result on nonlinear hyperbolic equations obtained in [Sattinger (1968)] (see also [Lions (1969)]). We point out the following problems. Problem 1.2 Investigate global well-posedness of the Cauchy problem for DNKG and DNW equations in the case when λ < 0 and k is odd. Problem 1.3 Study global well-posedness of Cauchy problem for the discrete ϕ4 -equation. Since DNKG and DNW equations are discrete counterparts of continuum KG and wave equations, it is natural to study the following Problem 1.4 tering?

When DNKG and DNW equations possess nonlinear scat-

Basic facts on nonlinear scattering theory (for classical continuum wave equations) can be found in [Reed (1976); Reed and Simon (1979)] and [Strauss (1989)].

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Chapter 2


2.1

Setting of problem

Consider periodic in time solutions of system (1.1): 0 m(n) q¨(n) = Un+1 q(n + 1) − q(n) − Un0 q(n) − q(n − 1) ,

(2.1)

with the configuration space e l2 . Such solutions are often called discrete breathers (see, e. g. [Aubry (1997)]). Throughout this section we make use the following assumptions (i) There exist M0 ≥ m0 > 0 such that m0 ≤ m(n) ≤ M0 ,

n ∈ Z.

(ii) The potential Un is a C 1 function on R and Un (0) = Un0 (0) = 0. Further restrictions will be imposed in subsequent sections. Note that for (local) well-posedness we need slightly more restrictive assumption (see Section 1.2, assumption (ii)). Recall that (2.1) can be written as the following operator differential equation in e l2 (see (1.6)) m¨ q = ∂ + G(∂ − q),

(2.2)

where G(q)(u) = Un0 q(n) , and ∂ + and ∂ − are the right and left differences, respectively. Now we introduce a variational framework that serves T -periodic (in e the space of all T -periodic in t time) problem for Eq. (2.1). Denote by H 27

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functions q(t) = {q(t, n)} ∈ H 1 (0, T ; e l2 ) such that q˙ = {q(t, ˙ n)} ∈ L2 (0, T ; l2 ). Endowed with the norm h i1/2 kqkHe = kqk2L2 (0,T ;el2 ) + kqk ˙ 2L2 (0,T ;l2 ) , e is a Hilbert space. A more explicit form for kqk e is the space H H (Z kqkHe =

T 2 kq(t)k ˙ l2

T

Z

k∂

dt +

0

−

q(t)k2l2

Z

2

|q(t, 0)| dt

dt +

0

)1/2

T

.

0

e that consists of all q ∈ H e such that Denote by H the subspace of H T

Z

q(t, 0) dt = 0.

(2.3)

0

e The space H is considered This is a closed 1-codimensional subspace of H. as a Hilbert space with the norm "Z

T

kqk =

2 kq(t)k ˙ l2 dt +

0

( =

k∂ − q(t)k2l2 dt

0

XZ

n∈Z

#1/2

T

Z

T

h

2 2 i q(t, ˙ n) + q(t, n) − q(t, n − 1) dt

0

This norm is equivalent to k · kHe , because on H the term T

Z

q(t, 0) 2 dt

0

can be estimated above by the term Z 0

T

2 q(t, ˙ 0) dt.

)1/2 .

(2.4)

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29

e we consider the functional On the space H Z XZ T 1 T 1/2 2 J(q) := km q(t)k ˙ Un ∂ − q(t, n) dt l2 dt − 2 0 n∈Z 0 (Z ) T X 2 m(n) = q(t, ˙ n) − Un q(t, n) − q(t, n − 1) dt . (2.5) 2 0 n∈Z

Proposition 2.1

e Assume (i) and (ii). Then J is a C 1 functional on H.

e Let Proof . First we prove that the functional J is well-defined on H. e q ∈ H. Since the embeddings H 1 (0, T ) ⊂ C [0, T ] ⊂ L∞ (0, T ) are continuous, we have1 for every n ∈ Z

q(·, n) − q(·, n − 1) ∞ ≤ c q(·, ˙ n) − q(·, ˙ n − 1) L2 L

+ q(·, n) − q(·, n − 1) L2

≤ c q(·, ˙ n) L2 + q(·, ˙ n − 1) L2

+ q(·, n) − q(·, n − 1) L2 . Hence, X

q(·, n) − q(·, n − 1) 2 ∞ ≤ c kqk2 ≤ c kqk2 . e L H

(2.6)

n∈Z

Assumption (ii) implies that there exists r0 > 0 such that for all n ∈ Z Un (r) ≤ cr2 if |r| ≤ r0 . (2.7) Due to (2.6) there exists N0 > 0 such that

q(·, n) − q(·, n − 1) ∞ ≤ r0 , L provided |n| > N0 . Therefore, Eqs. (2.5), (2.6) and(2.7) yield X Z T J(q) ≤ c kqk2 + q(t, n) − q(t, n − 1) U dt < ∞. n e H |n|≤N0 1 Everywhere

0

in this chapter the Lp -norms are taken over the interval (0, T ) unless otherwise stated.

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30



Now we prove that the Fréchet derivative of J exists and is continuous. e we have to prove that For h ∈ H, hJ 0 (q), hi =

T

Z

X ˙ m q(t), ˙ h(t) dt −

0

Z =

T

n∈Z

Z

T

Un0 ∂ − g(t, n) ∂ − h(t, n) dt

0

XZ T ˙ m q(t), ˙ h(t) dt + ∂ + Un0 ∂ − q(t, n) h(t, n) dt. (2.8)

0

n∈Z

0

In more details, 0

hJ (q), hi =

XZ n∈Z

T

˙ n) m(n) q(t, ˙ n) h(t,

0

h 0 q(t, n + 1) − q(t, n) + Un+1 i − Un0 q(t, n) − q(t, n − 1) h(t, n) dt. The derivative of the quadratic part of J is easy to treat. Hence, it suffices to check that hΦ0 (q), hi =

XZ n∈Z

T

Un0 ∂ − q(t, n) ∂ − h(t, n) dt,

0

where Φ(q) :=

XZ n∈Z

T

Un ∂ − q(t, n) dt.

0

e then Thus, we have to prove that if h → 0 in H, X Z Th Un ∂ − q(t, n) + ∂ − h(t, n) − Un ∂ − q(t, n) n∈Z 0 i − 0 − − Un ∂ q(t, n) ∂ h(t, n) dt = o khk . Using the Lagrange mean value theorem, we get, with some λn = λn (t) ∈ (0, 1),

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31

that the left-hand side here does not exceed X

0 −

Un ∂ q(·, n) + λn ∂ − h(·, n) − Un0 ∂ − q(·, n) 2 ∂ − h(·, n) L2 L

n∈Z

"

X

0 −

2 ≤ khk

Un ∂ q(·, n) + λn ∂ − h(·, n) − Un0 ∂ − q(·, n) 2

#1/2

L

n∈Z

=: khkB(h). Therefore, the result follows if B(h) → 0 as khk → 0. By Eq. (2.6), given ε > 0 there exists an integer Nε > 0 such that X

∂ − q(·, n) 2 ∞ < ε. L |n|≥Nε

If khk is small enough, we also have X

∂ − h(·, n) 2 ∞ ≤ ε. L n∈Z

Hence, by (ii), X

0 −

2

Un ∂ q(·, n) + λn ∂ − h(·, n) − Un0 ∂ − q(·, n) 2 L

|n|≥Nε

2 X

2

∂ − q(·, n) + λ∂ − h(·, n)

+ ∂ − q(·, n) L2

≤C

L2

|n|≥Nε

≤ Cε, with C > 0 independent of ε. Furthermore, Un0 is continuous, hence, uniformly continuous on compact sets and X

0 −

Un ∂ q(·, n) + λn ∂ − h(·, n) − Un0 ∂ − q(·, n) ≤ Cε, L2

|n| 0, by Eq. (2.6), for k large enough we have Let q (k) → q in H. X

∂ − q (k) (·, n) − ∂ − q(·, n) 2 ∞ ≤ ε L n∈Z

and, for some Nε , X

∂ − q(·, n) 2 ∞ ≤ ε. L |n|≥Nε

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32



By the uniform convergence of q (k) (t, n) on [0, T ], we have that X Z Th i Un0 ∂ − q (k) (·, n) − Un0 ∂ − q(·, n) ∂ − h(·, n) dt ≤ ε sup khk=1 0 |n| 0, 2 where Vn (0) = Vn0 (0) = 0 and Vn0 (r) = o(r) as r → 0; (iv) there exists θ > 2 such that

(iii) Un (r) = −

Vn0 (r) r ≥ θ Vn (r) ≥ 0 and there exists r0 > 0 such that Vn (r) > 0 if |r| ≥ r0 ; (v) m(n + N ) = m(n), c(n + N ) = c(n) and Vn+N (r) = Vn (r). In the spatial periodicity assumption (v) we always assume that N > 0 is the minimal period. Standard results on differential inequalities (see, e. g. [Hartman (2002)]) show that assumption (iv) implies Vn (r) ≥ d |r|θ − d0 ,

d > 0, d0 ≥ 0.

(2.9)

In fact, one can prove that for every d0 > 0 there exists d > 0 such that inequality (2.9) holds. Let us also point out that c(n) > 0 and periodicity imply the existence of c0 > 0 and c1 > 0 such that c0 ≤ c(n) ≤ c1 for all n ∈ Z. Under the assumptions imposed here, the interaction potential Un (r) has a strict local maximum at 0 and admits at least two local minima. This means that the interparticle forces are repulsive-attractive, i. e. repulsive for small displacements and attractive for large displacements. The main result of the section is Theorem 2.1 Assume that (i)—(v) hold. Then there exists T0 > 0 such that for every T > T0 Eq. (2.1) admits a nonconstant T -periodic e l2 -valued solution. To prove Theorem 2.1, we are going to find a critical point of the functional J on the space H. Due to Propositions 2.2 and 2.3, any such critical point is a solution of (2.1). Thus, we consider the restriction J|H of J to H. However, to simply the notation, we skip H here and still denote this restriction by J. In the

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35

case we consider, the functional J becomes Z Z

2 2 1X T 1 T

m1/2 q(t) ˙ l2 dt + c(n) ∂ − q(t, n) dt J(q) = 2 0 2 n∈Z 0 Z X T − Vn ∂ − q(t, n) dt. (2.10) 0

n∈Z

Moreover, Eq. (2.8) gives Z T XZ ˙ hJ 0 (q), hi = m q(t), ˙ h(t) dt + 0

−

n∈Z

XZ n∈Z

T

c(n) ∂ − q(t, n) ∂ − h(t, n) dt

0

T

Vn0 ∂ − q(t, n) ∂ − h(t, n) dt.

(2.11)

0

Let us introduce the operator L : H → H defined by Z T XZ T ˙ e2 dt + (Lq, h) = (m q, ˙ h) c(n) ∂ − q(t, n) ∂ − h(t, n) dt l 0

(2.12)

0

n∈Z

for all h ∈ H and the functional Φ(q) :=

XZ n∈Z

T

Vn ∂ − q(t, n) dt.

(2.13)

0

Then L is a bounded self-adjoint operator in H, J(q) =

1 (Lq, q) − Φ(q), 2

(2.14)

and hJ 0 (q), hi = (Lq, h) − hΦ0 (q), hi.

(2.15)

As the first step we apply the mountain pass theorem without (P S) (see Appendix C.2) to prove that the functional J on H admits a Palais-Smale sequence, i. e. a sequence q (k) ∈ H such that J(q (k) ) is convergent and J 0 (q (k) ) → 0. Proposition 2.5 Under assumption (i)—(v), for every T > 0 the functional J on H admits a Palais-Smale sequence q (k) . Moreover, kq (k) k is bounded from below and above by two positive constants and J(q (k) ) converges to the mountain pass level b.

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36



Proof . To prove the proposition we verify the assumptions of Theorem C.3. From Eq. (2.10) we see that there exists α > 0 such that XZ T α 2 J(q) ≥ kqk − Vn ∂ − q(t, n) dt. (2.16) 2 0 n∈Z

Assumption (iii) shows that given ε > 0 there exists rε > 0 such that Vn (r) ≤ εr2 if |r| ≤ rε . Let % = rε /c1/2 , where c is the constant from inequality (2.6). If kqk = %, then (2.6) yields

−

∂ q(·, n) ∞ ≤ rε , n ∈ Z. L Hence, 2 Vn ∂ − q(t, n) ≤ ε ∂ − q(t, n) and XZ n∈Z

0

T

Z Vn ∂ − q(t, n) dt ≤ ε

T

0

−

∂ q(t, n) 22 dt ≤ ε kqk2 . l

Taking ε small enough, we obtain from inequality (2.16) that J(q) ≥

α 2 % > 0 = J(0), 4

kqk = %.

Now we fix q such that q(t, 0) = ϕ(t) 6= 0 and q(t, n) ≡ 0 if n 6= 0. Then Z Z T λ2 m(0) T λ 2 J(λq) = ϕ(t) ˙ dt + c(1) + c(0) ϕ(t)2 dt 2 2 0 0 Z Th i − V1 λϕ(t) + V0 λϕ(t) dt. 0

Inequality (2.9) implies that J(λq) ≤ Aλ2 − B|λ|θ + B0 , with A > 0, B > 0, B0 ≥ 0. Since θ > 2, J(λq) < 0 for λ > 0 large enough.

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37

Thus, Theorem C.3 applies and there exists a Palais-Smale sequence q (k) at the level b defined by Eq. (C.3), i. e. J(q (k) ) → b, and b ≥ α%2 /4 > 0. We can assume that J(q (k) ) ≥ b/2 > 0. Since the potentials Vn are nonnegative, Eq. (2.10) implies that J(q) ≤ Kkqk2 ,

q ∈ H,

and, therefore, kq (k) k is bounded below by a positive constant. If remains to prove that kq (k) k is bounded above. Let ε > 0,

εk = J 0 (q (k) ) and k large enough so that J(q (k) ) ≤ b + ε/2. Then 2b + ε + εk kq (k) k ≥ 2J(q (k) ) − hJ 0 (q (k) ), q (k) i XZ T h = Vn0 ∂ − q (k) (t, n) ∂ − q (k) (t, n) n∈Z

0

i − 2Vn ∂ − q (k) (t, n) dt. Due to assumption (iv), we get 2b + ε + εk kq (k) k ≥

Z θ − 2 X T 0 − (k) Vn ∂ q (t, n) ∂ − q (k) (t, n) dt. θ 0 n∈Z

Equation (2.11) shows that the right hand part here is equal to "Z T

θ−2

m1/2 q˙(k) (t) 22 dt l θ 0 # XZ T − (k) 2 0 (k) (k) + c(n) ∂ q (t, n) dt − hJ (q ), q i n∈Z

0

i θ−2 h ≥ α kq (k) k2 − εk kq (k) k . θ

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Hence, α

θ − 1 (k) θ − 2 (k) 2 kq k ≤ 2b + ε + εk kq k, θ θ

which implies that kq (k) k is bounded above.

Actually, in the proof of Proposition 2.5 we have shown that the functional J possesses the mountain pass geometry. However, it does not satisfy the Palais-Smale condition and, therefore, the classical mountain pass theorem, Theorem C.1, does not apply. Indeed, suppose that q (k) ∈ H is a precompact Palais-Smale sequence. Then the sequence p(k) defined by p(k) (t, n) = q (k) (t, n + k) −

1 T

Z

T

q (k) (t, k) dt

0

is also a Palais-Smale sequence, but, obviously, no subsequence of p(k) converges. Proof of Theorem 2.1. Given T > 0, we shall first show that there exists a critical point of J in H, hence, a T -periodic solution. Next we prove that this solution is not constant, provided T is large enough. The proof is divided into several steps. Step 1 . Let q (k) be a Palais-Smale sequence that exists according to Proposition 2.5 and u

(k)

Z (n) =

T

h

2 i q˙(k) (t, n)2 + q (k) (t, n + 1) − q (k) (t, n) dt.

0

The real valued sequence u(k) = {u(k) (n)}n∈Z belongs to l1 and ku(k) kl1 = kq (k) k2 . Due to Proposition 2.5, 0 < c1 ≤ kq (k) k2 ≤ c2 . Step 2 . We prove that, up to a subsequence, there exists δ > 0 and nk such that u(k) (nk ) ≥ δ.

(2.17)

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39

Indeed, if not, then u(k) → 0 in l∞ . In this case for k large enough the norm

sup q (k) (·, n + 1) − q (k) (·, n) L∞ n∈Z

is small enough. Since Vn0 (r) = o(r) at 0, Eq. (2.11) gives Z T XZ T

(k) 2 2

q˙ (t) 2 dt + c0 ∂ − q (k) (t, n) dt hJ 0 (q (k) ), q (k) i ≥ m0 l 2 0 0 n∈Z

≥ α kq

(k) 2

k .

Since J 0 (q (k) ) → 0, then kq (k) k → 0. This contradicts the inequality 0 < c1 ≤ kq (k) k2 ≤ c2 , hence, proves inequality (2.17). Step 3 . Due to the periodicity assumption, the sequence of the form Z 1 T (k) q (t, pk N ) dt qe(k) (t, n) = q (k) (t, n + pk N ) − T 0 with pk ∈ Z, is also a Palais-Smale sequence at the same mountain pass level. Making such a shift, we can assume that in (2.17) we have 0 ≤ nk < N . Passing to a subsequence, we get the inequality u(k) (n0 ) ≥ δ

(2.18)

for some integer n0 , 0 ≤ n0 < N . Step 4 . Since the sequence q (k) is bounded in the Hilbert space H, we can assume that q (k) → q weakly in H. The compactness of Sobolev embedding implies that for every n ∈ Z ∂ − q (k) (·, n) → ∂ − q(·, n) strongly in L∞ (0, T ). This is local, or pointwise, convergence with respect to n. Since elements h ∈ H such that ∂ − h(t, n) 6= 0 for all, but finite, number of n ∈ Z are dense in H, this type of convergence of q (k) is enough to pass to the limit in J 0 . Indeed, in (2.11) the first two terms are continuous with respect to weak convergence of q, while

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40

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the last term is continuous with respect to local convergence stated above. Therefore, J 0 (q) = 0 and q is a critical point of J. Passing to the limit in (2.18), we obtain that Z Th 2 i q(t, ˙ n0 )2 + q(t, n0 + 1) − q(t, n0 ) dt ≥ δ > 0, 0

hence, q is a nonzero solution. Now we are going to prove that the critical point q is below the mountain pass level. Let gn (r) =

1 0 V (r) r − Vn (r). 2 n

Assumption (iv) shows that gn (r) ≥ 0, and we have that 1 0 (k) (k) (k) (k) J (q ), q b = lim J(q ) = lim J(q ) − k→∞ k→∞ 2 XZ T = lim gn ∂ − q (k) (t, n) dt. k→∞

n∈Z

0

Since gn (r) ≥ 0 and ∂ − q (k) (·, n) → ∂ − q(·, n) in L∞ (0, T ) for every n ∈ Z, we obtain that X Z T X Z − (k) b ≥ lim gn ∂ q (t, n) dt = k→∞

|n|≤n0

0

|n|≤n0

T

gn ∂ − q(t, n) dt.

0

On the other hand, J(q) = J(q) −

XZ T 1 0 hJ (q), qi = gn ∂ − q(t, n) dt. 2 0 n∈Z

Since g(r) ≥ 0 and n0 is an arbitrary integer, we have that J(q) ≤ b,

(2.19)

where b is the mountain pass level. Step 5 . To complete the proof, we show that the solution q constructed above is not a constant, provided T is large enough.

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41

Proposition 2.4 shows that for any nonzero constant solution q X J(q) = −T Un (θn ), where the sum is extended to a finite number of indices and θn is a nontrivial critical point of Un . Define −d0 = min min Un (r) n∈Z r∈R

and −d1 = max max Un (θnj ), n∈Z

j

where {θnj } is the set of all nontrivial critical points of Un . Obviously, d0 ≥ d1 and for any nonzero constant solution the critical value is not less than T d1 . Due to (2.19), the required result will follow if we prove that the mountain pass value is less then T d1 . Given η ∈ (0, 1), let us consider the pass {q (σ) , σ ∈ [0, σ]}, where q (σ) is defined by q (σ) (t, n) = 0

if n 6= 0

and q (σ) (t, 0) =

 

σ sin



0

2π t ηT

if 0 ≤ t ≤ ηT, if ηT ≤ t ≤ T.

It is not difficult to verify that for all σ large enough, J(q (σ) ) < 0 (see the proof of Proposition 2.5). By definition (see Appendix C.2), the mountain pass value does not exceed max J(q (σ) ). Obviously, XZ n∈Z

T

Un ∂ − q (σ) (t, n) dt ≥ −2ηT d0 .

0

Then, for a suitable choice of η and for all T large we have J(q (σ) ) ≤ and the proof is complete.

2σ 2 π 2 + 2ηT d0 < T d1 η2 T

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2.3 2.3.1

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Indefinite case Main result

In this section we consider the case when the quadratic part of the functional J is not positive definite. Here we keep assumptions (i), (ii), (iv) and (v). But now we allow c(n) to be of arbitrary sign, i. e. we replace (iii) by c(n) 2 (iii0 ) Un (r) = − r +Vn (r), where Vn (0) = Vn0 (0) = 0 and Vn0 (r) = o(r) 2 as r → 0. This assumption means that the interaction potential Un (r) can be either of the same type as in Section 2.2, i. e. repulsive-attractive, or purely repulsive (the last case takes place if c(n) ≤ 0). We start with some properties of the operator L defined by (2.12). If c(n) vanishes for some n = n0 , then it is clear that the operator L is not invertible. Indeed, the vector q ∈ H defined by q(n) =

0 if n < n0 1 if n ≥ n0

satisfies Lq = 0. This case will be considered separately later on, in Subsection 2.4.1. On the other hand, if c(n) > 0 for all n ∈ Z, then L is positive definite and the spectrum σ(L) ⊂ (0, +∞). This is the case of Section 2.2. Assume that c(n) 6= 0 for every n ∈ Z and c(n) is negative for at least one value of n. Let A = n ∈ Z | c(n) > 0 . Obviously A 6= Z. Let Y be the subspace of constant functions satisfying q(n) − q(n − 1) = 0

for all n ∈ A

and Z its orthogonal complement. The orthogonality condition means that h ∈ Z if and only if Z 0

T

∂ − h(t, n) dt = 0,

n∈ / A.

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43

Note that both spaces Y and Z are invariant with respect to L. Indeed, if q ∈ Y , then for every h ∈ Z we have XZ T ˙ n) + c(n) ∂ − q(t, n) ∂ − h(t, n) dt (Lq, h) = m(n) q(t, ˙ n) h(t, n∈Z

=

0 T

XZ n∈Z

c(n) ∂ − q(n) ∂ − h(t, n) dt

0

! X

=

n∈A

X

+

c(n) ∂ − q(n)

Z

T

∂ − h(t, n) dt

0

n∈A /

= 0. Hence, Lq ∈ Y . Since Y ⊥ Z, the subspace Z is also invariant. Let α = inf c(n) > 0 n∈A

(α is not defined when A = ∅) and β = − inf c(n) > 0. n∈Z

Lemma 2.1 Assume that c(n) is nowhere vanishing bounded sequence √ and T < π/ β. Then there exists λ > 0 such that (Lq, q) ≤ −λ kqk2 ,

q ∈ Y,

and (Lq, q) ≥ λ kqk2 ,

q ∈ Z.

In particular, L is an invertible operator in H. Proof . The case q ∈ Y is trivial, since on Y X X 2 (Lq, q) = c(n)k∂ − q(n)k2L2 = T 2 c(n) ∂ − q(n) n∈A /

n∈A /

and kqk2 =

X n∈A /

k∂ − q(n)k2L2 = T 2

X ∂ − q(n) 2 . n∈A /

Recall that every q ∈ Y is a constant function of t.

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Let q ∈ Z. Since q ⊥ Y , then for all n ∈ / A we have that Z T ∂ − q(t, n) dt = 0. 0

Therefore, for every n ∈ /A

−

2π

∂ q(·,

∂ − q(·, n) 2 , ˙ n) L2 ≥ L T as it follows by means of an elementary Fourier series argument. Hence, XZ T h 2 i (Lq, q) = q(t, ˙ n)2 + c(n) ∂ − q(t, n) dt 0

n∈Z

X

2

2

q(·, ≥ ˙ n) L2 + c(n) ∂ − q(·, n) L2 n∈A

+

X

2

2

− βT 2

∂ q(·,

2 + ε ∂ − q(·, n) 2 2 .

q(·, ˙ n) L2 − + ε ˙ n) L L 4π 2

n∈A /

Since

−

2

2

2

∂ q(·, ˙ n) L2 ≤ 2 q(·, ˙ n) L2 + q(·, ˙ n − 1) L2 , we have " X

#

2

−

2 βT 2

(Lq, q) ≥ 1− + 2ε q(·, ˙ n) L2 + α ∂ q(·, n) L2 2π 2 n∈A " # X

2

βT 2 2 + 1− + 4ε q(·, ˙ n) L2 + ε k∂ − q(·, n) L2 π2 n∈A,n+1 / ∈A / " # X

2

−

2 βT 2 + 1− + ε q(·, ˙ n) L2 + ε ∂ q(·, n) L2 . 2π 2

n∈A,n+1∈A /

Choosing ε > 0 small enough, we obtain (Lq, q) ≥ λ kqk2 , with some positive λ.

Remark 2.3 In [Arioli and Szulkin (1997)] it is shown that if c(n) is independent of n and β = −c(n) ≥ π 2 /T 2 , then 0 ∈ σ(L).

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45

Now we state the main result of the section. Recall that we assume that not all c(n) are positive. Theorem 2.2 Assume (i), (ii), (iii0 ), (iv) and (v), with c(n) 6= 0 for all n ∈ Z. Then for all T > 0 system (2.1) admits a nonzero T -periodic solution q ∈ H. The solution is nonconstant if c(n) < 0 for all n ∈ Z. If c(n) takes both signs, then there exist T0 > 0 and T1 > 0, where T0 depends on Vn and positive c(n), while T1 depends on min{c(n)}, such that the solution is nonconstant, provided T0 < T1 and T ∈ (T0 , T1 ). The proof is given in Subsection 2.3.3. Finally, we consider couple of examples. Example 2.1

Consider a monoatomic chain with c(n) = c < 0 and V (r) =

d p |r| , p

d > 0.

In the case p = 4 this is the FPU β-model. The potential V satisfies all the assumptions of Theorem 2.2 and, hence, for every T > 0 there exists a nonconstant T -periodic solution, i. e. a breather. Example 2.2

Consider a diatomoc chain (N = 2) with c(n) < 0 and Vn (r) =

d(n) p |r| , p

d > 0.

Theorem 2.2 gives the existence of breathers of arbitrary period T > 0. When p = 4 and ε2 =

m(0) m(1)

is sufficiently small, i. e. for a highly contrast FPU lattice, the existence of breathers was obtained in [Livi et. al (1997)]. These authors consider the system as a perturbation from the so-called anti-continuous limit2 , appropriately understood, with ε being a small parameter. In addition, their proof provides also the spatial exponential localization, i. e. that r(t, n) := q(t, n + 1) − q(t, n) ≤ Ce−α|n| , with some α > 0. We mention also recent paper [James and Noble (2004)] which extends the result of [Livi et. al (1997)] to the case of arbitrary mass ratio. This paper is based on a discrete spatial center manifold reduction. 2 Probably,

“decoupled limit” would be better.

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2.3.2

Periodic approximations

In the proof of Theorem 2.2 we employ the so-called periodic approximations. Let k be a positive integer. We shall consider solutions of Eq. (2.1) that are T -periodic in time and kN -periodic in spatial variable n ∈ Z. This means that we consider finite chains of particles with periodic boundary conditions. e k the Hilbert space of all functions q = q(t, n), t ∈ R, n ∈ Z, Denote by H T -periodic in t, kN -periodic in n and having finite norm kqk2ke =

kN −1 Z T X

h

2 i q(t, ˙ n)2 + ∂ − q(t, n) dt +

0

n=0

Z

T

2 q(t, 0) dt.

0

e k consists of all q such that The subspace Hk ⊂ H Z T q(t, 0) dt = 0. 0

e k , and on Hk the norm This is a closed 1-codimensional subspace of H kqk2k =

kN −1 Z T X n=0

h

2 i q(t, ˙ n)2 + ∂ − q(t, n) dt

0

is equivalent to k · kke . The corresponding inner product is denoted by (·, ·)k . Let Jk (q) :=

kN −1 Z T X n=0

h1 2

0

i m(n) q(t, ˙ n)2 − Un ∂ − q(t, n) dt.

(2.20)

e k , critical points of Jk As in Section 2.1, one can check that Jk is C 1 on H are solutions of (2.1) satisfying the periodicity condition and critical points of the restriction of Jk to Hk are critical points of Jk . So, in what follows we consider the functional Jk on the space Hk . The derivative of Jk on Hk is given by the formula ( kN −1 Z T X 0 ˙ n) + c(n) ∂ − q(t, n) ∂ − h(t, n) hJ (q), hi = m(n) q(t, ˙ n) h(t, k

n=0

−

Vn0

0

) − ∂ q(t, n) ∂ h(t, n) dt. −

(2.21)

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We have Jk (q) =

1 (Lk q, q)k − Φk (q), 2

hJk (q), hi = (Lk q, q)k − hΦ0k (q), hi,

(2.22) (2.23)

where the self-adjoint operator Lk in Hk is defined in the similar way as L (see Eq. (2.11)) and Φk (q) :=

kN −1 Z T X n=0

Vn ∂ − q(t, n) dt

0

(cf. Eq. (2.13)). Note that the norm of Lk is bounded by a constant independent of k. Denote by Yk the subspace of Hk that consists of constant (in t) functions q ∈ Hk such that q(n) − q(n − 1) = 0

for all n ∈ A,

and by Zk the orthogonal complement of Yk in Hk . The subspaces Yk ⊂ Hk and Zk ⊂ Hk are invariant subspaces of Lk . The statement of Lemma 2.1 is also valid for Lk , with λ independent of k, and Y and Z replaced by Yk and Zk , respectively. Lemma 2.2 tion.

The functional Jk on Hk satisfies the Palais-Smale condi-

Proof . Let q (j) be a Palais-Smale sequence at the level b. Then 1 0 (j) (j) hJ (q ), q i 2 k kN −1 Z T 1 1 X ≥ − V 0 ∂ − q (j) (t, n) 2 q n=0 0 n

b + 1 + εj kq (j) kk ≥ Jk (q (j) ) −

× ∂ − q (j) (t, n) dt, where εj → 0. By the assumptions on the potentials, we have 0 2 Vn (r) ≤ c1 |r| Vn0 (r) = c1 rVn0 (r) for all |r| ≤ 1 and, trivially, 0 Vn (r) ≤ |r| Vn0 (r) = rVn0 (r)

(2.24)

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if |r| ≥ 1. Let In,j = {t ∈ [0, T ] | ∂ − q (j) (t, n) ≤ 1} c and In,j = [0, T ] \ In,j . By Eq. (2.24), we have that

b + 1 + εj kq

(j)

kk ≥ c2

" kN −1 Z X

+ c In,j

2 Vn0 ∂ − q (j) (t, n) dt

In,j

n=0

Z

# 0 − (j) Vn ∂ q (t, n) dt .

Therefore, kN −1 Z X n=0

In,j

kN −1 Z X n=0

2 Vn0 ∂ − q (j) (t, n) dt ≤ c−1 c + 1 + εj kq (j) kk , 2

c In,j

0 − (j) c + 1 + εj kq (j) kk . Vn ∂ q (t, n) dt ≤ c−1 2

Let q (j) be the orthogonal projection of q (j) on Yk and qe(j) = q (j) − q (j) the orthogonal projection of q (j) on Zk . Since hJk0 (q), hi = (Lk q, h)k − hΦ0k (q), hi, by the version of Lemma 2.1 for Lk , we have that λ kq (j) k2k ≤ −(Lk q (j) , q (j) )k = −(Lq (j) , q (j) )k = −hJk0 (q (j) ), q (j) i − hΦ0k (q (j) ), q (j) i. Since, Jk0 (q (j) ) → 0, this implies that λ kq (j) k2k ≤ kq (j) k −

kN −1 Z T X n=0

0

Vn0 ∂ − q (j) (t, n) ∂ − q (j) (t, n) dt.

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The absolute value of the second term on the right does not exceed kN −1 X

" Z

1/2 Z 2 Vn0 ∂ − q (j) (t, n) dt

In,j

n=0

Z + c In,j

≤

1/2

In,j

0 − (j) Vn ∂ q (t, n) dt · q (j) (·, n) L∞

" kN −1 Z X n=0

2 ∂ − q (j) (t, n) dt

Vn0

− (j)

∂ q

2 (t, n) dt

In,j

#

#1/2 " kN −1 #1/2 X

∂ − q (j) (·, n) 2 2 L n=0

+ sup q (j) (·, n) L∞ n

kN −1 Z X n=0

h

c In,j

≤ c3 kq (j) kk b + 1 + εj kq (j) kk

0 − (j) Vn ∂ q (t, n) dt

1/2

i + b + 1 + εj kq (j) kk ,

because

sup q (j) (·, n) L∞ ≤ c0 kq (j) kk . n

Thus, we have shown that h 1/2 i λ kq (j) kk ≤ 1 + c3 b + 1 + εj kq (j) kk + b + 1 + εj kq (j) kk . Proceeding analogously for qe(j) , we obtain h 1/2 i λ ke q (j) kk ≤ 1 + c3 b + 1 + εj kq (j) kk + b + 1 + εj kq (j) kk . Since kq (j) k2k = kq (j) k2k + ke q (j) k2k , we have that h 1/2 i kq (j) kk ≤ C 1 + b + 1 + εj kq (j) kk + b + 1 + εj kq (j) kk . This inequality implies the boundedness of q (j) . Passing to a subsequence, we can assume that q (j) → q weakly in Hk . The Sobolev compactness embedding implies that ∂ − q (j) (·, n) → ∂ − q(·, n)

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strongly in L∞ (0, T ) for every n ∈ Z. Moreover, q (j) → q and qe(j) → qe weakly in Hk , and ∂ − q (j) (·, n) → ∂ − q(·, n), ∂ − qe(j) (·, n) → ∂ − qe(·, n) in L∞ (0, T ) for all n ∈ Z. This is enough to pass to the limit in hJk0 (q (j) ), hi = (Lk q (j) , h)k − hΦ0k (q (j) ), hi,

h ∈ Hk ,

and obtain that hJk0 (q), hi = (Lk q, h)k − hΦ0k (q), hi = 0,

h ∈ Hk ,

(2.25)

i. e. q is a critical point of Jk . Now we have (Lk q (j) , q (j) )k = (Lk q (j) , q (j) )k = hJk0 (q (j) ), q (j) i + hΦ0k (q (j) ), q (j) i. As before, passing to the limit, we get lim(Lk q (j) , q (j) )k = hΦ0k (q), qi = (Lk q, q)k = (Lk q, q)k . Similarly, lim(Lk qe(j) , qe(j) )k = (Lk qe, qe)k . Note that

2 (Lk q, q)k − (Lk qe, qe)k = |Lk |1/2 q k , where |Lk |1/2 is the square root of the

absolute

value of Lk defined by 1/2

means of spectral decomposition, and |Lk | g k is a Hilbert norm on Hk equivalent to the original norm. Thus,

|Lk |1/2 q (j) → |Lk |1/2 q k k and q (j) → q weakly. This implies the convergence q (j) → q in Hk and (P S) is proved. Remark 2.4 An argument similar to the beginning of the proof of Lemma 2.2 shows that for critical points of Jk on Hk there exists an estimate of the form kqkk ≤ ϕ Jk (q) ,

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where ϕ(r), r ≥ 0, is a continuous function independent on k and such that ϕ(0) = 0. Remark 2.5

Since for a critical point q we have

1 0 hJ (q), qi 2 k kN −1 Z T h i X 1 =− Vn ∂ − q(t, n) − Vn0 ∂ − q(t, n) ∂ − q(t, n) dt, 2 0 n=0

Jk (q) = Jk (q) −

assumption (iv) implies that any critical value is nonnegative. The same is true for J. √ Lemma 2.3 If T < π/ β, then there exists a constant ε0 > 0 independent of k such that for any nontrivial critical point q ∈ Hk of Jk we have kqkk ≥ ε0 . Proof . According to assumptions (iii0 ) and (v), there exists a continuous increasing function ψ(r), r ≥ 0 such that ψ(0) = 0 and Vn0 (r) r ≤ ψ |r| r2 . Let q = q + qe, where q ∈ Y and qe ∈ Z. Then λ kqk2k ≤

kN −1 Z T X n=0

0

0 − Vn ∂ q(t, n) ∂ − q(t, n) dt

−1 Z X

kN ≤ sup ψ ∂ − q(·, n) L∞ n∈Z

n=0

0

≤ ψ sup ∂ − q(·, n) L∞ kqk2k n∈Z ≤ ψ kqkk kqk2k . Similarly, 2 λ ke q k2k ≤ ψ kqkk ke q kk . So, we obtain λ kqk2k ≤ ψ kqkk kqk2k ,

T

− ∂ q(t, n) 2 dt

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because kqk2k = kqk2k + ke q k2k . Since q 6= 0, we get λ ≤ ψ kqkk , hence, kqkk ≥ ε0 = ψ −1 (λ).

Remark 2.6 Inequality kqk ≥ ε0 takes place also for any nontrivial critical point of J on H. Now we are going to prove the existence of nontrivial critical points of Jk by means of linking theorem (see Theorem C.4). √ Proposition 2.6 If T < π/ β, then the functional Jk has a nontrivial critical point qk ∈ Hk . Moreover, there exists a constant C > 0 independent of k such that kqk kk ≤ C and Jk (qk ) ≤ C. Proof . Without loss of generality we can assume that c(0) < 0. Let N = {q ∈ Z | kqkk = %0 }. From the assumptions it follows easily that Φk (q) = o kqk2k as kqkk → 0 uniformly in k. This together with Lemma 2.1 for Lk implies that there exists %0 > 0 independent of k such that inf Jk (q) > 0. N

Let z 0 ∈ Z, M = {q = y + z 0 | y ∈ Y, kuk ≤ %1 , s ≥ 0} and M0 = {q = y + z 0 | y ∈ Y, kykk = %1 and s ≥ 0, or kuk ≤ %1 , s = 0}, i. e. M0 = ∂M . Now we make a particular choice of z 0 , namely  2π  sin t if n = 0, 1, . . . , N − 1, z 0 (t, n) = T  0 if n = N, N + 1, . . . , kN − 1,

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and z 0 (t, n + kN ) = z 0 (t, n). For q = y + sz 0 ∈ M , we have s2 1 (Lk z 0 , z 0 ) + (Lk y, y) 2 2 Vn ∂ − (y + sz 0 )(t, n) dt.

Jk (q) = Jk (y + sz 0 ) = −

kN −1 Z T X n=0

0

For every d > 0 there exists a constant Cd > 0 such that Vn (r) ≥ −d + Cd |r|θ . Since y does not depend on t, this and Lemma 2.1 for Lk yield θ Z T 2π λ 2 0 0 Jk (q) ≤ − kykk + (Lk z , z )k + T d − Cd r0 − s · sin T t dθ 2 0 2 Z 2π 2 N 2 T 2π λ s cos t dt = − kyk2k + 2 T T 0 2 Z T 2π t dt + T d + c(0)s2 sin T 0 θ Z T r0 − s · sin 2π t dθ, − Cd T 0 where r0 = ∂ − q(0). Since c(0) < 0 and for every r0 θ θ Z T Z T sin 2π t dθ, r0 − s · sin 2π t dθ ≥ sθ T T 0 0 we obtain that λ 2π 2 N 2 Jk (q) ≤ − kyk2k + s + T d − Cd0 T sθ . 2 T This inequality shows immediately that for %1 large enough, Jk ≤ 0 on M0 . Moreover, 2 2π N 2 0 θ sup Jk (q) ≤ K := max s + T d − Cd T s . (2.26) s≥0 T M Applying the linking theorem (Theorem C.4), we obtain a nontrivial critical point qk ∈ Hk , with Jk (qk ) ≤ K. The estimate for kqk kk follows from Remark 2.4.

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2.3.3


Proof of main result

Now we are ready to prove Theorem 2.2. The idea is to obtain the solution as the limit of qk as k → ∞, where qk is the solution of spatially kN -periodic problem found in Proposition 2.6. √ Proof of Theorem 2.2. Step 1 . Assume T < π/ β. We have that there exist ε0 > 0 and a sequence nk that

−

∂ qk (·, nk ) ∞ ≥ ε0 (2.27) L for all k. Indeed, if not, then

vk := sup ∂ − qk (·, n) L∞ → 0. n∈Z

0

By assumptions (iii ) and (v), there exists an increasing continuous function ϕ(r), r ≥ 0, such that ϕ(0) = 0 and Vn0 (r)r ≤ ϕ |r| r2 . We have 1 0 hJ (qk ), qk i 2 k i ∂ − qk (t, n) ∂ − qk (t, n) − Vn ∂ − qk (t, n) dt

0 ≤ Jk (qk ) = Jk (qk ) − =

kN −1 Z T X n=0

≤

1 2

h1 2

0

kN −1 Z T X n=0

Vn0

Vn0 ∂ − qk (t, n) ∂ − qk (t, n) dt.

0

This implies that kN −1 X

−

1

∂ qk (·, n) 2 2 ≤ Cϕ(vk )kqk k2k . 0 ≤ Jk (qk ) ≤ ϕ(vk ) L 2 n=0

Since kqk kk is bounded, we obtain that Jk (qk ) → 0. Then, by Remark 2.4, kqk kk → 0. This contradicts the conclusion of Lemma 2.3. Note that, for any integer multiple p of N , Z 1 T qek (t, n) = qk (t, n + p) − qk (t, p) dt T 0 is also a critical point of Jk in H. Making such a shift, we can assume that 0 ≤ nk < N in (2.27). Passing to a subsequence, we can even assume that nk = n0 is independent of k.

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55

√ Step 2 . We still assume that T < π β. Since kqk kk is bounded, we can assume, passing to a subsequence, that qk (·, n) → q(·, n) weakly in H 1 (0, T ) and ∂ − qk (·, n) → ∂ − q(·, n) in C [0, T ] for all n ∈ Z. Inequality kqk kk ≤ C obviously implies that q ∈ H. It is also not difficult to show that q solves Eq. (2.1). Passing to the limit in (2.27) (remind that nk = n0 ) we obtain that

−

∂ q(·, n0 ) ∞ ≥ ε0 , L hence, q is a nontrivial solution. We have (see Remark 2.5) k(N −[ N 2 ])−1Z

Jk (qk ) =

X

n=−k[ N 2 ]

0

T

! − 1 0 − − V ∂ qk (t, n) ∂ qk (t, n) − Vn ∂ qk (t, n) dt, 2 n

where [x] denotes the integer part of x, and similarly ! ∞ Z T X − 1 0 − − J(q) = V ∂ q(t, n) ∂ q(t, n) − Vn ∂ q(t, n) dt. 2 n n=−∞ 0 These two formulas together with the convergence ∂ − qk (·, n) → ∂ − q(·, n) in C [0, T ] imply that J(q) ≤ lim Jk (qk ). k→∞

(2.28)

√ Step 3 . Suppose that c(n) < 0 for all n ∈ Z and T < π/ β. Then Y is the subspace of all constant functions in H. It is easily verified that J < 0 on Y \ {0}. Since critical values of J are nonnegative, the solution we have just obtained is nonconstant. √ If T ≥ π/ β, then the operators L and Lk may not be invertible and the procedure employed above does not work in this case. However, for every integer p > 0 a T /p-periodic function is also T -periodic. Therefore, we have a nonconstant T -periodic solution for all T > 0. Step 4 . If c(n) > 0 for some value of n, then Eq. (2.1) admits non zero √ constant solutions. We set T1 = π/ β. Now let us find T0 > 0. As in the

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proof of Theorem 2.1, Step 5, critical values of nonzero constant solutions are not less that T d1 , where −d1 = max max Un (θnj ) n∈Z

j

and {θnj } is the set of all critical points of Un . Calculating maximum in (2.26) and using (2.28), we obtain for the critical value J(q) the estimate θ+2

J(q) ≤ αT − θ−2 + T d, where α > 0 depends on d, but not on T and λ. Choose d < d. Then, for sufficiently large T , say T > T0 , θ+2

J(q) ≤ αT − θ−2 + T d < T d1 . Hence, the solution q cannot be constant. This completes the proof.

Remark 2.7 Since the conditions on T0 and T1 are independent of each other, there are potentials for which T0 < T1 and, hence, the problem has T -periodic solution for every T ∈ (T0 , T1 ). Since a T /p-periodic function (p > 0 integer) is T -periodic, we see that for every T ∈ (pT0 , pT1 ) there exists a nonconstant T -periodic solution. Thus, we have infinitely many bands for periods of nonconstant periodic solutions. 2.4 2.4.1

Additional results Degenerate case

In this section we consider the case when c(n) vanishes for some values of n ∈ Z. In this case, in addition to assumptions (i), (ii), (iii0 ), (iv) and (v), we suppose that (iii00 ) for every n ∈ Z the potential Vn (r) is strictly convex in a neighborhood of the origin. The situation is now more delicate because 0 ∈ σ(L). Nevertheless, we have the following result similar to Theorems 2.1 and 2.2. Theorem 2.3 Assume (i), (ii), (iii0 ), (iv) and c(n) = 0 for some value of n. In addition, we assume (iii00 ). Then for every T > 0 system (2.1) has a nonzero T -periodic solution q ∈ H. If c(n) ≤ 0 for all n ∈ Z, then the solution is nonconstant. If c(n) ≥ 0 for all n ∈ Z and c(n) > 0 for at least one value of n ∈ Z, then there exists T0 > 0 such that the solution

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57

is nonconstant provided T > T0 . If c(n) change sign, then there exists T0 > 0 and T1 > 0, where T0 depends on Vn and positive c(n), while T1 depends on min{c(n)}, such that the solution is nonconstant if T0 < T1 and T0 < T < T1 . The proof goes along the same lines as in Section 2.3. In particular, it uses kN -periodic in n approximations and passage to the limit as k → ∞. The first step, the existence of spatially kN -periodic solutions, is based on the linking theorem. However, in this case the operator Lk (see Subsection 2.3.2) has a nontrivial kernel. Nevertheless, the functional Jk possesses the linking geometry. Let Kk , Yk , Zk be the kernel, the negative and the positive subspaces √ of Lk , respectively. Assume that T < π/ β, with β defined in Section 2.3, and k is large enough. Then there exists λ > 0 independent of k such that (Lk q, q)k ≤ −λ kqk2k ,

q ∈ Yk ,

and (Lk q, q)k ≥ λ kqk2k ,

q ∈ Zk .

Moreover, Kk ⊕ Yk consists of constant in time functions. The space Yk is trivial if c(n) ≥ 0, but the kernel Kk is never trivial, since we assume that c(n) vanishes for some n ∈ Z. The linking geometry for Jk is generated by the space Kk ⊕ Yk and its orthogonal complement Zk . Even in the case when c(n) ≥ 0, this geometry does not reduce to the mountain pass geometry. The proof of (P S) condition for Jk is similar to that in Subsection 2.3.2. Thus, the linking theorem produces critical points qk ∈ Hk of Jk . The passage to the limit is based on uniform estimates for kqk kk and Jk (qk ). This requires a long technical work and assumption (iii00 ) is used here. For the detailed proof we refer to [Arioli and Szulkin (1997)]. Remark 2.8 At least in the nonnegative case (c(n) ≥ 0), assumption (iii00 ) can be weakened to the following: Vn (r) = 0 if and only if r = 0 [Arioli and Gazzola (1995)].

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2.4.2

Constrained minimization

Here we sketch briefly another approach to the existence of periodic solutions, namely, constrained minimization. We consider system (2.1) with Un (r) = −

c(n) 2 d(n) p r + |r| , 2 p

p > 2.

(2.29)

We assume that the coefficients m(n), c(n) and d(n) are N -periodic and strictly positive. Certainly, this case is covered by Theorem 2.1. The functional J on H defined by (2.5) now reads Z XZ T 2 1 T 2 kmq(t)k ˙ c(n) ∂ − q(t, n) dt J(q) = l2 dt + 2 0 n∈Z 0 Z T p 1X − d(n) ∂ − q(t, n) dt, p 0 n∈Z

while for Φ defined by (2.13) we have Z p 1X T d(n) ∂ − q(t, n) dt. Φ(q) = 2 0 n∈Z

1

These are C functionals on H. Consider the following minimization problem: 1 (Lq, q) : q ∈ H, Φ(q) = 1 . µ = inf 2

(2.30)

Theorem 2.4 Under the assumption above, there exists a minimizer q ∈ H, q 6= 0, of problem (2.30). Moreover, there exists T0 > 0 such that q is a nonconstant function of t if T > T0 . Under the assumptions imposed, the set q ∈ H, Φ(q) = 1 is a C 1 manifold. The Lagrange multiplier rule yield the existence of λ ∈ R (Lagrange multiplier) such that, for the minimizer q,

(Lq, h) = λ Φ0 (q), h , ∀h ∈ H. Taking h = q, we obtain that λ > 0. Using the homogeneity properties of (Lq, q) and Φ0 (q) it is easy to scale out the Lagrange multiplier and obtain a solution of the original problem. Actually, for any minimizer q of (2.30), 1 λ p−2 q is a critical point of J, hence, a solution of (2.1). Thus, Theorem 2.4 produces a T -periodic solution of Eq. (2.1) in the case when the potentials

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are of the form (2.29). The proof can be found in [Arioli and Chabrowski (1997)]. 2.4.3

Multibumps

Here we describe a multiplicity result for positive definite case obtained in [Arioli et. al (1996)]. Roughly speaking, the result states that, in a generic situation, the system possesses infinitely many nonconstant T -periodic solutions of multibump type, i. e. having most of their (finite) energy concentrated in a finite number of disjoint regions of the lattice. Let us come back to the assumptions of Section 2.2. (In [Arioli et. al (1996)] it is supposed, in addition, that the derivatives Vn0 are locally Lipschitz continuous, but this assumption is superfluous). For all α ≤ β we denote J α = q ∈ H | J(q) ≤ α , Jβ = q ∈ H | J(q) ≥ β , Jβα = q ∈ H | β ≤ J(q) ≤ α . Let K = q ∈ H \ {0} | J 0 (q) = 0 be the set of all nonzero critical points of J in H, K α = K ∩ J α,

K β = K ∩ Jβ ,

Kβα = K α ∩ Kβ .

In Section 2.2 it is shown that the functional J possesses the mountain pass geometry, with mountain pass level b > 0, and there exists a nontrivial critical point q ∈ H of J such that J(q) ≤ b. Moreover, q is a nonconstant function of t if T > T0 . The functional J is invariant under both a representation of Z denoted by ∗ and a representation of the unit circle S1 denoted by Ω. The first representation is defined by ∗ : Z × H → H,

(k, q) 7→ k ∗ q,

where (k ∗ q)(t, n) = q(t, n + kN ) −

1 T

Z

T

q(t, kN ) dt. 0

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Essentially, this is the spatial shift adjusted to live H invariant. The representation Ω is just the shift of time variable Ω : S1 × H → H,

Ω(τ, q)(t, n) = q(t + τ, n).

Let l be a positive integer, k = (k (1) , k (2) , . . . , k (l) ) ∈ Zl and q = (q (1) , q (2) , . . . , q (l) ) ∈ H l . We set k∗q =

l X

k (j) ∗ q (j) .

j=1 (1)

(2)

(l)

For a sequence k n = (kn , kn , . . . , kn ), by k n → ∞ we mean that for i 6= j we have (i) kn − kn(j) → ∞ as n → ∞. A critical point q is said to be a multibump solution of type (l, %), where l is a positive integer and % > 0, if there exist k = (k (1) , k (2) , . . . , k (l) ) ∈ Zl and q = (q (1) , q (2) , . . . , q (l) ),

q (j) ∈ K \ {0},

such that q belongs to the ball (in H) of radius % centered at k ∗ q. This means that k ∗ q is an approximate solution of (2.1). As it was pointed out in Section 2.2, due to Z invariance, the functional J does not satisfy the Palais-Smale condition. Nevertheless, the structure of Palais-Smale sequences is well-understood. Theorem 2.5 Under the assumptions of Section 2.2, let q (j) ∈ H be a Palais-Smale sequence for J at level c > 0. Then there exist a subsequence (j) still denoted by q (i) , l points q (j) ∈ K \ {0}, and l sequences of integers ki (j = 1, 2, . . . , l) such that kq (i) − k i ∗ qk → 0,

l X j=1

J(q (i) ) = c

fpu

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(2)

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(l)

and k i = (ki , ki , . . . , ki ) → ∞. Actually, the same result holds in the indefinite case of Section 2.3. When we are interesting in multiplicity of solutions, it is natural to consider geometrically distinct solutions. We say that the solutions q (1) ∈ H and q (2) ∈ H are geometrically distinct if q (2) does not belong to the orbit of q (1) under the action of the group Z×S1 , i. e. there exists no (k, τ ) ∈ Z×S1 such that q (2) (t) = (k ∗ q (1) )(t + τ ). Theorem 2.6 Under assumptions of Section 2.2 let T > T0 , where T0 is taken from Theorem 2.1. Then system (2.1) has infinitely many nonconstant geometrically distinct solutions. More precisely, assume that the following nondegeneracy condition holds: (N D) there exist α > 0 and a compact set K ⊂ H such that [ K b+α = k ∗ K, k∈Z

with k1 ∗ K ∩ k2 ∗ K = ∅ if k1 6= k2 . Then for every positive integer l, and for every a > 0 and % > 0 system (2.1) admits infinitely many geometrically distinct multibump solutions of lb+a type (l, %) in the set Jlb−a . For the proofs of Theorems 2.5 and 2.6 we refer to [Arioli et. al (1996)]. Note that if (N D) is not satisfied, then automatically J has infinitely many geometrically distinct critical points. 2.4.4

Lattices without spatial periodicity

First we consider the case when the lattice is harmonic at infinity. Theorem 2.7 Assume (i) of Section 2.1, (iv) of Section 2.2 and (iii0 ) of Section 2.3. Suppose that −C0 ≤ c(n) ≤ −c0 for some positive c0 and C0 , and that 1 0 V (r) → 0 r n

as |n| → ∞

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uniformly with respect to r in any bounded interval. Then for all T > 0 system (2.1) admits a T -periodic nonconstant solution. The proof can be found in [Arioli and Szulkin (1997)]. Here we only point out that in this case the functional J satisfies the Palais-Smale condition. The second result concerns lattices that are homogeneous at infinity. Theorem 2.8 Suppose that the potential Un (r) is given by Eq. (2.29). Assume that m(n) ≡ 1, there exists c0 , c and d such that either 0 < c0 ≤ c(n) ≤ c

and 0 < d < d(n)

or 0 < c0 ≤ c(n) < c

and 0 < d ≤ d(n),

and lim c(n) = c, lim d(n) = d

as |n| → ∞.

Then for every T > 0 system (2.1) has a nontrivial T -periodic solution. The solution is nonconstant if T is large enough. The proof given in [Arioli and Chabrowski (1997)] is based on the constrained minimization approach sketched in Subsection 2.4.2. 2.4.5

Finite lattices

We met already finite lattices with periodic boundary conditions in Subsection 2.3.2. They were used there to approximate infinite lattices. However, finite lattices are also interesting in their own rights and we mention here few results on such lattices. We consider system (2.1) assuming that m(n) is an N -periodic sequence and Un+N = Un for all n ∈ Z, where N ∈ Z, N > 0. Under this assumption it is natural to consider spatially periodic solutions, i. e. solutions satisfying the periodic boundary condition q(t, n + N ) = q(t, n)

for all n ∈ Z.

(2.31)

Theorem 2.9 Assume (i), (ii), (iii0 ), (iv) and (v), with c(n) 6= 0 for all n ∈ Z. Then for all T > 0 problem (2.1), (2.31) admits a nonzero T -periodic solution. The solution is nonconstant if c(n) < 0 for all n ∈ Z. If c(n) > 0 for all n ∈ Z, then there exists T0 > 0 such that the solutions

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is nonconstant if T > T0 . If c(n) takes both sings, then there exist T0 > 0 and T1 > 0, where T0 depends on Vn and positive c(n), while T1 depends on min{c(n)}, such that the solution is nonconstant, provided T0 < T1 and T ∈ (T0 , T1 ). The result follows basically from Proposition 2.6. Similar result holds for degenerate case when c(n) may vanish. In this case we have to assume, in addition, (iii00 ) (cf. Subsection 2.4.1). Another existence and multiplicity result for finite periodic lattices is the following Theorem 2.10 Assume that Un ∈ C 2 (R) for all n ∈ Z, m(n+N ) = m(n) and Un+N = Un . Suppose that there exists θ > 2, δ0 ≥ −1, δ0 < θ − 2, α > 0 and r0 > 0 such that 0 Un (r) ≤ α|r|1+δ0 , r ≤ −r0 , (2.32) r Un0 (r) ≥ θ Un (r),

r ≥ r0 .

(2.33)

Then for all T > 0 problem (2.1), (2.31) has infinitely many nonconstant T -periodic solution. Assumptions (2.32) and (2.33) mean, roughly speaking, that the potential Un (r) is superquadratic at +∞, while its growth rate at −∞ is of lower order. Certainly, in this theorem one can switch the roles of +∞ and −∞. Under some stronger assumptions the result of Theorem 2.10 was obtained in [Ruf and Srikanth (1994)] (without nonconstancy statement). The finial form of this theorem is a particular case of more general results found in [Tarallo and Terracini (1995)]. Example 2.3

The Toda potential (see Example 1.4) U (r) = ab−1 (e−br + br − 1)

satisfies the assumptions of Theorem 2.10. Example 2.4

Let U (r) = (r+ )α ± (r− )β ,

where r+ = max[r, 0] and r− = r − r+ . If α > β ≥ −1 and α > 2, then Theorem 2.10 applies, with θ = α and δ = β.

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Also one can consider finite lattices with fixed ends, i. e. solutions of (2.1) satisfying the Dirichlet boundary condition q(t, 0) = q(t, N + 1) = 0. Results similar to Theorems 2.9 and2.10 can be also obtained in this case see, e. g. [Ruf and Srikanth (1994)] . 2.5

Chains of oscillators

In this section we describe briefly the results on infinite chains of nonlinear oscillators obtained in [Bak and Pankov (2004); Bak (2004)]. We are looking for time periodic solutions of the equation q¨(n) = a(n) q(n + 1) + a(n − 1) q(n − 1) + b(n) q(n) − Vn0 q(n) , (2.34) where n ∈ Z (see Eq. (1.39)), with the following boundary condition lim q(t, n) = 0

(2.35)

n→±∞

at infinity. Actually, we consider (2.34) as a nonlinear operator differential equation q¨ = Aq + B(q) in the Hilbert space l2 , where (Aq)(n) = a(n) q(n + 1) + a(n − 1) q(n − 1) + b(n) q(n), (Bq)(n) = −Vn0 q(n) ,

n ∈ Z,

n ∈ Z.

The boundary condition is incorporated into the space l2 . In the notations of Section 1.4, b(n) = c(n) − a(n) − a(n − 1), with c(n) being the coefficient in the harmonic part of the potential Un . Note that problem (2.34)–(2.35) has a trivial solution q ≡ 0. We consider nonhomogeneous, but spatially periodic, chains. Precisely, we impose the following assumptions. (h0 ) The sequences a(n) and b(n) (equivalently, a(n) and c(n) )and the sequence of potentials Vn are N -periodic in n ∈ R, with N ≥ 1.

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(h1 ) The operator A is positive definite, i. e. there exists α0 > 0 such that (Aq, q) ≥ α0 kqk2 ,

q ∈ l2 .

(h2 ) For every n ∈ Z the potential Vn is C 1 , Vn (0) = Vn0 (0) = 0 and Vn0 (r) = 0 |r| as r → 0. (h3 ) There exists θ > 2 such that Vn0 (r) r ≥ θ Vn (r) ≥ 0, and there exists r0 > 0 such that Vn (r) > 0 if |r| ≥ r0 . Note that assumption (h0 ) implies that the operator A is a bounded linear operator in l2 . Moreover, A is self-adjoint because the coefficients a(n) and b(n) are real. Assumption (h1 ) means, roughly speaking, that the coefficient sequence b(n) (equivalently, c(n)) is large comparably to a(n). We have the following result – an analog of Theorem 2.1. Theorem 2.11 Assume (h0 )–(h3 ). Then there exists T0 > 0 such that for every T ≥ T0 equation (2.34) has a nonconstant T -periodic in time l2 -valued solution. The proof of the theorem relies upon variational arguments. Skipping details, we present here the basic idea only. 1 Let XT be the subspace in Hloc (R; l2 ) that consists of T -periodic functions, i. e. 1 XT := q ∈ Hloc (R; l2 ) : q(t + T ) = q(t) . This is a Hilbert space with the norm 1/2 kqkT = kqk2L2 (0,T ;l2 ) + kqk ˙ 2L2 (0,T ;l2 ) , induced from H 1 (0, T ; l2 ). Consider the functional # Z T" X 1 1 2 J(q) = Vn q(t, n) dt. kqk ˙ l2 + Aq(t), q(t) − 2 2 0 n∈Z

Under assumptions imposed above, J(q) is a C 1 functional on XT and its critical points are exactly T -periodic l2 -valued weak solutions of Eq. (2.34).

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It turns out to be that, due to positivity of A and superquadraticity of Vn at 0 and at infinity (assumptions (h2 ) and (h3 )), the functional J possesses the mountain pass geometry. But, like the functional J in Section 2.1, it does not satisfy the Palais-Smale condition. Applying Theorem C.4, one obtains a Palais-Smale sequence q (k) ∈ XT . Moreover, kq (k) kT is bounded below and above by positive constants, and J q (k) converges to the mountain pass level. Passing to a subsequence, one can assume that q (k) → q weakly in XT . Moreover, due to the Sobolev embedding theorem (Theorem A.1), q (k) (·, n) → q(·, n) in L∞ (R) for all n ∈ Z. Passing to the limit one obtains that q is a critical point of J in XT . Replacing q (k) by qe(k) (t, n) = q (k) (t, n + mk ), with appropriately chosen mk , we can achieve that q 6= 0. The final step is to show that q is nonconstant in t, provided T is large enough. This approach is similar to that of Section 2.2, but the details require a number of technical changes. Another approach to the same result, realized in [Bak and Pankov (2004)], is similar to that presented in Section 2.3, and is based on periodic approximations. The last means that we consider the periodic boundary condition q(t, n + kN ) = q(t, n)

(2.36)

instead of (2.35). More precisely, let Ek be the space of all kN -periodic sequences q(n). Endowed with the norm !1/2 kN −1 X q(n)2 , n=0

this is a kN -dimensional Hilbert space. Let XT,k be the space of all func1 tions in Hloc (R; Ek ) that are T -periodic in t, i. e. 1 XT,k := q ∈ Hloc (R; Ek ) : q(t + T ) = q(t) . This is a Hilbert space with respect to the H 1 (0, T ; Ek )-norm, denoted here by k · kT,k . Due to the periodicity of coefficients, the operator A acts also in all Ek . Moreover, A is positive definite in Ek with the same constant α0 as in (h1 ). Actually, the spectral theory of periodic difference operators (see

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Section 1.3 and, for detailed presentation, [Teschl (2000)]) tells us that the spectrum of A in Ek is contained in the spectrum of A in l2 . Let # Z T" kN −1 X 1 1 2 kq(t)k ˙ (Aq, q)Ek − Jk (q) = Vn q(t, n) dt. Ek + 2 2 0 n=0 The functional Jk is a C 1 functional on XT,k and its critical points are T -periodic in time solutions of problem (2.34), (2.36). Like J, the functional Jk possesses the mountain pass geometry. But, in contrast to J, Jk satisfies also the Palais-Smale condition. This follows from the fact that Ek is finite dimensional and the Sobolev embedding theorem. Hence, the mountain pass theorem (Theorem C.1) provides a nontrivial critical point q (k) which is a solution of (2.34), (2.36). Actually, q (k) is a nonconstant in time solution, provided T is large enough. As in the case of Palais-Smale sequence of mountain pass type for J, (k) kq kT,k is bounded below and above by positive constants. Therefore, passing to a subsequence, we can assume that q (k) (·, n) → q(·, n) weakly in H 1 (0, T ) and strongly in L∞ (R) for all n ∈ Z. The uniform bound for kq (k) kT,k implies that q ∈ XT . Moreover, q is a critical point of J. Replacing q (k) by qe(k) (·, n + mk ), with appropriate mk , we can obtain a nonzero limit q. In fact, q is nonconstant in time if T is large enough. Remark 2.9 Actually, in Theorem 2.11 a nontrivial solution exists for all T > 0. But, if T < T0 , this solution may be independent of t, i. e. a stationary solution. It follows from [Pankov and Zakharchenko (2001)] that under the assumptions of Theorem 2.11 nonzero stationary solutions do exist. In the particular case when Vn (r) =

d(n) p |r| , p

where p > 2 and d(n) > 0 is an N -periodic sequence, to solve (2.34) in the space of T -periodic functions one can use a constrained minimization similar to that discussed in Subsection 2.4.2. In this case Eq. (2.34) reads q¨(n) = a(n) q(n + 1) + a(n − 1) q(n − 1) + b(n) q(n) p−2 − d(n) q(n) q(n).

(2.37)

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Let 1 Q(v) := 2

Z

T

kvk ˙ 2l2 + (Av, v) dt

0

and 1 Ψ(v) := p

Z 0

T

! X

p d(n) v(t, n)

dt.

n∈Z

Then J(v) = Q(v) − Ψ(v). The functional Q is a continuous quadratic functional on XT , while Ψ is C 1 on XT . Given α > 0, consider the following minimization problem Iα = inf Q(v) : v ∈ XT , Ψ(v) = α . (2.38) Let u be a point of minimum in (2.37) (if it exists). Obviously, u 6= 0. Since both the functionals Q and Ψ are C 1 , then there exists a Lagrange multiplier λ ∈ R such that Q0 (u) = λΨ0 (u), or, explicitly, Z Th 0

i ˙ u(t), ˙ h(t) + Au(t), h(t) dt # Z T "X p−2 =λ d(n) un (t, n) un (t, n) h(t, n) dt 0

n∈Z

for all h ∈ XT . Testing with h = u, we obtain that λ=

2Iα >0 αp

and a straightforward calculation shows that 1

q = λ p−2 u is a solution to (2.38). The following result is proved in [Bak (2004)].

(2.39)

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Theorem 2.12 For every α > 0 problem (2.38) has a solution u ∈ XT . Furthermore, if T is sufficiently large, then u 6= const. Avoiding technical details, we explain only the basic idea of the proof. The functionals Q and Ψ satisfy Q(τ v) = τ 2 Q(v),

Ψ(τ v) = τ p Ψ(v),

τ > 0.

This implies immediately that problems (2.38), with different values of α, are equivalent and Iα = α2/p I1 .

(2.40)

Let u(k) be a minimizing sequence for (2.38), i. e. u(k) ∈ XT , Ψ(u(k) ) = α and Q(u(k) ) → α. It is readily verified that the sequence u(k) is bounded in XT . We define an element w(k) ∈ l1 by Z p d(n) T (k) (k) w (n) = u (t, n) dt. p 0 Then kw(k) kl1 = α. Applying concentration compactness Lemma B.3, we obtain that wk satisfies one of the statement (i)–(iii) of that lemma. Vanishing (statement (ii)) is impossible because otherwise α = kw(k) kl1 would go to zero. To rule out dichotomy (statement (iii)) one uses the inequality (subadditivity inequality) Iλ < Iα + Iλ−α ,

λ ∈ (0, α),

that follows from (2.40). Thus, we see that w(k) concentrates, i. e. satisfy property (iii) of Lemma B.3. This permits us to conclude that, after appropriate spatial shifts and passing to a subsequence, u(k) → u ∈ XT weakly in XT and strongly in Lp (0, T ; lp ). The functional Ψ is, obviously, continuous on Lp (0, T ; lp ), hence, Ψ(u) = α. The quadratic functional Q is positive definite and weakly lower semicontinuous. Therefore, Q(u) ≤ lim Q(u(k) ) = Iα . Hence, Q(u) = Iα and we conclude. Example 2.5 Consider a chain of identical oscillators with the coupling constant a ≡ a(n), c(n) ≡ c and the anharmonic potential Vn (r) ≡ V (r)

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that satisfies (h2 ) and (h3 ), for instance, the potential V (r) =

d p |r| p

with p > 2 and d > 0. Then the equation of motion reads q¨ = a∆d q + cq − d|q|p−2 q,

(2.41)

where (∆d q)(n) = q(n + 1) + q(n − 1) − 2q(n) is the one-dimensional discrete Laplacian. If c > 0, a > 0 and p = 4, this is the discrete ϕ4 -equation. It is well-known (and easy to verify) that 0 ≥ (∆d q, q) ≥ −4kqk2l2 , and 4 is the sharp constant. Therefore, for A = a∆d + c we have that (Aq, q) ≥ (c − 4a)kqk2l2 . Hence, A is positive defined if and only if a < c/4. Thus, if a < c/4, then equation (2.41) possesses nonconstant T -periodic solutions (breathers) whenever T large enough. In particular, this holds for the discrete ϕ4 equation with a < c/4. Example 2.6 When a > 0 and c = −m2 < 0 Eq. (2.41) becomes the modified nonlinear discrete Klein-Gordon equation. In this case the operator A = a∆d − m2 is negative defined. The potential V satisfies (h2 ) and (h3 ) if d > 0. If d < 0, then −V satisfies (h2 ) and (h3 ). In any case Theorem 2.11 does not apply. 2.6

Comments and open problems

Variational approach to time periodic solutions of FPU type system was introduced first in [Ruf and Srikanth (1994)]. These authors considered finite lattices with Toda-like potentials satisfying some one side conditions. They have obtained a number of results on existence of infinitely many T -periodic solutions. The approach of [Ruf and Srikanth (1994)] is based

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on minimax methods and a relative S1 -index. Another approach to similar problems based on a linking type arguments was suggested in [Tarallo and Terracini (1995)], and the results of [Ruf and Srikanth (1994)] were extended considerably. A result of such type, Theorem 2.10, is presented in Subsection 2.4.5. Those results motivated subsequent works on infinite lattices [Arioli and Chabrowski (1997); Arioli and Gazzola (1995); Arioli and Gazzola (1996); Arioli et. al (1996); Arioli et. al (2003); Arioli and Szulkin (1997)]. However, the following problem is still open. Problem 2.1 Find any result similar to Theorem 2.10, i. e. under asymmetric assumptions on the potentials, in the case of infinite lattices. Since Theorem 2.10 applies to any period of the form kN , k ≥ 1 an integer, a natural idea is to use periodic approximations in the spirit of Section 2.3. The key point here is to obtain appropriate uniform, with respect to k, a priori estimates for time periodic solutions that are spatially kN -periodic. A closely related problem is Problem 2.2 Extend the results of Sections 2.2–2.3 to the case of not everywhere defined potentials, like the singular Lennard-Jones potential. The results of Sections 2.1–2.3 are borrowed from [Arioli and Gazzola (1995); Arioli and Gazzola (1996); Arioli and Szulkin (1997)]. However, the proof of Theorem 2.2 is new. The approach developed in this chapter seems to be quite general and should work for more general lattices. For instance, let us suggest the following two problems. Problem 2.3 Extend the results of this chapter to the case of N dimensional lattices. Problem 2.4 Study time periodic solutions for lattices with second neighbor interactions. We mention here the paper [Srikanth (1998)] that deals with finite two dimensional lattices in the spirit of [Ruf and Srikanth (1994)]. The following problem seems to be interesting for both applications and mathematical development. Problem 2.5 Study time periodic oscillations of infinite spatially periodic lattices with impurities both in positive definite and indefinite cases.

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More precisely, this means that m(n) = m(n) + m0 (n), c(n) = c(n) + c0 (n) and Vn (r) = V n (r) + Vn0 (r), where m(n), c(n) and V n (r) are N -periodic in n, while m0 (n), c0 (n) and Vn0 tend to 0 as |n| → ∞. Theorem 2.8 could be considered as a prototype of results we expect to exist. Another interesting class of solutions consists of so-called homoclinics, i. e. solutions that satisfy the boundary condition q(−∞, n) = q(+∞, n) = 0. Formally, this corresponds to T = +∞. The following problem is of interest. Problem 2.6 Find homoclinic solutions to Eq. (2.1) with superquadratic potentials (the indefinite case is especially interesting). For results on homoclinics for finite dimensional second order Hamiltonian systems see [Omana and Willem (1992); Rabinowitz (1990); Rabinowitz (1996); Rabinowitz and Tanaka (1991)] and references therein. The class of asymptotically linear systems of the form (3.1), i. e. the systems with 0 −c (n) r + o(r) as r → 0 0 Un (r) = −c∞ (u) r + o(r) as r → ∞, is not covered by existing theory. So, we offer Problem 2.7 Study time periodic and homoclinic solutions to Eq. (2.1) in the case of asymptotically linear nonlinearities. Time periodic solutions of finite dimensional asymptotically linear Hamiltonian systems are considered in [Mawhin and Willem (1989)]. For homoclinics we refer to [Szulkin and Zou (2001)]. Multibump type solutions are shown to exist for a variety of problems (see, e. g. [Coti Zelati and Rabinowitz (1991); Coti Zelati and Rabinowitz (1991); Coti Zelati and Rabinowitz (1992); Rabinowitz (1993); Rabinowitz

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(1996)]). However, in the case of lattice systems Theorem 2.6 seems to be the only known result of such kind. Problem 2.8

Find multibump solutions in the indefinite case.

We expect that the idea of gluing on Nehari’s manifold [Li and Wang (2001)] together with the techniques of generalized Nehari’s manifold suggested in [Pankov (2004)] should work in this case. Certainly, this is not the only possible approach to the last problem. Discrete systems that consist of nonlinear oscillators, like FrenkelKontorova model, discrete Klein-Gordon and ϕ4 equations, etc., and their time periodic solutions (breathers) attracted a great attention in low dimensional nonlinear physics (see, e. g. [Aubry (1997); Braun and Kivshar (1998); Braun and Kivshar (2004); Flach and Willis (1998)] and references therein). The results of Section 2.5 are borrowed from [Bak and Pankov (2004); Bak (2004)]. To our best knowledge, these are the first results of such kind in the case on spatially nonhomogeneous systems of the form (2.34). Certainly, the results are far to be complete. First we mention Problem 2.9 Find and classify time periodic solutions that are at rest at spatial infinity to systems of the form (2.34) with potentials that possess many local maxima, like the Frenkel-Kontorova potential. In view of Example 2.6 it is interesting to study the case when the operator A is not positive definite and either Vn or −Vn satisfies (h2 ) and (h3 ). The operator Lq = −¨ q + Aq, with T -periodic boundary conditions, is a self-adjoint operator in L2 (0, T ; l2 ). Due to spatial periodicity, its spectrum σ(L) possesses the band structure. Therefore, generically σ(L) may have gaps. Suppose that 0 lies in a gap of σ(L). Then it is very likely that the approach of [Pankov (2004)] allows to find nontrivial critical points of the functional J. However, it is not clear a priori that the solution obtained is not constant. Notice that if A is positive definite, then L is positive definite. If, in addition, rV (r) ≤ 0, then it is easily seen that the only time periodic solution is 0 (cf. Section 4.4). So, this case is trivial. Problem 2.10 Do nonconstant T -periodic solutions exist in the case when 0 lies in a gap of σ(L) and either Vn or −Vn satisfies (h2 ) and (h3 )?

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Certainly, similar problem makes sense in the case of FPU lattices. Again, the case when the quadratic part is positive definite, while rV (r) ≤ 0, is trivial. Due to the analogy with gap solutions [Aceves (2000); de Sterke and Sipe (1994); Pankov (2004)], solutions of such kind can be called gap breathers. Commonly, breathers are considered as spatially exponentially localized objects. This important property is not yet studied for the solutions obtained in this chapter. Problem 2.11 Are the solutions obtained in Theorems 2.1, 2.2, 2.5 and 2.11 exponentially localized in the spatial variable? Exponential localization means that r(t, n) := q(t, n + 1) − q(t, n) ≤ Ce−α|n| in the case of FPU type systems and q(t, n) ≤ Ce−α|n| in the case of chains of oscillators, with α > 0 and C > 0. In the degenerate case of Subsection 2.4.1 we cannot expect exponential localization. Instead, we conjecture power localization, i. e. r(t, n) ≤ C|n|−α , α > 0, C > 0.

Finally, we would like to point out probably the most important problem that concerns time periodic oscillations of lattices systems. Problem 2.12 Investigate stability properties of time periodic solutions of spatially periodic infinite lattices and lattices with impurities for both FPU systems and chains of oscillators. This problem seems to be completely open. Here it could be useful the concept of Howland semigroup (see, e. g., [Chicone and Latushkin (1999); Howland (1974); Howland (1979)]).

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Chapter 3

Travelling Waves: Waves with Prescribed Speed

3.1

Statement of problem

In this chapter we consider monoatomic FPU lattices. Without loss of generality one can assume that all (identical) particles are of unit mass and, therefore, the corresponding equations of motion read q¨(n) = U 0 q(n + 1) − q(n) − U 0 q(n) − q(n − 1) , n ∈ Z, (3.1) where U is the potential of interaction between two adjacent particles (see Eq. (1.1)). Travelling wave is a solution of the form q(t, n) = u(n − ct),

(3.2)

where the function u(t), t ∈ R, is called the profile function,1 or simply profile, of the wave and the constant c, the speed of the wave, is always assumed to be positive. In the following we often do not distinguish the profile function and the wave itself. These are waves travelling to the right with the speed c. Making use Ansatz (3.2), we obtain the following equation for the profile function c2 u00 (t) = U 0 u(t + 1) − u(t) − U 0 u(t) − u(t − 1) . (3.3) This is a forward-backward differential-difference equation. One can rewrite Eq. (3.3) in terms of relative displacements. Let r(t) := u(t + 1) − u(t). 1 We

denote by t the independent variable of the profile function, but this does not mean time. 75

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This function is exactly the profile function for relative displacements because q(t, n + 1) − q(t, n) = r(n + ct). For the function r(t) we have the equation c2 r00 (t) = U 0 r(t + 1) + U 0 r(t − 1) − 2U 0 r(t) .

(3.4)

Note that the relative displacement profile r(t) is related to the profile function u(t) also by the following formula Z t+1 r(t) = u(s) ds. (3.5) t

In the following we shall consider two types of travelling waves: solitary waves and periodic waves. A solitary travelling wave is a travelling wave such that its relative displacement profile function r(t), or the derivative u0 (t) of the profile function (see Eq. (3.5)), vanishes at infinity. A periodic travelling wave is a travelling wave such that its relative displacement profile function r(t) (equivalently, u0 (t)) is a periodic function of t ∈ R, say, 2k-periodic. In general, the profile function u(t) of such wave is not periodic. In fact, it is of the form u(t) = hu0 i t + periodic function, where hu0 i =

Z

k

u0 (s) ds

−k

is the mean value of the 2k-periodic function u0 (t). Remark that for every solution u(t) of Eq. (3.3) u(t + α) + β is also a solution of Eq. (3.3) for all α ∈ R and β ∈ R. Therefore, in general travelling waves form two-parametric families. Notice also that looking for a wave of the form q(n, t) = u(n + ct),

c>0

travelling to the left with speed c > 0 we obtain the same equation (3.3) for the profile function u(t). Therefore, if u(t) is the profile function for a wave travelling to the right, then there is also a wave with the same profile travelling to the left with the same speed, and vice versa.

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Let u(t) be the profile function of 2k-periodic travelling wave (k ∈ R, k > 0). For the corresponding lattice solution q(n, t) the function q(t, ˙ n) = −cu0 (n − ct) is 2k/c-periodic in time, but only almost periodic with respect to the spatial variable n in general. The function q(t, ˙ n) is periodic in n only in the case of rational k/c. It is convenient to represent the interaction potential U (r) in the form a (3.6) U (r) = r2 + V (r), 2 where V (r) is the anharmonic part of the potential. In the case of harmonic potential (V (r) = 0), Eq. (3.4) becomes c2 r00 (t) = a r(t + 1) + r(t − 1) − 2r(t) . Making use the Fourier transform 1 rb(ξ) = √ 2π

Z

+∞

e−iξt r(t) dt,

−∞

we obtain −ξ 2 c2 rb(ξ) = 2a (cos ξ − 1) rb(ξ). Hence rb(ξ) = 0 almost everywhere, and thus r ≡ 0. Therefore, in the harmonic case nontrivial solitary waves do not exist. Now let a = c20 (c0 > 0) and V (r) = 0. Then, as it is mentioned on the end of Section 1.3, Eq. (1.37) for relative displacements, allows plane wave solutions r(t, n) = exp i(κn ± ωt) , with the dispersion relation2 ω = ±2c0 sin

κ . 2

Such solutions are non localized travelling waves. For either branch of this dispersion relation, the group velocity for linear wave propagation satisfies dω = c0 cos k ≤ c0 . dκ 2 2 Here

r(t, n) is the relative displacement of adjacent lattice sites, not the relative displacement profile function r(t).

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This is the reason to name c0 > 0 the speed of sound. In what follows we are interesting, primarily, in monotone waves, increasing and decreasing. This means that the corresponding wave profile is a monotone function or, equivalently, the relative displacement profile is either positive or negative. From the point of view of physics, increasing waves are expansion waves, while decreasing waves are compression waves. Also we present few results on nonmonotone waves.

3.2

Periodic waves

3.2.1

Variational setting

Consider 2k-periodic travelling waves. This means that we are looking for solutions of Eq. (3.3) with the boundary condition u0 (t + 2k) = u0 (t).

(3.7)

Let Xk be the Hilbert space defined by 1 Xk := {u ∈ Hloc (R) : u0 (t + 2k) = u0 (t), u(0) = 0},

with the inner product Z

k

u0 (t)v 0 (t) dt

(u, v)k := −k

1/2

and corresponding norm kukk = (u, u)k . Since every function that belongs 1 to Hloc (R) is continuous (see Theorem A.1), the condition u(0) = 0 is meaningful. By k · kk,∗ we denote the dual norm on Xk∗ , the dual space to Xk . Actually, Xk is a 1-codimensional subspace of the Hilbert space ek := u ∈ H 1 (R) : u0 (t + 2k) = u0 (t) , X loc with Z

k

−k

as the inner product.

u0 (t) v 0 (t) dt + u(0) v(0)

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ek we define the operator A by On X Z

t+1

(Au)(t) := u(t + 1) − u(t) =

u0 (s) ds.

(3.8)

t

ek into itself. Obviously, A acts from X Lemma 3.1 The operator A is a linear bounded operator from Xk into L2 (−k, k) ∩ L∞ (−k, k) satisfying kAukL∞ (−k,k) ≤ kukk and kAukL2 (−k,k) ≤ kukk . Proof . By the Cauchy-Schwatz inequality we have Z |Au(t)| =

t+1

t

Z u (s) ds ≤ 0

t+1

1/2 0 2 u (s) ds ≤ kukk

t

and Z

k

Au(t) 2 dt ≤

Z

k

−k

−k

Z

t+1

0 2 u (s) ds dt ≤ kuk2k .

t

Consider the functional Z Jk (u) :=

k

−k

c2 0 2 u (t) − U Au(t) dt. 2

We always assume that (i) the potential U is C 1 on R and U (0) = U 0 (0) = 0. Proposition 3.1 Under assumption (i) the functional Jk is well-defined on Xk . Moreover, Jk is C 1 and Z k h i 0 c2 u0 (t)h0 (t) − U 0 Au(t) Ah(t) dt. hJk (u), hi = −k

Proof . We have Jk (u) =

c2 (u, u)k − Φk (u), 2

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where Z

k

Φk (u) :=

U Au(t) dt.

−k

Hence, we have to consider only the functional Φk . Since for any u ∈ Xk the function Au is continuous, Φk (u) < ∞. A straightforward calculation shows that the Gateaux derivative of Φk exists and is given by Z k hΦ0k (u), hi = U 0 Au(t) Ah(t) dt. −k

Finally, we check the continuity of Φ0k . Let khkk ≤ 1 and un → u in Xk . Then Aun → Au uniformly on [−k, k] and

0 Φk (un ) − Φ0k (u), h ≤ kAhkL1 · U 0 (Aun ) − U 0 (Au) L∞

≤ (2k)1/2 kAhkL2 · U 0 (Aun ) − U 0 (Au) L∞

≤ (2k)1/2 U 0 (Aun ) − U 0 (Au) ∞ , L

where all Lp -norms are taken over [−k, k]. Since U 0 (Aun ) → U 0 (Au) uniformly on [−k, k], we conclude. Proposition 3.2 Under assumption (i) any critical point of Jk is a classical, i. e. C 2 , solution of Eq. (3.3) satisfying (3.7). Proof . Let ϕ(t) be any 2k-periodic C ∞ function. Then h(t) = ϕ(t) − ϕ(0) ∈ Xk . If u is a critical point of Jk , we have Z k h i 0= c2 u0 (t)h0 (t) − U 0 u(t + 1) − u(t) h(t + 1) − h(t) dt −k k

Z =

−k k

Z =

h

i c2 u0 (t)ϕ0 (t) − U 0 u(t + 1) − u(t) ϕ(t + 1) − ϕ(t) dt

h i c2 u0 (t)ϕ0 (t) − U 0 u(t)−u(t − 1) − U 0 u(t + 1)−u(t) ϕ(t) dt.

−k

Hence u is a weak solution of (3.3). Since u(t) and U 0 (r) are continuous, Eq. (3.3) implies that u ∈ C 2 .

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Remark 3.1 Taking the test function h(t) ≡ t, we obtain that any critical point of Jk satisfies the following additional identity Z k c2 u(k) − u(−k) = U 0 u(t + 1) − u(t) dt. −k

3.2.2

Monotone waves

Here we present a result on existence of monotone supersonic waves. We impose the following conditions: c20 2 r + V (r), where c0 ≥ 0, V is C 1 on R, V (0) = V 0 (0) = 0 20 and V (r) = o(|r|) as r → 0,

(i0 ) U (r) =

and either (ii+ ) there exist r0 > 0 and θ > 2 such that V (r0 ) > 0 and , for r ≥ 0, we have 0 ≤ θV (r) ≤ rV 0 (r), or (ii− ) there exist r0 < 0 and θ > 2 such that V (r0 ) > 0 and, for r ≤ 0, we have 0 ≤ θV (r) ≤ rV 0 (r). Assumption (ii+ ) can be written as the differential inequality rθ+1

d −θ r V (r) ≥ 0, dr

r > 0.

Integration shows that V (r) ≥ a0 rθ

for r > r0 ,

with a0 = r0−θ V (r0 ). Together with assumption (i0 ) this implies that V (r) ≥ a1 (rθ − r2 ),

r ≥ 0.

(3.9)

Similarly in the case of (ii− ) the last inequality holds for r ≤ 0. We are interesting in 2k-periodic travelling waves having either nondecreasing or nonincreasing profile. Theorem 3.1

Assume (i0 ) and k ≥ 1.

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(a) Under assumption (ii+ ) for every c > c0 there exists a nontrivial nondecreasing 2k-periodic travelling wave uk ∈ Xk . (b) Under assumption (ii− ) for every c > c0 there exists a nontrivial nonincreasing 2k-periodic travelling wave uk ∈ Xk . Moreover, in both cases there exist constants δ > 0 and M > 0, independent of k, such that the critical value Jk (uk ) satisfies 0 < δ ≤ Jk (uk ) ≤ M. The proof is based on a version of mountain pass theorem (see Theorem C.2). Let Z t 0 u (s) ds. (P u)(t) = 0

It is readily verified that P maps continuously the space Xk into itself and P Xk consists of nondecreasing functions. Since we are looking for monotone waves, we can assume that V (r) ≡ 0 for r < 0 in case (a) and V (r) = 0 for r > 0 in case (b). (Equally well one can assume in both cases that V (r) is an even function). In particular, this means that the modified potential satisfies 0 ≤ θV (r) ≤ rV 0 (r)

for all r ∈ R.

In the following we consider case (a) only. Case (b) is similar. Lemma 3.2 Under assumptions of Theorem 3.1 there exists δ > 0 and % > 0 such that Jk (u) ≥ δ if kukk = %. Furthermore, there exists ek ∈ P Xk such that kek kk > % and Jk (ek ) = J1 (e1 ) ≤ 0. Proof . Assumption (i0 ) implies that, given ε > 0, there exists % > 0 such that V (r) ≤ εr2 if |r| ≤ %. If kukk ≤ %, then, by Lemma 3.1, kAukL∞ ≤ %, and Z k 2 c20 c2 − c20 − 2ε c 0 2 2 2 |u | − |Au| − ε|Au| dt ≥ kuk2k . Jk (u) ≥ 2 2 2 −k Choosing ε small enough, we obtain the first statement of the lemma. To construct the function ek , we first choose a function v ∈ P X1 such that v(t) = 0 for 0 ≤ t ≤ 1, v 0 (−1) = 0 and Av(t0 ) > 0 for some t0 . Note

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that v 0 can be nonzero only on intervals of the form (2l − 1, 2l), l ∈ Z. Since Av ≥ 0, by (3.9) we have that Z 1 2 Z 1 c c20 2 0 2 2 θ J1 (τ v) ≤ τ |v | + a1 − |Av| dt − τ a1 |Av|θ dt 2 2 −1 −1 2 c 2 2 2 kvk1 + a1 kAvkL2 − τ θ a1 kAvkθLθ . ≤τ 2 Since θ > 2, we obtain that J1 (τ v) → −∞ as τ → +∞. Hence, we can fix e1 = τ0 v satisfying J1 (e1 ) ≤ 0 and ke1 k1 > %. Now we define ek ∈ Xk by ek (t) = e1 (t) if |t| ≤ 1 and e0k (t) = 0 if 1 ≤ t ≤ k. (Extending e0k to R by 2k-periodicity, we define ek ∈ Xk uniquely). Obviously, e0k can be nonzero only on intervals (2kl − 1, 2kl), l ∈ Z. We see immediately that kek kk = ke1 k1 . Moreover, e1 (t + 1) − e1 (t), t ∈ [−2, 0], (Aek )(t) = ek (t + 1) − ek (t) = 0, t ∈ [−k, k] \ [−2, 0]. Consequently, Z k Z V Aek (t) dt = −k

0

V Aek (t) dt =

−2

Z

−1

V e1 (t + 1) − e1 (t) dt

−2

Z

0

V e1 (t + 1) − e1 (t) dt

+ −1

Z

1

V e1 (t − 1) − e1 (t − 2) dt

= 0

Z

0

V e1 (t + 1) − e1 (t) dt

+ −1

Z

1

=

V Ae1 (t) dt

−1

because the difference e1 (t + 1) − e1 (t) is 2-periodic. In particular, we have that Jk (ek ) = J1 (e1 ) ≤ 0. Remark 3.2

In fact, Jk (sek ) = J1 (se1 ) for all s ∈ R.

Lemma 3.3

Assume (i0 ) and c > c0 . If the potential satisfies θV (r) ≤ rV 0 (r),

r ∈ R,

with θ > 2, then the functional Jk satisfies the Palais-Smale condition.

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Proof . Let un ∈ Xk be a Palais-Smale sequence at the level a. Then, for n large enough, kJk0 (un )kk,∗ ≤ 1 and |Jk (un )| ≤ a + 1. Hence, a+1+

1 1 kun k ≥ Jk (un ) − hJk0 (un ), un i θ θ Z k 1 1 = − c2 |u0n |2 − c20 |Aun |2 dt 2 θ −k Z k 1 0 V (Aun ) Aun − V (Aun ) dt. + −k θ

Due to the assumption on V (r) the second integral is nonnegative and, by Lemma 3.1, we obtain that 1 1 1 a + 1 + kun kk ≥ − (c2 − c20 ) kun k2k . θ 2 θ Hence, the sequence un is bounded in Xk . The boundedness of un implies that, up to a subsequence, un → u fk and, by the compactness weakly in Xk , hence, Aun → Au weakly in X 2 of Sobolev embedding (Theorem A.1), in L (−k, k). A straightforward calculation shows that Z k 2 2 c kun − ukk = c2 (u0n − u0 )2 dt −k

= hJk0 (un ) − Jk0 (u), un − ui + c20 kAun − Auk2L2 Z k 0 + V (Aun ) − V 0 (Au) (Aun − Au) dt. −k

The first and the second terms on the right obviously converge to 0. By Lemma 3.1, Aun is bounded in L∞ (−k, k). Hence, V 0 (Aun ) − V 0 (Au) is bounded in L∞ (−k, k). Since, Aun → Au strongly in L2 (−k, k), the last integral term also tends to zero. Therefore, we conclude that kun −ukk → 0, which proves the lemma. Proof of Theorem 3.1. Case (a). To prove the existence we apply Theorem C.2. Due to Lemmas 3.2 and 3.3, the only we have to verify is that Jk (P u) ≤ J(u) for every u ∈ Xk . We have Z t+1 Z t+1 Z t+1 0 0 . u (s) ds ≥ u (s) ds (AP u)(t) = (P u)0 (s) ds = t

t

Hence, (AP u)(t) ≥ (Au)(t) ≥ (Au)(t).

t

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Since the modified potential V (r) is nondecreasing on R, then Z

k

Jk (P u) =

h i 2 c2 (P u)0 − c2 (AP u)2 − V (AP u) dt

−k k

Z =

2 02 c |u | − c20 (AP u)2 − V (AP u) dt

−k k

Z ≤

2 02 c |u | − c20 (Au)2 − V (Au) dt

−k

= Jk (u). Theorem C.2 implies the existence of a nontrivial critical point uk ∈ P Xk of Jk such that Jk (uk ) ≥ δ, with δ > 0 from Lemma 3.2, and Jk (uk ) ≤ max Jk (sek ). s∈[0,1]

By Remark 3.2, Jk (sek ) = J1 (se1 ) and Jk (uk ) ≤ M := max J1 (se1 ). s∈[0,1]

Case (b) is similar, with P replaced by −P .

Remark that the assumption c > c0 means that the wave speed is greater then the speed of sound and, hence, the wave is supersonic. Remark 3.3 It is not clear whether monotone waves obtained in Theorem 3.1 are strictly monotone. 3.2.3

Nonmonotone and subsonic waves

First we present a version of Theorem 3.1 that concerns the existence of not necessary monotone waves. Theorem 3.2 Assume that a (i00 ) U (r) = u2 + V (r), where V is C 1 on R, V (0) = V 0 (0) = 0 and 2 V 0 (r) = o(|r|) as r → 0, (ii0 ) for some r0 ∈ R we have that V (r0 ) > 0 and there exists θ > 2 such that θV (r) ≤ rV 0 (r),

r ∈ R.

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Let c2 > max(a, 0). Then for every k ≥ 1 there exists a nontrivial 2kperiodic travelling wave uk ∈ Xk . Moreover, the corresponding critical value Jk (uk ) satisfies 0 < δ ≤ Jk (uk ) ≤ M, with δ > 0 and M > 0 independent of k. Proof . The proof is similar to that of Theorem 3.1. Just use the standard mountain pass theorem (Theorem C.1) instead of Theorem C.2. Now let us come back to assumption (i0 ) and consider the case when 0 < c ≤ c0 , i. e. the case of subsonic waves. Theorem 3.3 Assume (i0 ), (ii+ ) and (ii− ). Then for every k ≥ 1 and c ∈ (0, c0 ] there exists a 2k-periodic travelling wave uk ∈ Xk . Proof . We sketch briefly how to apply the linking theorem (see Theorem C.4). We have to check that the functional Jk possesses the linking geometry and satisfies the Palais-Smale condition. Note first that the space Xk splits into the orthogonal sum of the onedimensional subspace generated by the function h0 (t) = t and the space 1 of all 2k-periodic functions from Xk with zero mean value. Consider Hk,0 the operator L defined by Lu = c2 u00 − c20 Au, with 2k-periodic boundary conditions. Elementary Fourier analysis shows that L is a self-adjoint operator in L2 (−k, k) bounded below and that L has discrete spectrum which accumulates at +∞. The eigenvalues and eigenfunctions can be calculated explicitly, but we do not use this fact. We mention only that all eigenvalues, λj , with nonconstant eigenfunctions are 1 double. Denote by h± j ∈ Hk,0 linearly independent pairs of eigenfunctions with the eigenvalues λj . 1 Let Z be the subspace of Hk,0 generated by the functions h± j with λj > 0 ± and Y be the subspace of Xk generated by the functions hj with λj ≤ 0 and the function h0 . Note that dim Y < ∞. It is readily verified that Y ⊥Z and Xk = Y ⊕ Z. A straightforward verification shows that Qk (y + z) = Qk (y) + Qk (z),

y ∈ Y, z ∈ Z,

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where Qk is the quadratic part of the functional Jk , Z 1 k 2 02 Qk (u) = c |u | − c20 |Au|2 dt. 2 −k Since on Z the quadratic form Qk is positive definite, i. e. Qk (u) ≥ α kuk2k , with α > 0, the argument from the beginning of the proof of Lemma 3.2 shows that Jk (u) ≥ δ > 0 on N = {u ∈ Z : kukk = r} provided r > 0 is small enough. Now we fix any z ∈ Z, kzkk = 1, and set M = {u = y + λz : y ∈ Y, kukk ≤ %, λ ≤ 0}. We have to prove that Jk (u) ≤ 0 on M0 = ∂M provided % is large enough. Recall that M0 = {u = y+λz : y ∈ Y, kukR = % and λ ≥ 0, or kukk ≤ % and λ = 0}. We have 2

Z

k

Jk (y + λz) = Qk (y) + λ Qk (z) −

V A(y + λz) dt.

−k

Due to assumptions (i0 ), (ii+ ) and (ii− ), there exists a constant C > 0 such that V (r) ≥ C|r|θ − 1. Therefore, Jk (y + λz) ≤ λ2 α0 + 2k − Cky + λzkθLθ , where α0 = Qk (z). Since %2 = ky + λzk2k = kyk2 + λ2 ,

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we have λ2 ≤ %2 . Furthermore, on finite dimensional spaces all norms are equivalent. Hence, ky + λzkLθ ≥ c ky + λzkk = c% and Jk (y + λz) ≤ α0 %2 + 2k − C%θ . Since θ > 2, the right hand part here is negative if % is large enough. Hence, Jk (y + λz) ≤ 0. If u ∈ M0 , kukk ≤ % and λ = 0, then u = y ∈ Y and, obviously, Jk (u) ≤ 0. Thus, we see that Jk possesses the linking geometry. Finally, we verify the Palais-Smale condition (PS). Let un ∈ Xk be a Palais-Smale sequence at some level a. Choose β ∈ (θ−1 , 2−1 ). For n large we have a + 1 + βkun k ≥ Jk (un ) − βhJk0 (un ), un i Z k " 1 1 2 0 2 = − β c (un ) − − β c20 (Aun )2 2 2 −k # + βV 0 Aun ) Aun − V (Aun ) dt

1 1 2 2 ≥ − β c kun k − − β c20 kAun k2L2 2 2 Z k + (βθ − 1) V (Aun ) dte −k 1 1 ≥ − β c2 kun k2k − − β c20 kAun k2L2 2 2 + C(βθ − 1)kAun kθLθ − C0 .

(3.10)

Since θ > 2, we have kAun k2L2 ≤ CkAun k2Lθ ≤ K(ε) + εkAun kθLθ , where K(ε) → ∞ as ε → 0. Choosing ε small enough, we see that the Lθ term in (3.10) absorbs the L2 term and a + 1 + βkun kk ≥

1 − β c2 kun k2k + C(βθ − 1)kAun kθLθ − C0 . 2

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Since βθ > 1, a + 1 + βkun kk ≥

1 − β c2 kun k2k − C0 . 2

This implies that un is bounded. Now the relative compactness of un in Xk follows exactly as in the proof of Lemma 3.3. Remark 3.4 The linking geometry in Theorem 3.3 is not uniform with respect to k. Therefore, we cannot derive any uniform bound for solutions obtained in that theorem. 3.3

Solitary waves

3.3.1

Variational statement of the problem

In a sense, the case of solitary waves is a limit case of the setting considered in Section 3.2 when k = ∞. Let X be the Hilbert space 1 X := {u ∈ Hloc (R) : u0 ∈ L2 (R), u(0) = 0}

endowed with the inner product Z

+∞

u0 (t) v 0 (t) dt

(u, v) = −∞

and corresponding norm kuk = (u, u)1/2 . Note that the condition u(0) = 0 1 in the definition of X is meaningful because every element of Hloc (R) is a continuous function. As usual k · k∗ stands for the dual norm on the dual space X ∗ . The space X is a 1-dimensional subspace of the Hilbert space 1 e := u ∈ Hloc X (R) : u0 ∈ L2 (R) , with the inner product Z

+∞

u(0) v(0) +

u0 (t) v 0 (t) dt.

−∞

It is readily verified that the linear operator A defined by Z t+1 (Au)(t) := u(t + 1) − u(t) = u0 (s) ds t

e into itself. acts from X The following statement is similar to Lemma 3.1.

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e into L2 (R)∩L∞ (R). Moreover, Lemma 3.4 The operator A acts from X e the function Au is continuous, (Au)(±∞) = 0, i. e. for u ∈ X lim (Au)(t) = 0,

t→±∞

kAukL∞ (R) ≤ kuk and kAukL2 (R) ≤ kuk. Proof . The proof is similar to the proof of Lemma 3.1. The only novelty concerns (Au)(±∞) = 0. We have that Z t+1 1/2 Z t+1 0 0 2 (Au)(t) ≤ u (s) ds ≤ u (s) ds , t

t

and this implies the required since u0 ∈ L2 (R).

We remark also that the map Z (P u)(t) =

t

0 u (s) ds

0

acts continuously from X into itself. On the space X we consider the functional Z ∞ 2 c 0 J(u) := u (t) − U Au(t) dt. 2 −∞ We impose the following assumption: (i000 ) the potential U is C 1 on R, U (0) = U 0 (0) = 0 and for some R > 0 0 U (r) < ∞. sup (3.11) r |u|≤R Actually, (3.11) is the restriction on the behavior of V 0 near r = 0. Hence, if it holds for some R > 0, then so is for every R > 0. Note that (i000 ) is slightly stronger than (i). Proposition 3.3 Under assumption (i000 ) the functional J is well-defined on X. Moreover, J is C 1 and Z +∞ h i hJ(u), hi = c2 u0 (t) h0 (t) − U 0 Au(t) Ah(t) dt. −∞

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Proof . Since c2 (u, u) − Φ(u), 2

J(u) = where

Z

+∞

U Au(t) dt,

Φ(u) = −∞

we have to consider only the functional Φ. Due to assumption (i000 ) (in particular, we use here (3.11)), for every R>0 U (r) sup 2 < ∞. r |r|≤R By Lemma 3.4, there exists a constant C > 0 depending on kAukL∞ , hence, on kuk, such that 2 U Au(t) ≤ C Au(t) . This implies immediately that Φ(u) is finite. A direct calculation shows that the Gateaux derivative of Φ exists and is given by hΦ0 (u), hi =

Z

+∞

U 0 Au(t) Ah(t) dt.

−∞

To complete the proof we have to verify that Φ0 is continuous. Let khk ≤ 1 and un → u in X. Then Aun → Au in X, in L2 (R) and also in L∞ (R). We have, by Lemma 3.4,

0 Φ (un ) − Φ0 (u), h ≤ kAhkL2 U 0 (Aun ) − U 0 (Au) L2

≤ U 0 (Aun ) − U 0 (Au) L2 . Here Lp -norms are taken over R. Assumption (i000 ) implies that there exists a constant C > 0 such that 0 U (r) ≤ C|r|,

|r| ≤ R = max kAukL∞ , kAun kL∞ .

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Now Z

0

U (Aun ) − U 0 (Au) 2 2 ≤ L

a

Z

a

2 0 U Aun (t) − U 0 Au(t) dt −a Z 2 0 2 0 + U Aun (t) + U Au(t) dt |t|≥a

≤

2 0 U Aun (t) − U 0 Au(t) dt −a Z h i Aun (t) 2 + Au(t) 2 dt. +C |t|≥a

Let ε > 0. Since Aun → Au in L2 (R), we can choose a > 0 independent of n and such that the second integral above is less than ε. Since Aun → Au uniformly on [−a, a], then the first integral above is less than ε, provided n is large enough. Hence, for n large

0

U (Aun ) − U 0 (Au) 2 2 ≤ ε + Cε, L and we conclude.

Proposition 3.4 Under assumption (i000 ) a function u ∈ X is a critical point of J if and only if u is a classical, i. e. u ∈ C 2 (R), solution of Eq. (3.3). Proof . Let ϕ ∈ C0∞ (R). Then h(t) = ϕ(t) − ϕ(0) ∈ X. If u is a critical point of J, we see as in the proof of Proposition 3.2 that u is a weak solution of (3.3) and, moreover, u ∈ C 2 (R). Hence, u is a classical solution. Conversely, if u ∈ X is a classical solution of Eq. (3.3), then Z ∞h i hJ(u), hi = c2 u0 (t) h0 (t) − U 0 Au(t) Ah(t) dt = 0 −∞

for every h ∈ X such that h0 ∈ C0∞ (R). Since C0∞ (R) is dense in L2 (R), we conclude. The next proposition deals with simple general properties of solitary waves. See Section 4.5 for further information. Proposition 3.5

Assume (i000 ). Then for any travelling wave u ∈ X

(a) u0 (∞) = lim u0 (t) = 0; t→∞

(b) for the displacement profile r(t) = u(t + 1) − u(t) we have r(∞) = lim r(t) = 0. t→∞

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Proof . Since Au(t) ∈ L∞ (R), assumption (i000 ) implies that the right hand part of Eq. (3.3) belongs to L2 (R). Hence, u00 ∈ L2 (R) and u0 ∈ H 1 (R). By the Sobolev embedding theorem (Theorem A.1), u0 ∈ C0 (R) and statement (a) is proved. Statement (b) follows from Lemma 3.4. 3.3.2

From periodic waves to solitary ones

To pass to the limit as k → ∞, we need some preliminary results. Actually, we shall consider such limits along sequences kn → ∞. Therefore, to simplify the notation every time when we use an expression like “a sequence uk ” this means that there is a sequence kn → ∞ and uk = ukn . So, this does not mean that k is integer. Lemma 3.5 (a) Assume (i00 ), (ii0 ) and c2 > max(a, 0). Then there exists a constant ε1 > 0 independent of k such that for any nontrivial critical points uk ∈ Xk of Jk and u ∈ X of J ε1 ≤ (c2 − a)kuk k2k ≤

2θ Jk (uk ) θ−2

and ε1 ≤ (c2 − a)kuk2 ≤

2θ J(u). θ−2

(b) Under assumptions (i0 ), (ii+ ) (resp., (ii− )) and c > c0 the statement of part (a) holds for nontrivial critical points uk ∈ P Xk (resp., uk ∈ −P Xk ) and u ∈ P X (resp., u ∈ −P X) of Jk and J, with a = c20 . Proof . Part (b) follows from part (a), with a = c20 , if we modify V (r) so that the new potential coincides with V (r) for r > 0 (resp., r < 0) and vanishes for r < 0 (resp., r > 0). Let us prove the upper bound for Jk . By assumption (ii0 ), 1 Jk (uk ) = Jk (uk ) − hJk0 (uk ), uk i 2 Z k 1 0 = V (Auk ) Auk − V (Auk ) dt −k 2 Z θ−2 k ≥ V (Auk ) dt. 2 −k

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Hence, (c2 − a) kuk k2k = 2Jk (uk ) + 2

Z

k

V (Auk ) dt −k

≤

2θ Jk (uk ). θ−2

To obtain the lower bound, we assume on the contrary that there exists a sequence of nontrivial critical points ukn ∈ Xkn such that kukn kkn → 0. (It is not necessary that kn → ∞). By Lemma 3.1, kAukn kL∞ (−kn ,kn ) → 0 and assumption (i0 ) implies that 2 0 V Aukn (t) Aukn (t) ≤ εn Aukn (t) , where εn → 0 as n → ∞. Since hJkn (ukn ), ukn i = 0, we have c2 kukn k2kn =

Z

kn

a|Aukn |2 + V 0 (Aukn ) Aukn dt

−kn

≤ (a + εn ) kukn k2kn . Since c2 > a, this is a contradiction and the lemma is proved. The case of J is similar.

Remark 3.5 In the proof of Lemma 3.5 we have only used the fact that hJk (uk ), uk i = 0 (resp., hJ(u), ui = 0). Hence, the statements of that lemma remain true for nonzero elements uk ∈ Xk (resp., u ∈ X) satisfying the last identities. Lemma 3.6 (i00 ) and c2 > max(a, 0). Let uk ∈ Xk be a sequence

0 Assume

such that Jk (uk ) k,∗ → 0 and kuk kk is bounded. Then either kuk kk → 0, or for any r > 0 there exist η > 0, a subsequence of uk , still denoted by uk , and ζk ∈ R such that Z ζk +r Auk (t) 2 dt ≥ η. ζk −r

Proof . Assume that Z

ζ+r

lim sup

k→∞ ζ∈R

ζ−r

Auk (t) 2 dt = 0.

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Let ϕk ∈ C0∞ (R) be a function such that 0 ≤ ϕk (t) ≤ 1,

ϕk (t) = 1

if |t| ≤ k,

ϕk (t) = 0 if |t| ≥ k + 1 0 and ϕk (t) ≤ C, where C > 0 is independent of k. We set fk (t) = ϕk (t) Auk (t). It is readily verified that fk ∈ H 1 (R), kfk kH 1 is bounded and Z ζ+r fk (t) 2 dt = 0. lim sup k→∞ ζ∈R

ζ−r

Then Lemma B.2 implies that kfk kLp (R) → 0 for all p > 2. Since kAuk kLp (−k,k) ≤ kfk kLp (R) , we see that kAuk kLp (−k,k) → 0 for all p > 2. Let εk = kJk0 (uk )kk,∗ → 0. Then we have Z k 2 0 2 hJk0 (uk ), uk ik = c (uk ) − a|Auk |2 − V 0 (Auk )Auk dt ≤ εk kuk kk . −k

By Lemma 3.1, kAuk kL∞ (−k,k) ≤ C. Fix any p > 2. Then assumption (i00 ) implies that for every ε > 0 there exists a constant Cε > 0 such that 0 V (r)r ≤ εr2 + Cε |r|p , |r| ≤ C. Now we have that Z c2 kuk k2k ≤

k

a |Auk |2 + V 0 (Auk ) Auk dt + εk kuk kk

−k k

Z ≤

(a + ε)|Auk |2 + Cε |Auk |p dt + εk kuk kk

−k

= (a + ε) kAuk k2L2 (−k,k) + Cε kAuk kpLp (−k,k) + εk kuk kk .

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If a < 0 we can take ε > 0 small enough so that a + ε < 0 and obtain c2 kuk k2k ≤ Cε kAuk kpLp (−k,k) + εk kukk which implies that kuk kk → 0. If a > 0, using Lemma 3.1 we obtain that (c2 − a − ε) kuk k2k ≤ Cε kAuk kpLp (−k,k) + εk kukk . Since c2 > a, we can choose ε > 0 so small that c2 − a − ε > 0 and complete the proof as above. Proposition 3.6 (a) Assume (i00 ), (ii0 ) and c2 > max(a, 0). Let uk ∈ Xk be a sequence of nontrivial critical points of Jk such that the critical values Jk (uk ) are uniformly bounded. Then there exist a nontrivial critical point u ∈ X of J and a sequence ζk ∈ R such that a subsequence of uk (· + ζk ) − uk (ζ) converges to u uniformly on compact intervals together with first and second derivatives, i. e. in C 2 (R). (b) Under assumptions (i0 ), (ii+ ) (resp., (ii− )) and c2 > c20 the same statement holds for nontrivial critical points uk ∈ P Xk (resp., uk ∈ −P Xk ), with u ∈ P X (resp., u ∈ −P X). Proof . By Lemma 3.5, kuk kk is bounded above and below by two positive constants. This means that kuk kk does not converge to 0 and Lemma 3.6 shows that, along a subsequence, Z ζk +r |Auk |2 dt ≥ η (3.12) ζk −r

for some r > 0, η > 0 and ζk ∈ R. Let u ek (t) = uk (t + ζk ) − uk (ζk ). Then ke uk kk = kukk and since Jk is invariant under translations and adding constants, Jk (e uk ) = Jk (uk ) and Jk0 (e uk ) = 0. Moreover, since ke uk kk is bounded, there exist a subsequence, 1 still denoted by u ek , that converges to a function u ∈ Hloc (R) weakly in 1 1 Hloc (R), i. e. weakly in H (a, b) for every finite interval (a, b). e Indeed, we have u First, we check that u ∈ X. e0k → u0 weakly in L2loc (R). Hence, for every a < b Z b Z b 0 2 0 2 u (t) dt ≤ lim inf u ek (t) dt ≤ lim inf ke uk k2k ≤ C. a

a

Passing to the limit as a → −∞ and b → +∞, we obtain the result.

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By the compactness of Sobolev embedding, Ae uk → Au strongly in i. e. uniformly on finite intervals, and in L2loc (R). This and Eq. (3.12) shows that Z r |Au|2 dt ≥ η > 0, L∞ loc (R),

−r

hence, u 6= 0. 0 0 ∞ Since Ae uk → Au in L∞ loc (R), then U (Auk ) → U (Au) in Lloc (R). Let ϕ ∈ C ∞ (R),

ϕ(0) = 0,

ϕ0 ∈ C0∞ (R).

For k large, supp Aϕ ⊂ [−k, k]. For such k, let ϕk ∈ Xk be the primitive function of the 2k-periodic extension of ϕ0 |[−k,k] . Then Z ∞ 2 0 0 hJ 0 (u), ϕi = c u ϕ − U 0 (Au) Aϕ dt Z−∞ 2 0 0 = c u ϕ − U 0 (Au) Aϕ dt supp Aϕ Z 2 0 0 = lim c u ek ϕ − U 0 (Ae uk ) Aϕ dt k→∞

supp Aϕ k c2 u e0k −k

Z = lim

k→∞

− U 0 (Ae uk ) Aϕ dt

= 0. Hence, u is a nontrivial solution of Eq. (3.3). The right hand side of Eq. (3.3) for u ek converges in L∞ loc (R) to the right hand side of that equation for u. Therefore, u e00k → u00 , hence, u e0k → u0 and ∞ u ek → u, in Lloc (R). In particular, u(0) = 0 and x ∈ X. This proves part (a). Part (b) follows from part (a). Just replace V (r) by a function that coincides with V (r) for r > 0 (resp., r < 0) and vanishes for r < 0 (resp., r > 0), and note that the limit of a sequence of monotone functions is a monotone function. Remark 3.6 Proposition 3.6 is still valid if, instead of of

a sequence

critical points, we consider a sequence uk ∈ Xk such that Jk0 (uk ) k,∗ → 0 and Jk (uk ) is a bounded sequence. Combining Proposition 3.6 with Theorems 3.1 and 3.2, we obtain the following existence results.

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Theorem 3.4 (a) Assume (i0 ), (ii+ ) and c > c0 . Then there exists a nontrivial nondecreasing solitary wave u ∈ X. (b) Assume (i0 ), (ii− ) and c > c0 . Then there exists a nontrivial nonincreasing solitary wave u ∈ X. Theorem 3.5 Assume (i00 ), (ii0 ) and c2 > max(a, 0). Then there exists a nontrivial solitary wave u ∈ X. The next result is an important supplement to Theorem 3.4 (cf., however, Remark 3.3). Proposition 3.7 Under the assumptions of Theorem 3.4 suppose, in addition, that r · U 0 (r) > 0 for all r > 0 (resp. r < 0). Then any nontrivial monotone wave u ∈ X is strictly monotone. In particular, this is so if either c0 > 0, or r · V 0 (r) > 0 for all r > 0 (resp. r < 0). Proof . We consider only the case of nondecreasing waves. The case of nonincreasing waves is similar. Let u ∈ X be a nontrivial nondecreasing solitary wave. Then there exists t0 ∈ R such that u0 (t0 ) > 0. Hence, if t0 − 1 ≤ t ≤ t0 , then t+1

Z

u0 (s) ds > 0

Au(t) =

(3.13)

t

because the interval of integration contains the point t0 . In Section 4.5, Theorem 4.10 and Remark 4.6, it is shown that r(t) = Au(t), u0 (t) and u00 (t) decay exponentially at infinity.3 Integrating Eq. (3.3) over the interval (−∞, t), we obtain that 1 u (t) = 2 c 0

Z

t

U 0 Au(s) ds.

t−1

Together with (3.13), this shows that u0 (t) > 0 whenever t0 −1 ≤ t ≤ t0 +1. Iterating this argument n times, we obtain that the derivative u0 (t) is positive on the interval [t0 − n, t0 + n]. This completes the proof. In Theorems 3.4 and 3.5 solitary waves, i. e. critical points of J, are obtained as local limits of mountain pass critical points of Jk . Note that the functional J itself also possesses the mountain pass geometry. 3 Note

that Section 4.5 is independent on the previous material.

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Confining ourself to the setting of Theorem 3.5, let αk = inf max Jk γ(s) γ∈Γk s∈[0,1]

and α = inf max J γ(s) γ∈Γ s∈[0,1]

be mountain pass values of Jk and J, respectively. Here n o Γk = γ ∈ C [0, 1]; Xk : γ(0) = 0, γ(1) = ek , n o Γ = γ ∈ C [0, 1]; X : γ(0) = 0, γ(1) = e , e ∈ X and e ∈ Xk are arbitrary elements such that J(e) < 0 and Jk (ek ) < 0. Note that α is not necessary a critical value of J. Proposition 3.8

Under assumptions of Theorem 3.5 we have

J(u) ≤ lim sup αk ≤ α ≤

inf

sup J(τ v).

v∈X\{0} τ >0

Proof . In Theorem 3.5 the critical point u ∈ X is obtained as the limit of an appropriate sequence uk ∈ Xk of mountain pass critical points of the functionals Jk in the topology of C 1 (R), i. e. uniformly on finite interval together with first derivatives. Let k0 ≥ 1 and k ≥ k0 . Then Z k 1

Jk (uk ) = Jk (uk ) − Jk (uk ), uk = g(Auk ) dt, 2 −k where g(r) =

1 0 V (r) r − V (r). 2

Due to assumption (ii0 ), g(r) ≥ 0. Hence, Z k0 Jk (uk ) ≥ g(Auk ) dt. −k0

Since Auk → Au in

L∞ loc (R),

we obtain that Z

k0

lim sup Jk (uk ) = lim sup αk ≥

g(Au) dt. −k0

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Passing to the limit as k0 → ∞ yield Z

+∞

g(Au) dt ≤ lim sup αk . −∞

Since 1 0 J(u) = J(u) − J (u), u = 2

Z

+∞

g(Au) dt, −∞

we get J(u) ≤ lim sup αk . Now let γ ∈ Γ. Consider the pass γk ∈ Γk uniquely defined by:   d d γ(s)(t) , |t| ≤ k − 1, γk (s)(t) = dt  0, dt k − 1 < |t| ≤ k. (Here s ∈ [0, 1] is the parameter of the pass). Then it is not difficult to see that lim Jk γk (s) = J γ(s)

k→∞

uniformly with respect to s ∈ [0, 1]. This immediately implies the inequality lim sup αk ≤ α. Finally, for every v ∈ X \ {0} there exists τ0 > 0 such that J(τ v) < 0 for all τ ≥ τ0 . Considering the path that consists of the segment {τ v : 0 ≤ τ ≤ τ0 } and a path joining the points τ0 v and e inside the domain where J is negative, we obtain the last inequality of the proposition. Similar result takes place in the setting of Theorem 3.4. The only we need is to choose e and ek to be increasing (respectively, decreasing) and take the infimum in the last inequality of Proposition 3.8 among all nonzero v ∈ P X (respectively, v ∈ −P X).

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3.3.3

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Global structure of periodic waves

The convergence result of Proposition 3.6 shows, in particular, that locally a periodic wave profile looks like an appropriate single solitary wave profile. Here we present a refined version of that result, which shows that a periodic wave profile, with large period, on the wavelength interval looks like a sum of finite number of solitary profiles. Actually, it is convenient to consider not exact periodic waves, i. e. critical

points

of Jk , but approximate waves, i. e. sequences uk ∈ Xk such that Jk (uk ) k,∗ → 0 as k → ∞. Proposition 3.9 In addition to assumptions (i00 ) and (ii0 ), with c2 > max(a, that V ∈ C 2 (R). Let uk ∈ Xk be a sequence

0 0), suppose

satisfying Jk (uk ) k,∗ → 0 and Jk (uk ) → α > 0 as k → ∞. Then there exist a finite number of critical points ui ∈ X of J, i = 1, 2, . . . , l, and ζki ∈ R, i = 0, 1, . . . , l − 1, l, with ζkl = 0, such that l X

J(ui ) = α

i=1

and, as k → ∞ along a sequence,

l X

i i 0

uk (· + ζk0 ) − u (· + ζk )

i=1

→ 0.

L2 (−k,k)

Proof . By Remark 3.6, there exist a critical point u = u1 ∈ X of J and ζk ∈ R such that, along a subsequence, u ek (t) = uk (t + ζk ) − uk (ζk ) converges to u in the sense of Proposition 3.6. Let ϕk ∈ C ∞ (R) be a sequence with the following properties: ϕk (0) = 0,

ϕ0k ∈ C0∞ (R),

supp ϕk ⊂ [−k + 1, k − 1]

and ϕk → u strongly in X. Obviously, such a sequence exists. Since J is C 1 , then J 0 (ϕk ) → 0 and J(ϕk ) → J(u). Denote by ϕ ek ∈ Xk the function such that (ϕ ek )0 is the 2k-periodic extension of ϕ0k |[−k,k] to R. For any η ∈ Xk we set  −k − 1 ≤ t ≤ k + 1,  η(t), ηk (t) := (Tk η)(t) := η(k + 1), t ≥ k + 1,  η(−k − 1), t < −k − 1.

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This is a kind of cut-off. The function ηk is well-defined because η is continuous. It is easy that Tk η ∈ X and kTk ηk ≤ 2 kηkk . We have

0 ek ), η = J 0 (ϕk ), ηk ≤ J 0 (ϕk ) ∗ kηk k Jk (ϕ

≤ 2 J 0 (ϕk ) ∗ kηkk .

Therefore, Jk0 (ϕ ek ) k,∗ → 0.

Setting vk = u fk − ϕ ek , we want to show that Jk0 (vk ) k,∗ → 0. First, we note that, for η ∈ Xk ,

Jk0 (vk ), η = Jk0 (e uk ) − Jk0 (ϕ ek ), η Z k 0 + V (Ae uk )−V 0 (Aϕk )−V 0 (Ae uk −Aϕ) Aη dt. (3.14) −k

Since ϕk → u in X, for any ε > 0 we can find k0 > 0 such that Aϕk (t) ≤ ε for t ∈ / B := [−k0 , k0 ] and Z

|Aϕk |2 dt ≤ ε2 .

R\B

Let Bk = [−k, k] \ B. Since Aϕk is bounded in L∞ (R), assumption (i00 ) yields 0 V Aϕk (t) ≤ C Aϕk (t) , with C > 0 independent of k. Then Z

Bk

0 V (Aϕk ) Aη ≤ C

Z

!1/2 |Aϕk |2 dt

kAηkL2 (−k,k) ≤ Cε kηkk .

R\B

Using the fact that V ∈ C 2 and Ae uk and Aϕk are bounded in L∞ (R), we obtain 0 V (Ae uk − θAϕk ) Aϕk ≤ C |Aϕk |, uk − Aϕk ) − V 0 (Ae uk ) = V 00 (Ae

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with θ ∈ [0, 1]. Hence, Z Z 0 0 V (Ae uk − Aϕk ) − V (Ae uk ) Aη dt ≤ C |Aϕk | |Aη| dt Bk Bk Z 1/2 2 ≤C |Aϕk | dt kηkk Bk

≤ Cε kηkk . ∞ 1 Since Ae uk → Au in L∞ loc (R), Aϕk → Au in L (R) and V ∈ C (R), we see that Z 0 0 0 V (Ae uk ) − V (Aϕk ) − V (Ae uk − Aϕk ) Aη dt ≤ Cε kηkk , B

provided k is large enough. Thus, given ε > 0 the integral term in Eq. (3.14) is by Cεkηkk provided k is large enough. This means that

estimated

J 0 (vk ) → 0. k k,∗ Now a straightforward calculation shows that Jk (e uk ) = Jk (vk ) + Jk (ϕ ek ) + (vk , ϕ ek )k Z k − V (Avk − Aϕk ) − V (Avk ) − V (Aϕk ) dt.

(3.15)

−k

The integral term here can be estimated as above and, hence, tends to zero. A trivial calculation shows that (vk , ϕ ek )k = (Tk vk , ϕk ). 0 Since vk0 → 0 in L∞ loc (R), so is (Tk vk ) . Moreover, Tk vk → 0 weakly in X because Tk vk is bounded in X. Since ϕk is strongly convergent in X, we have that (Tk vk , ϕk ) → 0, hence, (vk , ϕ ek )k → 0. Now Eq. (3.15) implies that

α = lim Jk (e u)k = J(u) + lim Jk (vk ), i. e. Jk (vk ) → α − J(u). Moreover, u is a nontrivial critical point of J, hence, J(u) 0, by

Lemma 3.5.

> 0

Since Jk (vk ) k,∗ → 0, the beginning of the proof of Proposition 3.6 shows that either Jk (vk ) → 0 and kvk kk → 0, or Jk (vk ) and kvk kk is bounded below by a positive constant independent of k. In the first case we are done, with l = 1. In the second case, by Lemma 3.5, 0 < ε2 :=

θ−2 ε1 ≤ J(u) < α, 2(θ − 2)

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with ε1 from Lemma 3.5. Note that ε2 is independent of uk and u. Now we can repeat the arguments above, with uk replaced by vk and α replaced by α − J(u) < α, to get u2 . After a finite number, not greater that α/ε2 , of such steps we conclude. 3.3.4

Examples

Here we consider several examples. Example 3.1

In the case of Toda lattice the potential is given by U (r) = ab−1 e−br + br − 1 ,

ab > 0.

If b > 0 and c > ab, then for every k ≥ 1, Eq. (3.1) has a nontrivial nonincreasing 2k-periodic travelling wave and also a nontrivial decreasing solitary wave. If b < 0 and c > ab, then there exist a nontrivial nondecreasing 2k-periodic travelling wave and a nontrivial increasing solitary wave. Certainly, these results are known [Toda (1989)]. Example 3.2

Consider the potential U (r) =

c20 2 c1 r + r2n+1 , 2 2n + 1

c0 > 0, c1 6= 0.

If n = 1, this is the FPU α-model. Let c > c0 . If c1 > 0, then there exist nontrivial nondecreasing travelling waves, both 2k-periodic, with k ≥ 1, and solitary. If c1 < 0, then there exist nontrivial 2k-periodic and solitary travelling waves, nonincreasing in this case. Actually, due to Proposition 3.7, the solitary waves are strictly monotone. Example 3.3

Let U (r) =

c20 2 c1 2n r + r , 2 2n

c0 > 0, c1 > 0.

In the case when n = 2 we obtain the FPU β-model. Note that the potential U (r) is even. If c > c0 , then for every k ≥ 1, there exists a pair ±uk ∈ Xk of nontrivial travelling waves, one nondecreasing and the other nonincreasing. Also there exists a pair ±u ∈ X of nontrivial strictly monotone solitary waves. If 0 < c ≤ c0 , then for every k ≥ 1 there exists a pair ±uk ∈ Xk of 2k-periodic subsonic travelling waves with the speed c.

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Example 3.4

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Consider the potential U (r) =

m n X a 2 X r + ai |r|i + bi |u|i , 2 i=3 i=m+1

with ai ≤ 0, bi ≥ 0 and bn > 0. If c2 > max(0, a), then Theorems 3.2 and 3.5 apply and, hence, we obtain nontrivial periodic and solitary travelling waves, not necessary monotone. 3.4 3.4.1

Ground waves: existence and convergence Ground waves: periodic case

Consider a travelling wave that has the profile u ∈ Xk , i. e. a 2k-periodic wave in our terminology. We say that u is a periodic ground wave if the corresponding critical value Jk (u) is minimal possible among all nontrivial critical values of the functional Jk . First we study general, not necessary monotone waves, in the setting of Subsection 3.2.3 and postpone the discussion of monotonicity to Subsection 3.4.3. Since the functional Jk satisfies the Palais-Smale condition, the existence of ground waves in this case is almost trivial and we have Proposition 3.10 Assume (i00 ) and (ii0 ) from Theorem 3.2. Let c2 > max(a, 0). Then for every k ≥ 1 there exists a 2k-periodic ground wave uk ∈ Xk . Moreover, any 2k-periodic ground wave uk satisfies δ ≤ Jk (uk ) ≤ M, with δ > 0 and M > 0 independent of k. Proof . Fixed k ≥ 1, let α be the infimum of nontrivial critical values of Jk . By Theorem 3.2, 0 < α < +∞. Then there exists a sequence u(n) of critical points of Jk such that the sequence Jk (u(n) ) of corresponding critical values converges to α. Obviously, u(n) form a Palais-Smale sequence. Since Jk satisfies the Palais-Smale condition, the sequence u(n) is precompact in Xk and any its limit point is a critical point of Jk at the level α, hence, a ground wave. Remark 3.7 In addition to the assumptions of Proposition 3.10, suppose that the interaction potential is even. This implies that the functional Jk is even. Using standard multiplicity results for even functionals [Rabinowitz (1986); Willem (1996)], one can show that, for every k ≥ 1, there exists an

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infinite sequence u(n) ∈ Xk of critical points, such that the critical values Jk (u(n) ) go to infinity. We do not present the details because no counterpart of this result is known in the case of solitary waves. Now we are going to present a more explicit description of ground waves. With this aim we impose the following assumption (N ) V ∈ C 2 and there exists µ ∈ (0, 1) such that 0 < r−1 V 0 (r) ≤ µV 00 (r),

r 6= 0.

An elementary integration by parts shows that (N ) implies (ii0 ), with θ=

1+µ > 2. µ

Moreover, the function V 0 (r)/r is monotone decreasing from +∞ to 0 on (−∞, 0) and monotone increasing from 0 to +∞ on (0, +∞). Let us introduce the functional Ik (u) := hJk0 (u), ui =

Z

k

2 0 2 c (u ) − a(Au)2 − V 0 (Au) Au dt.

(3.16)

−k

This is the so-called Nehari functional of Jk . The same argument as in the proof of Proposition 3.1 shows that, under assumptions (i00 ) and (N ), the functional Ik is a C 1 functional on Xk and hIk0 (u), hi

Z

k

=

2 0 0 2c u h − 2a Au Ah − V 00 (Au) Au Ah − V 0 (Au) Ah dt

−k

(3.17) for all h ∈ Xk . Now we define the Nehari manifold by Sk = u ∈ Xk \ {0} : Ik (u) = 0 .

(3.18)

Assumption (N ) implies that Sk is a C 1 submanifold of codimension 1 in Xk . Indeed, this follows from the implicit function theorem. For, we have to check only that Ik0 (u) 6= 0 for every u ∈ Sk . But, for any such u, a

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straightforward calculation shows that Z k 2 0 2 0 c (u ) − a(Au)2 − V 00 (Au) (Au)2 dt hIk (u), ui = −k k

Z ≤ Z =

2 0 2 c (u ) − a(Au)2 − µ−1 V 0 (Au) Au dt

−k k

c2 (u0 )2 −a(Au)2 −V 0 (Au) Au−(µ−1 −1)V 0 (Au) Au dt

−k

= −(µ−1 − 1)

Z

k

V 0 (Au) Au dt,

−k

where we have used the definition of Sk and assumption (N ). Since u 6= 0, the last integral above is positive. Hence, hIk0 (u), ui < 0 and Ik0 (u) 6= 0. This also shows that if u ∈ Sk , then the line Ru is transverse to the tangent space ker Ik0 (u) to the manifold Sk at the point u. Another important property of the Nehari manifold Sk is that, under our assumption, for any nonzero u ∈ Xk there exists a unique τ > 0 such that τ u ∈ Sk . In particular, Sk 6= ∅. Indeed, for τ > 0 we have " # Z k

Ik (τ u) = τ 2 c2 kuk2k − akAuk2L2 (−k,k) − τ −1

V 0 (τ Au) Au dt .

−k

Since c2 kuk2k − akAuk2L2 (−k,k) > 0, the monotonicity property of V 0 (r)/r implies the required. We consider the restriction Jk |Sk of the functional Jk to the Nehari manifold Sk . Note that Z k 1 0 V (Au) Au − V (Au) dt on Sk . (3.19) Jk (u) = −k 2 Proposition 3.11 Under assumptions (i00 ) and (N ), let c2 > max(a, 0). Then any critical point of Jk |Sk is a critical point of Jk . Proof . A straightforward calculation shows that, for any u ∈ Xk , d Jk (τ u)|τ =1 = Ik (u). dτ

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Therefore, if u ∈ Sk , then Jk0 (u) vanishes on the line Ru. But Jk0 (u) vanishes also on the tangent space ker Ik0 (u) to Sk at the point u because u is a critical point of Jk |Sk . Since the subspaces ker Ik0 (u) and Ru a transverse, then Jk0 (u) = 0 and u is a critical point of entire functional Jk . Remark 3.8

Obviously, all nontrivial critical points of Jk belong to Sk .

Due to Proposition 3.11, it is natural to consider the following minimization problem mk = inf{Jk (u) : u ∈ Sk }.

(3.20)

Equation (3.19) together with (ii0 ), a consequence of (N ), shows that mk ≥ 0. In fact, we have Lemma 3.7 Under assumptions (i00 ) and (N ), let c2 > max(a, 0). Then there exist constants ε1 > 0 and ε2 > 0, independent of k, such that kukk ≥ ε1 and Jk (u) ≥ ε2 for all u ∈ Sk . In particular, mk ≥ ε2 > 0. Proof . Similar to the proof of Lemma 3.6 (see Remark 3.5).

Now we summarize geometric properties of the Nehari manifold. The manifold Sk separates two open subsets Dk+ ∪ {0} and Dk− , i. e. Xk = Dk+ ∪ {0} ∪ Dk− ∪ Sk and Sk = ∂ Dk+ ∪ {0} = ∂Dk− , where Dk+ = {u ∈ Xk : Ik (u) > 0} and Dk− = {u ∈ Xk : Ik (u) < 0}. Moreover, Dk+ ∪ {0} is a star-shaped set that contains a ball centered at the origin. The functional Jk restricted to any transverse ray {τ v, τ > 0}, v ∈ Sk , attains its maximum exactly at v ∈ Sk . Since all nontrivial critical points of Jk lie in Sk , the importance of problem (3.20) is that any minimum point is a 2k-periodic ground wave.

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Now let ek be the function from Lemma 3.2, Jk (ek ) ≤ 0. Denote by αk the mountain pass critical value αk := inf max Jk γ(y) γ∈γk t∈[0,1]

of Jk , where n o Γk := γ ∈ C [0, 1]; Xk : γ(0) = 0, γ(1) = ek . This is exactly the critical value from Theorem 3.2. Lemma 3.8 Assume (i00 ) and (N ), with c2 > max(a, 0). Then αk = mk and any mountain pass critical point is a solution of (3.20). Proof . Since αk is a critical value with the critical point uk ∈ Sk , then mk ≤ αk . Let v0 ∈ Xk , kv0 kk = 1. Then, as in the proof of Lemma 3.2, we see that there exists τ0 > 0 such that Jk (τ v) < 0 for all τ ≥ τ0 and all v ∈ span{v0 , ek }, with kvkk = 1. This means that one can join τ0 v0 and ek by a path γ 0 such that Jk is negative on γ 0 . Combining γ 0 with the segment {τ v0 , 0 ≤ τ ≤ τ0 }, we obtain the path that belongs to Γk . This implies that αk ≤ αk0 =

inf

sup Jk (τ v).

v∈Xk {0} τ >0

Now observe that, for any v ∈ Xk , v 6= 0, the function Jk (τ v0 ) of τ > 0 attains its maximum value at some point τ0 > 0 and we have 0=

d 1 Jk (τ v)|τ =τ0 = Ik (τ0 v). dt τ0

Hence, τ0 v ∈ Sk . This shows that αk0 = mk = inf{Jk (u) : u ∈ Sk } and we conclude. 3.4.2

Solitary ground waves

Here we consider solitary ground waves, i. e. nontrivial solitary travelling waves with profile u ∈ X such that the critical value J(u) is minimal possible among all nontrivial critical values of J. Since the functional J does not satisfy the Palais-Smale condition, the existence of such waves is not trivial and to prove it we employ the Nehari manifold approach. Therefore, we still assume (N ) and (i0 ), with c2 > max(a, 0).

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The Nehari functional for J reads Z +∞ 2 0 2 I(u) := hJ 0 (u), ui = c (u ) − a(Au)2 − V 0 (Au) Au dt

(3.21)

−∞

and the Nehari manifold is defined by S := u ∈ X \ {0} : I(u) = 0 .

(3.22)

Note that Z

∞

J(u) = −∞

1 0 V (Au) Au − V (Au) dt =: Ψ(u) 2

on S.

(3.23)

Under assumptions (i00 ) and (N) geometric properties of I and S are the same as in the case of Ik and Sk . In particular, S is a C 1 -submanifold of X of codimension 1. All nontrivial critical points of J belong to S. Moreover, we have Proposition 3.12 Under assumptions (i00 ) and N , let c2 > max(a, 0). Then any critical point of J|S is a critical point of J. Proof uses the same argument as the proof of Proposition 3.11.

As in the case of Sk , for every u ∈ X, u 6= 0, there exists a unique τ = τ (u) > 0 such that τ u ∈ S. Moreover, τ (u) depends continuously on u ∈ X. Consider the minimization problem m = inf{J(u) : u ∈ S}

(3.24)

analogous to problem (3.20). Due to Proposition 3.12, any minimizer of (3.24) if it exists is a ground wave. Let e1 be the function from Lemma 3.2. Denote by e ∈ X the function defined by e0 |[−1,1] = e01 |[−1,1] and e0 (t) = 0 if |t| > 1. Then J(e) ≤ 0. Let α be the mountain pass value α := inf max J γ(s) , γ∈Γ s∈[0,1]

where n o Γ := γ ∈ C [0, 1]; X : γ(0) = 0, γ(1) = e . Note that a priori it is not known that α is a critical value of J because J does not satisfy (PS).

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Lemma 3.9

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Under assumptions of Proposition 3.12 we have that α = m.

Proof is identical to the proof of Lemma 3.8.

Now we are ready to prove the main result of this section that gives, in particular, the existence of solitary ground waves. Theorem 3.6 Assume (i00 ) and N , with c2 > max(a, 0). Let uk ∈ Xk be a sequence of periodic ground waves. Then there exist a sequence ζk ∈ R and a solitary ground wave u ∈ X such that, along a subsequence k → ∞, the functions u ek = uk (· + ζk ) − uk (ζ) satisfy lim ke u0k − u0 kL2 (−k,k) = 0.

k→∞

(3.25)

Moreover, m = J(u) and mk → m. Proof . The existence of periodic ground waves follows from Proposition 3.10. The same proposition and Lemma 3.6 imply that kuk kk is bounded. Now, applying Proposition 3.6, we see that there exist ζk and a 1 nontrivial critical point u ∈ X of J such that u ek → u in Hloc (R). To prove that the limit u is actually a ground wave, first note that for any v ∈ S and ε > 0 there exists vk ∈ Sk such that Jk (vk ) ≤ J(v) + ε if k is large enough. To show this, take a sequence ϕk ∈ C ∞ (R) such that supp ϕ0k ⊂ [−k + 1, k − 1] and ϕk → v in X. Denote by ϕk ∈ Xk the function such that (ϕk )0 = ϕ0k on [−k + 1, k − 1]. Let τk > 0 be a unique number such that vk = τk ϕk ∈ Sk . In fact, we have also that τk ϕk ∈ S. Hence, τk → 1 and τk ϕk → v. Therefore, Jk (vk ) = J(τk ϕk ) → J(v) and we obtain the required. As consequence, since mk ≤ Jk (vk ), we have that lim sup mk ≤ J(v) + ε k→∞

for all v ∈ S. Hence, lim sup mk ≤ m. k→∞

Let g(r) =

1 0 V (r) r − V (r). 2

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Since g(r) ≥ 0, then, by (3.19), Z k Z mk = g(Ae uk ) dt ≥ g(Ae uk ) dt −k

B

for any bounded interval B and k large enough. We know that Auk → Au in L∞ uk ) → g(Au) in L∞ loc (R). Therefore, g(Ae loc (R), and the integral in the right hand part of the inequality above converges to Z g(Au) dt. B

Since B is an arbitrary interval, this implies that Z +∞ lim sup mk ≥ g(Au) dt. k→∞

−∞

As u is a critical point of J, we have that u ∈ S and, by (3.23), Z +∞ g(Au) = J(u) ≥ m. −∞

This implies that lim sup mk ≥ J(u) ≥ m k→∞

and therefore lim mk = m = J(u).

k→∞

Thus u is a solitary ground wave. The convergence statement (3.25) follows from Proposition 3.9. Indeed, since J(u) is the lowest nontrivial critical point of J, we have that l = 1 in that proposition and we are done. 3.4.3

Monotonicity

Now we consider the case of even interaction potential. Proposition 3.13 Assume (i00 ), with a = c20 , and (N ). Suppose that V (r) is an even function and c > c0 . Then every ground wave, periodic or solitary, is either nondecreasing or nonincreasing. Proof . We consider the case of solitary waves. The remaining case of periodic waves is quite similar.

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Let u ∈ X be a ground solitary wave. Since the potential is even, the function −u is also a ground wave. Remind that t

Z

0 u (s) ds.

P u(t) = 0

Suppose that u is not monotone. Then P u 6= ±u. By the definitions of operators A and P , Z (AP u)(t) =

t+1

0 u (s) ds

t

and Au(t) ≤ (AP u)(t). Moreover, since u(t) is not a monotone function, the last inequality is strict for all t in an open subset of R. By Theorem 3.6, u ∈ S is a minimizer of problem (3.24). Since (P u)0 = |u0 |, we have that Z

+∞

0 = I(u) =

2 0 2 c (u ) − c20 |Au|2 − V 0 (Au) Au dt

−∞

Z

+∞

= −∞ +∞

Z >

n o 2 c2 (P u)0 − c20 |Au|2 − V 0 |Au| |Au| dt o n 2 c2 (P u)0 − c20 |AP u|2 − V 0 (AP u) (AP u) dt

−∞

= I(P u). We have used here the strict monotonicity of V 0 (r)r. Thus, P u ∈ / S, but there exists τ > 0 such that v = τ P u ∈ S. Moreover, since I(P u) < 0, we have that τ < 1. A straightforward calculation shows that

0 r 1 V (r) r − V (r) = V 00 (r) − r−1 V 0 (r) . 2 2

Hence, due to assumption (N ), the function g(r) :=

1 V (r) r − V (r) 2

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is decreasing if r ≤ 0 and increasing if r ≥ 0. Therefore, Z +∞ Z +∞ Z J(u) = Ψ(u) = g(Au) dt = g |Au| dt > −∞

−∞

+∞

g(AP u) dt

−∞

= Ψ(P u). Further, Z

+∞

Ψ(v) =

Z

+∞

g(τ AP u) dt < −∞

g(AP u) dt = Ψ(P u). −∞

Hence, Ψ(v) < J(u) = m. Since v ∈ S, this mean that u is not a minimizer of (3.24). Contradiction. Thus, we have proved that u is nondecreasing. Remark 3.9 Actually, in Proposition 3.14 solitary ground waves are strictly monotone as it follows from Proposition 3.8. 3.5 3.5.1

Near sonic waves Amplitude estimate

Here we consider travelling waves with the speed c > c0 close to the speed of sound c0 . We start with the following statement which is a key point of the present section. Proposition 3.14 Assume (i0 ), (ii+ ) and (ii− ). In addition, suppose that there exist a0 > 0 and q > 2 such that V (r) ≥ a0 |r|q ,

r ∈ R.

(3.26)

Let c > c0 and uk = uk,c ∈ Xk (resp. u = uc ∈ X) denote any nontrivial 2k-periodic (resp. solitary) travelling wave with critical value not greater than the mountain pass level of Jk (resp. J). Then there exist constants c and C such that for every c ∈ (c0 , c] q+4

kuk k2k ≤ C (c2 − c20 ) q−2 , provided k is large enough (k ≥ k0 (c)), and q+4

kuk2 ≤ C (c2 − c20 ) q−2 .

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Proof . We present the proof in the case when c0 > 0. The case c0 = 0 is similar and simpler. By Proposition 3.8, for every v ∈ X, v 6= 0, J(u) ≤ sup J(τ v) := ϕ(v). τ >0

We have that J(τ v) ≤

τ2 2

Z

+∞

2 0 2 c (v ) − c20 |Av|2 dt − a0 τ q

−∞

Z

+∞

|Av|q dt

−∞

= τ 2 Q(v) − a0 τ q Φ(v).

(3.27)

Now we make a particular choice of v. Namely, let ε ∈ (0, 1/2] and v = vε be a function such that its Fourier transform vb is given by 1 if |ξ| ≤ 2ε vb(ξ) = 0 if |ξ| > 2ε. An elementary calculation shows that Z +∞ 2 sin 2εt 1 eitξ vb(ξ) dξ = √ . v(t) = √ 2π −∞ 2π t (In order to work in the space X, we have to consider v(t) − v(0) instead of v(t), but this does not change Q and Φ). Change of variable x = 2εt shows that Z +∞ sin(x + 1) sin x q 2q q−1 q−1 (2ε) Φ(v) = p x + 1 − x dx = a1 ε , (3.28) (2π)q −∞ where a1 > 0. Next we estimate Q(v). Integrating by parts, we obtain Z +∞ Z +∞ 1 2 0 0 2 Q(v) = c v · v dt − c0 Av · Av dt 2 −∞ −∞ Z 1 +∞ = − c2 v 00 + c20 (A∗ Av) · v dt 2 −∞ Z 1 +∞ = Lv · v dt, 2 −∞ where (A∗ v)(t) = v(t) − v(t − 1)

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is the operator formally adjoint to A and (Lv)(t) := −c2 v 00 (t) + c20 (A∗ Av)(t) = −c2 v 00 (t) + c20 v(t + 1) + v(t − 1) − 2v(t) . The operator L is a pseudodifferential operator with the symbol ξ σL (ξ) = c2 ξ 2 − 4c20 sin2 . 2 This follows from easy formulas σA (ξ) = eiξ − 1, σA∗ (ξ) = 1 − e−iξ and ξ σA∗ A (ξ) = (1 − e−iξ )(eiξ − 1) = −4 sin2 . 2 We have that Z Z 1 +∞ 1 +∞ 2 2 2 ξ Lv · v dt = c ξ − 4c0 sin vb2 (ξ) dξ Q(v) = 2 −∞ 2 −∞ 2 Z ξ 1 2ε c2 ξ 2 − 4c20 sin2 dξ. = 2 −2ε 2 (The last formula can be also obtained by means of a straightforward calculation). For |x| ≤ ε 2 x2 x4 2 sin x ≥ |x| − = x2 − |x|3 + ≥ x2 1 − |x| ≥ (1 − ε)x2 . 2 4 Therefore, if |ξ| ≤ 2ε, then c2 ξ 2 − 4c20 sin2

ξ ≤ c2 ξ 2 − c20 (1 − ε)ξ 2 = [c2 − c20 + εc20 ]2 ξ 2 . 2

Choosing ε=

c2 − c20 , c20

(3.29)

8 2 (c − c20 )ε3 = a2 (c2 − c20 )4 , 3

(3.30)

we obtain that Q(v) ≤ (c2 − c20 )

Z

2ε

ξ 2 dξ =

−2ε

where a2 > 0 is independent of c.

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117

Equations (3.27)–(3.30) yield ϕ(v) ≤ a2 (c2 − c20 )4 τ 2 − a3 (c2 − c20 )q−1 τ q . Elementary calculus shows that the right hand part here attains its maximum value at the point

2a2 2 τ0 = (c − c20 )5−q qa3

1 q−2

5−q

= a4 (c2 − c20 ) q−2 .

Hence, ϕ(v) ≤ a2 a24 (c2 − c20 )

2(5−q) q−2 +4

= a5 (c2 − c20 )

2(q+1) q−2

.

(3.31)

Now Lemma 3.6 shows that kuc k2 ≤ a6 (c2 − c20 )

2(q+1) q−2 −1

q+4

= a6 (c2 − c20 ) q−2

and the second inequality of the proposition is proved. Proposition 3.8 shows us that Jk (uk ) ≤ αk ≤ 2ϕ(v), provided k is large enough, and the first inequality of the proposition follows from inequality (3.31) and Lemma 3.6 as above. Remark 3.10 Arguing as in proof of Proposition 3.14, with J replaced by Jk , and taking as the test function v(t) = t, considered as an element of Xk , one can obtain a similar estimate for kuk kk for all k ≥ 1, but with C dependent on k. 3.5.2

Nonglobally defined potentials

Here we consider the case when the potential is of the form U (r) =

c20 2 r + V (r), 2

where c0 ≥ 0 and V (r), the unharmonic part, is defined in a neighborhood of r = 0. We shall assume a local version, (i0loc ), of (i0 ), i. e. that V ∈ C 1 in neighborhood of 0, V (0) = V 0 (0) = 0, and V 0 (r) = o |r| as r → 0. We use also local versions of (ii+ ) and (ii− ) refereed to as (ii+ loc ) and (ii− ), respectively, i. e. in those assumptions we restrict r and r to a 0 loc neighborhood of 0.

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− Theorem 3.7 Assume (i0loc ) and (ii+ loc ) (resp., (iiloc ). Suppose, in addition, that inequality (3.26), with a0 > 0 and q > 2, takes place in a right (resp., left) neighborhood of 0. Then there exists c > c0 such that, given c ∈ (c0 , c), for every k large enough (k ≥ k0 (c)) there exists a nondecreasing (resp. nonincreasing) travelling wave uk ∈ Xk and an increasing (resp., decreasing) solitary travelling wave u ∈ X, and, up to a subsequence and shifts, uk → u in C 2 (R).

Proof . Passing to the symmetrized potential, we may assume without loss of generality that the potential is even and satisfies both (ii+ loc ) and − (iiloc ), and inequality (3.26) holds in a whole neighborhood of 0. Thus, we can treat increasing and decreasing waves simultaneously. Now let us introduce a new even potential Ve defined on whole R. Fix δ1 > δ0 > 0 such that V (r) is defined on [−δ1 , δ1 ] and let q1 = max[q, θ]. Let F (r) = drq1 . We fix d > 0 so that F (r) ≥ V (r) on [δ0 , δ1 ] (obviously, this is possible). Since θ ≥ q1 , we have that θF (r) ≤ q1 F (r) = F 0 (r)r. Choose a smooth function η(r) such that 1 for 0 ≤ r ≤ δ0 η(r) = 0 for r ≥ δ1 and η 0 (r) ≤ 0. Define the potential Ve by Ve (r) = η(r) V (r) + 1 − η(r) F (r) (we mean that η(r) V (r) = 0 if r ≥ δ1 ) and Ve (−r) = Ve (r). We have that Ve 0 (r) r = η(r) V 0 (r) r + 1 − η(r) F 0 (r) r + η 0 (r) V (r) − F (r) r, for r ≥ 0. Since F (r) ≥ V (r) on [δ0 , δ1 ], we obtain Ve 0 (r) r ≥ η(r) V 0 (r) r + 1 − η(r) F 0 (r) r ≥ η(r) θ V (r) + 1 − η(r) θF (r) = θ Ve (r). It is also easy to see that Ve satisfies (3.26), with another constant a0 > 0.

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Applying Theorems 3.1 and 3.6, we obtain travelling waves uk and u with desired convergence property for the system with the potential Ve . By Proposition 3.14, if c is close enough to c0 (and k is large enough), then kuk kk ≤ δ1 ,

kuk ≤ δ1 .

Hence kAuk kL∞ (−k,k) ≤ δ1 ,

kAukL∞ (R) ≤ δ1 .

Therefore, Ve (Auk ) = V (Auk ) and Ve (Au) = V (Au), i.e. uk and u are solutions of the original system. − Now we show that assumptions (ii+ loc ) and (iiloc ) are generic.

Proposition 3.15 Suppose that V is C n , n > 2, in a neighborhood of 0 and V (0) = V 0 (0) = . . . = V (n−1) (0) = 0, V (n) (0) = a0 6= 0. (a) If n is odd and a0 > 0 (resp., a0 < 0), then V satisfies (ii+ loc ) (resp., (ii− )), with θ < n sufficiently close to n. loc − (b) If n is even and a0 > 0, then V satisfies both (ii+ loc ) and (iiloc ), with θ < n sufficiently close to n. Proof . We consider case (b) only (case (a) is similar). By Taylor’s formula, we have near r = 0 a0 n r + rn ϕ(r), V (r) = n! a0 V 0 (r) = rn−1 + rn−1 ψ(r), (n − 1)! where ϕ(r) = o(1) and ψ(r) = o(1). Then, in a neighborhood of 0, we have θa0 a0 θ V (r) = rn + θϕ(r) ≤ rn (θ + ε) . n! n! Similarly, V 0 (r) r = rn

a0 a0 (1 − ε) + ψ(r) ≥ rn (n − 1)! (n − 1)!

near the origin. Taking ε > 0 such that θ + ε = n(1 − ε), i. e. ε=

n−θ , n+1

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we obtain the result. Example 3.5 Example 1.5)

In the case of generalized Lennard-Jones potential (see 2 U (r) = a (d + r)−k − d−k ,

where a > 0, d > 0 and k is a positive integer, a straightforward calculation shows that c20 = U 00 (0) = 2ad−2(k+1) k 2 and V (0) = V 0 (0) = V 00 (0) = 0, V 000 (0) < 0. Proposition 3.15 and Theorem 3.7 imply now the existence of decreasing solitary waves for every speed c > c0 sufficiently close to c0 . Also there exist corresponding periodic waves of large periods.

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Chapter 4

Travelling Waves: Further Results

4.1

Solitary waves and constrained minimization

4.1.1

Statement of problem

Here we present another approach to solitary waves borrowed from [Friesecke and Wattis (1994)]. Consider the following problem. (F W ) Minimize the average kinetic energy Z 1 +∞ 0 2 u (t) dt T(u) := 2 −∞ subject to the constraint that the average potential energy is fixed Z +∞ U(u) := U u(t + 1) − u(t) dt = K, −∞

where K > 0 is a given number . Problem (F W ) is set up on the space X. The travelling wave equation (3.3) is easily recognized as the EulerLagrange equation of problem (F W ), with the inverse square, c−2 , of the wave speed corresponding to the Lagrange multiplier. Now let us summarize the main result of this section. Theorem 4.1 Let U ∈ C 2 (R), U ≥ 0 in a neighborhood of zero, U (0) = 0 and U superquadratic on at least one side, i. e. r−2 U (r) increases strictly with |r| for all r ∈ Λ, where either Λ = (−∞, 0) or Λ = (0, ∞). Then there exists K0 ≥ 0 such that for every K > K0 the system possesses a nontrivial solitary travelling wave u = uK ∈ X, with U(uK ) = Kand some speed c = cK > 0. The waves uK have the following properties: 121

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(p1) They are monotone functions, increasing if Λ = (0, ∞) and decreasing if Λ = (−∞, 0). (p2) uK ∈ C 2 (R), and uK (t + 1) − uK (t) → 0 and u0K (t) → 0 as t → ±∞ (localization). (p3) They are supersonic, i. e. c2K > c20 = U 00 (0). (p4) They are solutions of problem (F W ), with U replaced by the symmetrized energy +∞

Z e U(u) :=

h

i e u(t + 1) − U e u(t) dt, U

−∞

where e (r) = U

U (r), U (−r),

r∈Λ r∈ / Λ.

If in addition U satisfies either U 00 (0) = 0

(4.1)

or the following nondegeneracy condition at zero: U (r) =

1 00 U (0)r2 + ε|r|p + o |r|p , 2

r ∈ Λ, r → 0,

(4.2)

for some ε > 0 and 2 < p < 6, then K0 can be taken to be zero. Property (p2) follows immediately from Proposition 3.5, since assumption (i000 ) is obviously satisfied. From the results of Section 4.5 it follows that travelling waves obtained in Theorem 4.1 are exponentially localized. The proof of the main part of the theorem is given in Subsections 4.1.2– 4.1.4. Remark 4.1 In addition to the assumptions of Theorem 4.1, suppose that U ∈ C 3 near 0 and 000

U (0)

>0 0 for r ∈ (−δ, δ) \ {0} for some δ > 0. For every K ≥ 0 the set AK is nonempty and, hence, the minimum value TK is well-defined. Moreover, TK is a monotone nondecreasing and continuous function of K ∈ [0, ∞), and TK > 0 for all K > 0. Proof . Pick λ0 ∈ (0, δ). Then U (λ0 ) > 0. Let us introduce the the function  t≤0  0, vλ,l (t) := λt, 0 ≤ t ≤ l  λl, t ≥ l, with λ ≥ 0 and l ≥ 1. Obviously vλ,l ∈ X and Z 1 U(vλ,l ) = (l − 1)U (λ) + 2 U (λt) dt. 0

In particular, we have that U(vλ0 ,l ) → ∞ as l → ∞. The functional U is continuous, while vλ,l ∈ X depends continuously on λ and l. Therefore, the continuous function U(vλ,1 ) of λ ∈ [0, λ0 ] attains each intermediate value in the segment 0, U(vλ0 ,1 ) , while the continuous function U(vλ0 ,l ) of l ≥ 1 attains each intermediate value in U(vλ0 ,1 ), ∞ . This proves the first statement of the lemma. Now let us show that TK is nondecreasing with K. Let α ≤ K and v ∈ AK . Since 0 if λ = 0 U(λv) = K if λ = 1, there exists λ0 ∈ [0, 1] such that U(λ0 v) = α. Hence Tα ≤ T(λ0 v) = λ20 T(v) ≤ T(v).

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Since v is an arbitrary element of AK , we obtain that Tα ≤ TK , i. e. TK increases with K. Next, we prove that TK depends continuously on K. In view of monotonicity of TK , it suffice to show that there exist η(δ) → 0 as δ → 0 such that TK+δ − TK ≤ η(δ) for all K ≥ 0 and δ > 0. Fix K and δ. Given ε > 0, pick vK ∈ AK such that TK (vK ) ≤ TK + ε, and consider the function vK (t), v(t) := vK (t) + vλ,1 (t − a − 1),

t≤a t ≥ a,

with vλ,1 defined before, and λ and a > 0 to be specified later. Then v ∈ X and Z a−1 Z a U(v) = U (AvK ) dt + U (Av) dt + U(vλ,1 ). −∞

a−1

Here we use the notation (Av)(t) = v(t + 1) − v(t) introduced in Section 3.2. Clearly the first term tends to K and the second term to zero as a → ∞. Thus Z 1 U(v) → K + U(vλ,1 ) = K + 2 U (λt) dt 0

as a → ∞. Assuming that 0 0 near zero. This proves the continuity of TK . It remains to show that TK > 0 if K > 0. Let v ∈ X and kvk ≤ C. By Lemma 3.5, |Av(t)| ≤ C. Choose C1 > 0 such that |U 00 (r)| ≤ C1 on [−C, C]. Since U 0 (0) = 0, we have that Z +∞ Z +∞ U (Av) dt ≤ C1 |Av|2 dt ≤ C1 kvk2 . (4.3) −∞

−∞

Choose v ∈ AK such that T(v) ≤ TK + ε, with 0 < ε < 1. Since kvk2 = 2T(v), taking C = 2(TK + 1), we obtain from (4.3) that K ≤ 2C1 T(v). This implies that TK > 0 and we conclude.

To address the question of existence of minimizers we use the concentration compactness principle of P.-L. Lions [Lions (1984)] in the following form (cf. Lemma B.1). Lemma 4.2 Under the assumptions of Lemma 4.1, let un be a sequence 1 in Hloc (R) such that (un )0 is bounded in L2 (R) and Z +∞ U n dt = K, −∞

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where K > 0 is fixed and U n (t) = U Aun (t) . Then there exists a subsequence, still denoted U n , satisfying one of the following three possibilities: (a) (compactness) There exists yn ∈ R such that U n (· + yn ) is tight, i. e. for every ε > 0 there exists R > 0 such that ! Z Z yn −R

+∞

U n dt ≤ ε.

+ −∞

yn +R

(b) (vanishing) For every R > 0 y+R

Z

U n dt = 0.

lim sup

n→∞ y∈R

y−R

1 (c) (dichotomy) There exists α ∈ (0, K) and un1 , un2 ∈ Hloc (R), with the 2 derivatives in L (R), such that

dist supp (un1 )0 , supp (un2 )0 → ∞, Z

+∞

n 0 2

(u )

+∞

Z dt −

−∞

2 (un1 )0 dt

Z

+∞

−

−∞

2 (un2 )0 dt ≥ 0,

−∞

and for every ε > 0 there exists nε ≥ 1 with the properties that for n ≥ nε

n

U − (U1n + U2n ) 1 ≤ ε, L (R) Z

+∞

−∞

U1n dt − α ≤ ε

and Z

+∞

−∞

U2n

dt − (K − α) ≤ ε,

where Ujn (t) = U Aunj (t) , j = 1, 2. Proof . The concentration function Z

y+R

Qn (R) := sup y∈R

y−R

U n dt

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is a monotone nondecreasing function of R ≥ 0, and Qn (R) ≤ K for all R ≥0. Passing to a subsequence, we may assume that Qn (R) →Q(R) pointwise, and the function Q(R) is monotone nondecreasing and Q(R) ≤ K. Let α := lim Q(R) ∈ [0, K]. R→∞

We note that α = 0 implies (a) and α = K implies (b). This follows also from Lemma B.1. Case (c) that corresponds to α ∈ (0, K) does not follow immediately from Lemma B.1 and require more work. Suppose that α ∈ (0, K). Fix ε > 0 and choose R > 0 such that Q(R − 1) > α − ε. Then, for n large, Qn (R − 1) > α − ε and, hence, there exists yn ∈ R such that Z yn +(R−1) U n dt > α − ε. (4.4) yn −(R−1)

Furthermore, since limR→∞ Q(R) = α, we can find Rn → ∞ such that Qn (Rn + 1) < α + ε.

(4.5)

Now for 1 R1n ∈ R, (Rn − R) , 3

2 R2n ∈ R + (Rn − R), Rn 3

(R1n and R2n to be specified later), define  n n n  u (yn − R1 ), t ≤ yn − R1 , n n n u1 (t) := u (t), yn − R1 ≤ t ≤ yn + R1n ,  n u (yn + R1n ), t ≥ yn + R1n ,  n n n  u (t) − u (yn − R2 ), n u2 (t) := 0,  u(t) − un (yn + R2n ),

t ≤ yn − R2n , yn − R2n ≤ t ≤ yn + R2n , t ≥ yn + R2n .

Then it is readily verified that Z +∞ Z +∞ Z 2 2 (un )0 dt − (un1 )0 dt − −∞

−∞

Z

Z

yn +R1n

+ yn −R2n

and dist supp (un1 )0 , supp (un2 )0 → ∞.

2 (un2 )0 dt

−∞

yn −R1n

=

+∞

yn +R2n

! 2 (un ) dt ≥ 0

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It remains to verify the other statements of (c). This can be achieved by 1 an appropriate choice of R1n and R2n . First, note that for every u ∈ Hloc (R), 0 with ku kL2 (R) ≤ C, one has 2 U Au(t) ≤ C1 Au(t) ≤ C1

Z

t+1

(u0 )2 dt,

(4.6)

t

with an appropriate constant C1 > 0 depending on C. For notational convenience let us drop the superscript n. Using (4.6), we have that Z yn −R1 Z yn −R1 Z t+1 2 U1 dt ≤ C1 u01 (s) ds dt yn −R1 −1 yn −R1 −1 t ! Z yn −R1 Z t+1 2 0 = C1 u1 (s) ds dt yn −R1 −1

Z

max[t,yn −R1 ]

yn −R1 +1

2 u0 (t) dt.

≤ C1 yn −R1

Similarly, yn +R1

Z

yn +R1

Z U1 dt ≤ C1

yn +R1 −1

Z

2 u0 (t) dt,

yn +R1 −1

yn −R2

Z

yn −R2

U2 dt ≤ C1 yn −R2 −1

2 u0 (t) dt

yn −R2 −1

and Z

yn +R2

Z

yn +R2 +1

U2 dt ≤ C1 yn +R2 −1

2 u0 (t) dt.

yn +R2

Since Z

∞

2 u0 (t) dt ≤ C 2 ,

−∞

we obtain that Z

min

R1 ∈ R,R+ 13 (Rn −R)

yn −R1

Z

yn +R1

U1 dt ≤ δ(n)

+ yn −R1 −1

!

yn +R1 −1

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129

and Z min

R2 ∈ R+ 32 (Rn −R),R

yn −R2

!

yn +R2

Z

U2 dt ≤ δ(n),

+

yn −R2 −1

yn +R2 −1

where 1 →0 n − R) − 1 (R 3

δ(n) = C1 C 2 1

as n → ∞. Choosing R1 and R2 so that the above minima are attained at R1 and R2 respectively, we have that Z

+∞

yn −R1

Z |U − U1 − U2 | dt =

!

yn +R1

Z

|U − U1 | dt

+

−∞

yn −R1 −1

Z

yn +R1 −1

yn −R2

+

!

yn +R2

Z

|U − U2 | dt

+ yn −R2 −1

Z

yn +R2 −1

yn −R1 −1

+

Z

yn +R1 −1

!

+

U dt

yn −R2

Z

yn +R1

yn +Rn +1

≤ 2δ(n) +

Z

yn +R−1

U dt − yn −Rn −1

U dt yn −R+1

≤ 2δ(n) + (α + ε) − (α − ε) = 2δ(n) + 2ε. Moreover, Z

+∞

Z

yn −R1

Z

U1 dt = −∞

!

yn +R1

+

Z

yn +R1 −1

U1 dt +

yn −R1 −1

yn +R1 −1

U dt

(4.7)

yn −R1

and Z

+∞

Z

yn −R2

U2 dt =

Z

yn +R2

!

+

−∞

yn −R2 −1

Z

U2 dt yn +R2 −1

yn −R2 −1

+

Z

∞

+ −∞

! U dt.

yn +R2

(4.8)

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The first terms in both (4.7) and (4.8) do not exceed δ(n). Due to (4.4) and (4.5), the second terms lie, respectively, in the interval "Z # n Z yn +R−1

yn +R +1

U dt ⊂ (α − ε, α + ε)

U dt, yn −Rn −1

yn −R+1

and in " Z K−

yn +Rn +1

Z U dt, K −

yn −Rn −1

#

yn +R−1

U dt ⊂ (K − α) − ε, (K − α) + ε .

yn −R+1

Since δ(n) → 0 as n → 0, (c) follows, and the proof is complete.

Lemma 4.3 Under the assumptions of Lemma 4.1 let K > 0 be fixed. Then the following two statements are equivalent: 1) Dichotomy does not occur, i. e. no minimizing sequence un ∈ AK of T satisfies (c) of Lemma 4.2. 2) Subadditivity inequality TK < Tα + TK−α

for all a ∈ (0, K)

(4.9)

holds. Proof . First suppose that (4.9) holds. Let un ∈ AK be a sequence satisfying (c), with some α ∈ (0, K). Then defining αn := U(un1 ) and βn := U(un2 ), letting n → ∞ and using the continuity of TK (Lemma 4.1) we obtain TK ≥ lim inf T(un1 ) + T(un2 ) ≥ lim inf (Tαn + Tβn ) n→∞

n→∞

= Tα + TK−α that contradicts (4.9). Now assume that (4.9) does not hold, i. e. there exists α ∈ (0, K) such that TK ≥ Tα + TK−α . It suffices to construct a minimizing sequence un ∈ AK satisfying (c). Pick minimizing sequences unα ∈ Aα and unβ ∈ AK−α . Arguing similarly to the proof of the continuity of TK in Lemma 4.1, we may assume without loss that supp (unα )0 and supp (unK−α )0 are contained in some interval (−Rn , Rn ). Then un (t) := unα (t + Rn + n) − unK−α (t − Rn − n) + Cn , where Cn is chosen so that un (0) = 0, has the required property.

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Lemma 4.4 Suppose that U is as in Lemma 4.1. Let K > 0 be fixed. Then the following three statements are equivalent: 1) Vanishing does not occur, i. e. no minimizing sequence un ∈ AK satisfies (b) of Lemma 4.2. 2) There exists ε = ε(K) > 0 such that for every minimizing sequence un ∈ A K lim inf kAun kL∞ (R) > ε. n→∞

3) The following energy inequality U 00 (0) TK < K

(4.10)

holds. Proof . For ε > 0, let TK,ε := inf T(u) : u ∈ Ak , kAukL∞ (R) ≤ ε . Since the function vε,L defined in the proof of Lemma 4.1 belongs to the above set, provided ε > 0 is small enough and L = L(ε) is appropriately chosen, the TK,ε is well-defined at least for all small ε > 0. We shall show that lim TK,ε =

ε→0

K , U 00 (0)

(4.11)

where the limit is to be understood as +∞ in case U 00 (0) = 0. First we estimate this limit from bellow. Let un ∈ AK be an arbitrary sequence such that kAun kL∞ (R) → 0. Then Z K= U (Aun ) dt R

Z Z +∞ t+1 2 1 ≤ sup U 00 (r) (un )0 ds dt 2 r∈[−an ,an ] −∞ t = sup U 00 (r) T(un ), r∈[−an ,an ]

where an = kAun kL∞ (R) → 0. Therefore, K ≤ U 00 (0) lim inf T(un ) n→∞

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and, in particular, K ≤ U 00 (0) lim inf TK,ε . ε→0

To obtain an upper bound we use the function vλ,l defined in the proof of Lemma 4.1. Recall that T(vλ,l ) =

1 2 λ l 2

and Z U(vλ,l ) = (l − 1) U (λ) + 2

l

U (λt) dt. 0

Pick λ0 so small that U(vλ,1 ) ≤ K for all λ ∈ (0, λ0 ). Then for every λ ∈ (0, λ0 ) there exists l = l(λ) ≥ 1 such that U(vλ,l ) = K. Furthermore, it is easy to verify that l(λ) → ∞ as λ → 0. Now R1 (l − 1) U (λ) + 2 0 U (λt) dt (l − 1) U (λ) K ≤ . (4.12) ≤ 1 2 1 2 T(vλ,l ) 2 λ l 2 λ l Since U (λ) =

1 00 U (0) λ2 + o(λ2 ) 2

and l = l(λ) → ∞ as λ → 0, both the left-hand part and the right-hand part of (4.12) tend to U 00 (0) as λ → 0. Hence, K = U 00 (0) lim T(vλ,l ) ≥ U 00 (0) lim sup TK,ε λ→0

ε→0

and we obtain (4.11). This proves the equivalence 2) ⇔ 3). It remains to prove that 1) ⇔ 2). The implication “⇒” is trivial. To verify “⇐” it suffices to show that, given ε > 0 and C > 0, there exist ε1 = ε1 (ε, C) > 0 and R = R(ε, C) > 0 such that all u ∈ X, with kuk ≤ C and kAukL∞ (R) ≥ ε, satisfy Z

y+R

sup y∈R

U (Au) dt > ε1 . y−R

Pick y ∈ R such that U Au(y) ≥ ε. Obviously, such a y exists. By Lemma 3.5, kAukL∞ (R) ≤ C.

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Letting C0 :=

0 U (r) ,

sup r∈[−C,C]

we obtain for t ≥ y U Au(t) − U Au(y) ≤ C0 (Au)(t) − (Au)(y) Z t 0 u (s + 1) + u0 (s) ds ≤ C0 y

√ ≤ 2CC0 t − y. Hence, √ U Au(t) ≥ ε − 2CC0 t − y.

(4.13)

This implies the required, with R=

ε 2CC0

2

and Z ε1 =

R

√ (ε − 2CC0 s) ds.

0

The proof is complete. 4.1.3

The minimization problem: existence

The technicalities of the previous subsection give the possibility to obtain the existence of solution to problem (F W ). We start with the following Proposition 4.1 Let U be as in Lemma 4.1 and K > 0. Assume that the subadditivity inequality (4.9) and the energy inequality (4.10) hold. Then T attains its minimum on AK . Proof . This is a consequence of standard arguments in the calculus of variations and of Lemmas 4.2–4.4. Take any minimizing sequence un ∈ AK for the functional T. Lemmas 4.2–4.4 show that, passing to a subsequence, one can assume that un satisfies property (a) of Lemma 4.2. Since T and U are invariant under translations and under adding constants, we can replace un (t) by un (t + yn ) − un (yn ). Hence, we can assume that yn = 0.

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Obviously, un is bounded in X and, passing to a subsequence, we can assume that un converges weakly to u ∈ X. Since the norm in a Hilbert space is weakly lower semicontinuous, we have that T(u) ≤ inf T(v) : v ∈ Ak = TK . It remains to prove that U(u) = K. By property (a), with yn = 0, it is enough to show that for every R > 0 Z R Z R n lim U (Au ) dt = U (Au) dt, n→∞

−R

−R

but this follows immediately from the compactness of the Sobolev embedding H 1 (−R, R) ⊂ L∞ (−R, R) (see Theorem A.1) which implies that un → u strongly in L∞ (−R, R) for every R > 0. The next step is to study for which potentials the subadditivity inequality and the energy inequality hold. For, we shall need the following elementary lemma that goes back to [Lions (1984)]. Lemma 4.5

Let h : [0, K] → R. Suppose that K K ,1 < θ ≤ , 2 α

h(θα) ≤ θh(α),

0 K0 . Proof . Since r−2 U (r) increases strictly with |r| and tends to U 00 (0)/2 as r → 0, we have that U (r) >

1 00 U (0) r2 2

(4.14)

for all r 6= 0. Fix λ > 0 and consider the function vλ,l defined in the proof of Lemma 4.1. We have that Z 1 U(vλ,l ) = (l − 1) U (λ) + 2 U (λt) dt > (l − 1) U (λ) 0

and U 00 (0) T(vλ,l ) =

1 00 U (0) λ2 l. 2

Hence, there exists l0 ≥ 1 such that for l ≥ l0 we have, using (4.14), that (l − 1) U (λ) >

1 00 U (0) λ2 l. 2

Therefore, for all l ≥ l0 U(vλ,l ) > U 00 (0) T(vλ,l ). Let K0 = U(vλ,l0 ). Then the last inequality implies immediately that K > U 00 (0) TK for all K ≥ K0 . The energy inequality is proved. Now we claim that, with this definition of K0 , the subadditivity holds for K ≥ 2K0 . We check that for all K ≥ 2K0 the function h(α) = Tα satisfies the assumption of Lemma 4.5. Fix α ∈ (0, K) and θ ∈ (1, K/α]. We consider the case α ≥ K/2 only, the other case being easier. By Lemma 4.4 and the boundedness of minimizing sequences, there exist ε > 0 and C > 0 such that Tα = inf T(u) : u ∈ Aα,ε,C ,

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where Aα,ε,C = u ∈ X : U(u) = α, kAukL∞ (R) ≥ ε, kuk ≤ C . By (4.14), there exists α0 > 0 such that Z U Au(t) dt ≤ α0 {t : |Au(t)|≥ε/2}

for all u ∈ Aα,ε,C . Let hε i √ U (λr) θ0 := min : |r| ∈ , C , λ ∈ 1, θ . λ2 U (r) 2 By the superquadraticity of U , we have that θ0 > 1. Now take v ∈ Aα,ε,C . Since U(λv) = α at λ = 1 and √ U(λv) = U θv ≥ θ U(v) = θα √ √ at λ = θ, there exists λ = λ(θ, v) ∈ 1, θ such that U(λv) = θα. In fact, λ
0. Remark 4.3 In [Friesecke and Wattis (1994)] it is shown that if U is subquadratic, i. e. r−2 U (r) is nonincreasing with |r|, then, for every K > 0, neither energy inequality nor the subadditivity inequalities hold, and the minimum of T on AK is not attained. Now we discuss the nondegeneracy assumption (4.2). Proposition 4.3 Let U be as in Lemma 4.1 and superquadratic. In addition, let U satisfy nondegeneracy condition (4.2), with ε > 0 and p ∈ (2, 6). Then subadditivity and energy inequalities (4.9) and (4.10) hold for all K > 0. Therefore, by Proposition 4.1, the functional T attains its minimum on AK for all K > 0. Proof . As in the proof of Proposition 4.2, it suffices to prove that the energy inequality holds for all K > 0. Let 1 wλ,β = λ tanh β z − + C, 2 where C is a constant such that wλ,β (0) = 0 so that wλ,β ∈ X. First we calculate T(wλ,β ). A straightforward calculation shows that 2λ tanh β2 sech2 (βt) . ϕ(t) := Awλ,β (t) = 1 − tanh2 (βt) tanh2 β2 We expand ϕ using β 1 3 2 tanh =β− β + O(β 5 ) 2 12 and 1 = 1 + x + O(x2 ). 1−x We have 1 3 1 ϕ=λ β− β + O(β 5 ) 1 + β 2 tanh2 (βt) + O(β 4 ) sech2 (βt) 12 4 1 1 3 = λ β + β 3 tanh2 (βt) − β + O(β 5 ) sech2 (βt) 4 12

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as β → 0. Since 0 wλ,β = λβ sech2 (βt),

we obtain that 1 T(wλ,β ) = λ2 β 2

Z

+∞

sech4 (x) dx.

−∞

Now we show that, given K > 0, there exists a λ = λ(β) such that U(wλ,β ) = K. To calculate U(wλ,β ), we need first to estimate the term βλ(β). By (4.14), λ(β) does not exceed the solution λ0 (β) of the equation Z +∞ 2 1 00 U (0) Awλ0 ,β (t) dt = K. 2 −∞ The left-hand part simplifies to 1 00 U (0) λ20 β 2 + O(β 4 ) 2

Z

+∞

sech4 (βt) dt

−∞

1 = U 00 (0) λ20 β 1 + O(β 2 ) 2

Z

+∞

sech4 (x) dx.

−∞

Let r = r(β) := λ0 (β)β and C1 =

1 00 U (0) 2

Z

+∞

sech4 x ds.

−∞

We obtain the equation for r: C1 1 + O(β 2 ) r = K. Hence, r(β) =

K K = 1 + O(β 2 ) , 2 C1 C1 1 + O(β ) s

λ0 (β) =

r(β) = β

s

K + O(β 1/2 ), C1 β

and βλ(β) ≤ βλ0 (β) = O(β 1/2 ).

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139

Knowing that the order of magnitude of λ is at most β −1/2 , we can expand U(wλ,β ). For ϕ = Awλ,β , we have 1 00 U (0) ϕ2 + ε|ϕ|p + o |ϕ|p 2 1 2 00 1 4 1 4 2 2 6 = λ U (0) β + β tanh (βt) − β + O(β ) sech2 (βt) 2 2 6 + ελp β p + o(β p ) sech2 (βt) + o λp β p sech2p (βt) .

U (ϕ) =

This gives us the following expression for U(wλ,β ): Z +∞ 1 U(wλ,β ) = λ2 β U 00 (0) sech4 x dx − β 2 C2 + O(β 2 ) 2 −∞ Z +∞ p + ελp β p−1 sech2p x dx + o β − 2 −1 , −∞

where Z

+∞

C2 = −∞

1 1 tanh2 x − 2 2

sech4 x dx.

Since U(wλ,β ) = K, we obtain that s K + o(β −1/2 ). λ(β) = C1 β Finally, we have to check the inequality U 00 (0) T(wλ,β ) < U(wλ,β ).

(4.15)

We obtain that 1 2 3 00 λ β U (0) C2 + O(β 2 ) 2 Z +∞ p p p−1 + ελ β sech2p x dx + o β 2 −1 −∞ p p 2 2 = a1 β + o(β ) + a2 β 2 −1 + o β 2 −1 ,

U(wλ,β ) − U 00 (0)T(wλ,β ) =

where a1 , a2 are constants, depending on K, C1 and C2 , but independent of β and λ, and a2 > 0. On the other hand, one calculates C2 = −4/45 and, hence, a1 < 0. This implies immediately that (4.15) is valid for β small enough, provided p/2 − 1 < 2, i. e. p < 6. The proof is complete.

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4.1.4


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Proof of main result

First we show that the minimizers of problem (F W ) solve the travelling wave equation. Lemma 4.6 Let U be as in Lemma 4.1. Suppose in addition that U (r) is increasing for r > 0 and decreasing for r < 0. Let K > 0 and let u ∈ AK be a minimizer of T on AK . Then u ∈ C 2 (R) and satisfies the travelling wave equation (3.3) with some c > 0. Proof . The quadratic functional T is continuous on X, hence, of class C 2 . It is not difficult to verify that U is also a C 2 functional (cf. the proof of Proposition 3.3), and, for every h ∈ X, Z +∞ 0 hU (v), hi = U 0 Av(t) Ah(t) dt. −∞ 0

Since U (r) r > 0 for all r 6= 0, we have that hU0 (v), vi > 0 for all v ∈ X, v 6= 0. Hence, U0 (v) 6= 0. By the implicit function theorem, this shows that AK is a C 1 submanifold of X. Applying the Lagrange multiplier rule, we obtain that there exists λ ∈ R such that T0 (u) = λU0 (u), or, in more details, Z +∞ −∞

u0 (t)h0 (t) dt = λ

Z

+∞

U 0 Au(t) A(h) dt

(4.16)

−∞

for all h ∈ X. Taking h = u, we see that λ > 0. Equation (4.16) means that u is a weak solution of (3.3), with c = λ−1/2 (see Proposition 3.4). Proposition 3.4 implies also that u ∈ C 2 . The proof is complete. Now we show that, in the case of symmetric potential, the minimizers constructed in Subsection 4.1.2 are monotone functions. Lemma 4.7 Suppose that the interaction potential U satisfies the assumptions of Lemma 4.1 and, in addition, U is symmetric, i. e. U (r) = U (−r), and superquadratic. Let K > 0 and u ∈ AK be a minimizer of T on AK . Then u is a monotone function, i. e. either Au ≥ 0 or Au ≤ 0.

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Travelling Waves

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141

Proof . Consider the function t

Z u e(t) := P u(t) =

0 u (s) ds.

0

Let ϕ = Au and ϕ e = Ae u. We claim that ϕ(t) e = ϕ(t) .

(4.17)

It is clear that ϕ e ≥ |ϕ|. Suppose for contradiction that ϕ(t e 0 ) > ϕ(t0 ) at some point t0 . By continuity of ϕ and ϕ, e we have that ϕ e > |ϕ| on some neighborhood of t0 . By symmetry and strict monotonicity of U , on this neighborhood U (e u) > U (u) and, hence, U(e u) > U(u). As consequence, there exists λ ∈ (0, 1) such that U(λe u) = U(u) = K, so that λe u ∈ AK . But, by definition of u e, T(e u) = T(u), and we have that T(λe u) = λ2 T(e u) < T(e u) = Tu which contradicts the fact that u is a minimizer and proves (4.17). Now suppose that u is not monotone. By Lemma 4.6, u ∈ C 2 (R). Therefore, there exist a, b ∈ R such that u0 (a) < 0 and u0 (b) > 0. For definiteness, we assume that a < b. By (4.17), Z t

t+1

Z 0 u (s) ds =

t

t+1

u (s) ds 0

for all t ∈ R. Hence, on each interval [t, t + 1] of length one, either u0 ≥ 0 or u0 ≤ 0. This implies that there exist t1 , t2 ∈ (a, b) such that t2 − t1 ≥ 1 and u0 ≡ 0 on [t1 , t2 ]. Choose t1 and t2 so that, in addition, dist ti , t ∈ R : u0 (t) 6= 0 = 0,

i = 1, 2.

Let w(t) :=

u e(t), t ≤ t1 u e t + (t2 − t1 ) , t > t1 .

(4.18)

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Since t2 − t1 ≥ 1, we have that Z t1 U(w) − U(e u) = U u e t + 1 + (t2 − t1 ) − u e(t) dt t1 −1 Z t1

− t1 −1 t2

Z −

t2 −1

Z

t1

h

=

e(t1 ) − u e(t) dt U u U u e(t + 1) − u e(t1 ) dt

i U r1 (t) + r2 (t) − U r1 (t) − U r2 (t) dt,

t1 −1

where r1 (t) = u e(t1 ) − u e(t) ≥ 0, r2 (t) = u e t + 1 + (t2 − t1 ) − u e(t1 ) ≥ 0. By superquadraticity of U , we have that U (r1 + r2 ) ≥ U (r1 ) + U (r2 ) for all r1 , r2 ≥ 0, with equality if and only if one of the ri is zero. Now (4.18) implies the existence of t0 ∈ [t1 − 1, t1 ] such that both r1 (t0 ) > 0 and r2 (t0 ) > 0. Hence, U(w) > U(e u). Now there exists λ ∈ (0, 1) such that U(λw) = U(e u) = K and T(λw) = λ2 T(w) < T(w) = T(e u) = T(u), that contradicts the fact that u is a minimizer. The proof is complete. Proof of Theorem 4.1. Applying Propositions 4.2, 4.3, Remark 4.2 and Lemma 4.6 to the symmetrized potential, we obtain in this case all the statements of Theorem 4.1, except (p3) and (p1). The last holds in a weaker form: u is a nondecreasing (resp. nonincreasing) function. By Lemma 4.7, the minimizer u we consider is a monotone function and, therefore, either u or −u is a solution of the original problem. Property (p1), i. e. strict monotonicity, follows from Proposition 3.8. If remains to prove (p3), i. e. the fact that the wave is supersonic. Since r−2 U (r) increases with |r|, differentiating we obtain 1 0 U (r) r ≥ U (r), 2

r ∈ Λ.

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Travelling Waves

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143

Now Eq. (4.16), with h = u, c2 = λ−1 , and the energy inequality, which holds by Proposition 4.2, imply that Z +∞ 1 U(u) c2 = U 0 (Au) Au dt ≥ > U 00 (0). 2T(u) −∞ T(u) This completes the proof of the theorem. 4.1.5

Lennard-Jones type potentials

Here we discuss singular potentials, like the Lennard-Jones potential (see Examples 1.5 and 3.5) 2 U (r) = a (d + r)−6 − d−6 , r > −d, where a > 0 and d > 0. We start with the following result that concerns the case of solitary waves with small averaged potential energy. Theorem 4.2

Let U (r) = U0 (r)

if r > −d,

with some d > 0, and U (r) = +∞

if r ≤ −d.

Assume that U0 ∈ C 2 (−d, ∞), U0 ≥ 0, U0 (0) = 0, U0 is superquadratic on (−d, 0), U000 (0) 6= 0 and that (4.2) holds. Then there exists K0 ≥

1 00 U (0) d2 2

such that for every K ∈ (0, K0 ) the system admits a solitary wave u = uK ∈ X with the averaged potential energy U(u) = K, with the property −d ≤ AuK (t) ≤ 0,

t ∈ R,

and properties (p2)–(p4) of Theorem 4.1. Proof . The proof relies on a cutoff argument which makes Theorem 4.1 applicable. Fix ε ∈ (0, d/2) and choose a cutoff function η ∈ C ∞ (R) such that 0 on (−∞, −d + ε) η= 1 on (−d + 2ε, ∞)

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and η 0 ≥ 0. Let p be the exponent from assumption (4.2). Choosing C = Cε so large that Wε (r) :=

1 00 U (0) r2 + C|r|p > U (r) 2

on (−d + ε, 0), we define a new potential on the whole of R by Wε (r), r ≤ −d Uε (r) := η(r) U (r) + 1 − η(r) Wε (r), r > −d. The superquadraticity of U and W , and the fact that η 0 (U − W ) ≤ 0

on (−∞, 0)

imply immediately that

Uε (r) r2

0 < 0,

r < 0,

which means that Uε is superquadratic on (−∞, 0). Also it is clear that Uε satisfies (4.2). Hence, Theorem 4.1 applies and for all K > 0 the system with the modified potential Uε possesses a travelling wave u = uK = uε,K such that Uε (u) = K. Now we shall show that, for K
−d + 2ε,

t ∈ R.

Since Uε = U on (−d + 2ε, ∞), this completes the proof. By Lemma 3.5 and energy inequality (4.10), kϕkL∞ (R) ≤ kuk = (2Tε,K )1/2
0. Then, after passing to a subsequence, one of the three following statements holds true: (i) (concentration) there is a sequence {yk } ⊂ R such that for every ε > 0 there exists R > 0 with the property that Z yk +R ρk (x) dx ≥ λ − ε; yk −R

(ii) (vanishing) Z

yk +R

lim sup

ρk (x) dx = 0

k→∞ y∈R

yk −R

for all R > 0; (iii) (dichotomy) there exist α ∈ (0, and sequences of compactly sup λ) (1) (2) ported nonnegative functions ρk , ρk ⊂ L1 (R) such that i h (1) (2) dist supp ρk , supp ρk → ∞ as k → ∞,

(1) (2) lim ρk − ρk + ρk

L1

k→∞

(1) lim ρk L1 = α

k→∞

173

= 0,

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and

(2) lim ρk L1 = λ − α.

k→∞

Proof. See [Lions (1984); Chabrowski (1997)]. Lemma B.2

Let r > 0. If {uk } is a bounded sequence in H 1 (R) and if Z y+r uk (x) 2 dx → 0, k → ∞, sup y∈R

y−r

then uk → 0 in Lp (R) for all p > 2. Proof. See [Lions (1984)]. Now we present discrete versions of Lemmas B.1 and B.2 (see [Pankov and Zakharchenko (2001)]). Lemma B.3 Let {vk } be a sequence of nonnegative elements of l1 such that kvk kl1 → λ > 0. Then there exists a subsequence, still denoted by vk , such that one of the three following possibilities holds: (i) (concentration) there is a sequence {mk } of integers such that for every ε > 0 there exists an integer r > 0 with the property that m k +r X

vk (n) ≥ λ − ε;

n=mk −r

(ii) (vanishing) lim kvk kl∞ = 0;

k→∞

(iii) (dichotomy) there exist α ∈ (0, λ) and two sequences (2) vk ⊂ l0 such that h i (1) (2) dist supp vk , supp vk → ∞ as k → ∞,

(1) (2) lim vk − vk + vk 1 = 0,

k→∞

l

(1) lim vk l1 = α

k→∞

(1)

vk

,

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Concentration Compactness

and

(2) lim vk l1 = λ − α.

k→∞

Lemma B.4

Let {vk } be a bounded sequence in l2 such that lim kvk kl∞ = 0.

k→∞

Then vk → 0 in lp , 2 < p ≤ ∞.

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Appendix C


C.1

Differentiable functionals

Let ϕ be a real valued functional on a Banach space X. We say that ϕ has a Gateaux derivative f ∈ X 0 at u ∈ X if for every h ∈ X lim

t→∞

1 ϕ(u + th) − ϕ(u) − hf, thi = 0. t

Here, as usual hf, hi stands for the value of the linear functional f ∈ X 0 at h ∈ X, i. e. h·, ·i is the canonical bilinear form on X 0 × X. The Gateaux derivative of f at u is denoted by ϕ0 (u). The functional ϕ has a Fréchet derivative f 0 ∈ X at u ∈ X if 1 ϕ(u + h) − ϕ(u) − hf, hi = 0. h→0 khk lim

(C.1)

Any Fréchet derivative is a Gateaux derivative. Therefore, for Fréchet derivatives we use the same notation as for Gateaux derivatives. If X is a Hilbert space with the inner product (·, ·) and ϕ has a Gateaux derivative at u ∈ X, the gradient, ∇ϕ(u), is defined by

∇ϕ(u), h = ϕ0 (u), h

∀h ∈ X.

The functional ϕ is said to be C 1 if the Fréchet derivative ϕ0 exists and is continuous, i. e. the map u 7→ ϕ0 (u) from X into X 0 is continuous. Actually, if ϕ has a continuous Gateaux derivative on X, then ϕ is C 1 . We say that the functional ϕ of class C 1 has a second Gateaux derivative 177

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L ∈ L(X, X 0 ) at u ∈ X if, for every h, v ∈ X, lim

t→0

1 0 ϕ (u + th) − ϕ0 (u) − t Lh, v = 0. t

The second Gateaux derivative at u is denoted by ϕ00 (u). Here L(E, F ) denotes the Banach space of all linear bounded operators from E into F . The functional ϕ has a second Fréchet derivative L ∈ L(X, X 0 ) at u ∈ X if 1 ϕ0 (u + h) − ϕ0 (u) − Lh = 0. h→0 khk lim

Certainly, any second Fréchet derivative is a second Gateaux derivative and we apply the same notation ϕ00 (u). The functional ϕ is said to be C 2 if the second Fréchet derivative ϕ00 (u) exists and is continuous in u ∈ X. In fact, it is enough to assume that the second Gateaux derivative exists and is continuous. A point u ∈ X is called a critical point of ϕ if ϕ has the Gateaux derivative ϕ0 (u) and ϕ0 (u) = 0. The value of the functional ϕ at a critical point is called a critical value of ϕ.

C.2

Mountain pass theorem

Let ϕ be a C 1 functional on a Banach space X. We say that ϕ satisfies the Palais-Smale condition (for shortness, condition (P S)) if any sequence {un } in X such that the sequence {ϕ(un )} is bounded and ϕ0 (un ) → 0 contains a convergent subsequence. The simplest mountain pass theorem reads (see [Rabinowitz (1986); Willem (1996)]) Theorem C.1 Let ϕ be a C 1 functional on a Banach space X. Assume that there exist e ∈ X and r > 0 such that kek > r, β := inf ϕ(u) > ϕ(0) ≥ ϕ(e), kuk=r

(C.2)

and ϕ satisfies the Palais-Smale condition. Let b := inf max ϕ(γ(t)) γ∈Γ t∈[0,1]

(C.3)

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where n o Γ := γ ∈ C [0, 1], X : γ(0) = 0, γ(1) = e .

(C.4)

Then b is a critical value of ϕ and b ≥ β. The following theorem of mountain pass type can be found in [Berestycki et. al (1995)]. Theorem C.2 Under the assumptions of Theorem C.1 let P : X → X be a continuous mapping such that ϕ(P u) ≤ ϕ(u)

for all u ∈ X,

P (0) = 0 and P (e) = e. Then there exists a critical point u ∈ P X (the closure of P X) of ϕ with the critical value b. Remark C.1 Typically, in applications the functional ϕ has a local minimum at 0, a trivial critical point. In this case Theorem C.1 gives the existence of a nontrivial critical point. There is also a version of the mountain pass theorem without condition (P S). Theorem C.3 Suppose that all the assumptions of Theorem C.1 except condition (P S) are satisfied. Let b be defined by (C.3), (C.4). Then there exists a sequence {un } ⊂ X such that lim ϕ(un ) = b

n→∞

and lim ϕ0 (un ) = 0,

n→∞

i. e. {un } is a Palais-Smale sequence at the level b. We say that the functional ϕ possesses the mountain pass geometry if ϕ satisfies the assumptions of Theorem C.1 except (P S). The number defined by Eq. (C.4) is called the mountain pass value (level).

C.3

Linking theorems

Let H be a Hilbert space decomposed into the direct orthogonal sum H = Y ⊕ Z. Let % > r > 0 and let z ∈ Z be such that kzk = r. Define M := u = y + λz : y ∈ Y, kuk ≤ %, λ ≥ 0

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and M0 := u = y +λz : y ∈ Y, kuk = % and λ ≥ 0, or kuk ≤ % and λ = 0 , i. e. M0 is the boundary ∂M of M . Let N := u ∈ Z : kuk = r . Consider a functional ϕ on H and suppose that β := inf ϕ(u) > α := sup ϕ(u). u∈N

(C.5)

u∈M0

In this situation we say that the functional ϕ possesses the linking geometry. Theorem C.4 Suppose that the functional ϕ of class C 1 satisfies the Palais-Smale condition and possesses the linking geometry. Let n o b := inf sup ϕ γ(u) , (C.6) γ∈Γ u∈M

where Γ := γ ∈ C(M ; H) : γ = id on M0 .2

(C.7)

If dim Y < ∞, then b is a critical value of ϕ and β ≤ b ≤ sup ϕ(u). u∈M

Remark C.2 The additional assumption dim Y < ∞ in Theorem C.4 implies that the suprema in (C.5) and (C.6) are attained and, hence, can be replaced by maxima. Under further restrictions on the functional ϕ one can drop the assumption dim Y < ∞ in the previous theorem. Theorem C.5 Suppose that the functional ϕ possesses the linking geometry and satisfies the following assumptions: 1 (i) ϕ(u) = (Au, u)+b(u), where u = u1 +u2 ∈ Y ⊕Z, Au = A1 u1 +A2 u2 , 2 A1 : Y → Y and A2 : Z → Z are linear bounded self-adjoint operators; (ii) the functional ϕ is weakly continuous and uniformly differentiable on bounded sets, i. e. the limit in (C.1) is uniform in u ∈ B for every bounded set B ⊂ H; 2 As

usual, id denotes the identity map.

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(iii) if {un } ∈ H is a sequence such that ϕ(un ) is bounded above and ϕ0 (un ) → 0, then it is bounded. Then b defined by (C.6) and (C.7) is a critical value and β ≤ b ≤ sup ϕ(u). u∈M

For the proof see [Benci and Rabinowitz (1979)]. Remark C.3 There is an extension of Theorem C.5 in the spirit of Theorem C.3 (see [Kryszewski and Szulkin (1998)] and [Willem (1996)]; a simplified proof is contained in [Bartsch and Ding (1999)]).

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Appendix D

Difference Calculus

Let l be the vector space of all real, or complex, two-sided sequences, i. e. functions on Z. Difference operations are linear maps from l into itself. The simplest are left and right shifts defined by S − u (n) := u(n − 1) (D.1) and S + u (n) := u(n + 1),

(D.2)

respectively. Operations of left and right differences are defined by ∂ − u (n) := u(n) − u(n − 1)

(D.3)

and ∂ + u (n) := u(n + 1) − u(n),

(D.4)

respectively. In other words, ∂+ = I − S− and ∂ − = S + − I. Here I stands for the identity operation. The following identity is useful h i ∂ − a∂ + u = ∂ + S − a ∂ − u , 183

(D.5)

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where a ∈ l. We mention also the product rules ∂ + (f g)(n) = f (n) ∂ + g (n) + g(n + 1) ∂ + f (n), ∂ − (f g)(n) = f (n) ∂ − g (n) + g(n − 1) ∂ − f (n). We do not consider general difference operations here. Instead, we restrict ourself to the case of divergence form difference operations of second order. These are operations of the form Ru = ∂ − a∂ + u + bu (D.6) where a ∈ l and b ∈ l are given sequences (coefficients). Alternatively, h i Ru = ∂ + S − a ∂ − u + bu, (D.7) as it follows from Eq. (D.5). Abel’s summation by parts formula reads n X

g(j) ∂ + f (j) = g(n) f (n + 1) − g(m − 1) f (m)

j=m

−

n X

∂ − g (j) f (j).

(D.8)

j=m

It implies that the operations ∂ + and −∂ − are formally adjoint, i. e. X X g(j) ∂ + f (j) = − ∂ − g (j) f (j) j

j

for all f ∈ l and g ∈ l0 (see Appendix A.1 for the definition of l0 ). As consequence, if the coefficient sequences in Eq. (D.6) are real, then the operation R is formally self-adjoint, i. e. X X g(j) Rf (j) = Rg (j) f (j) j

j

for all f ∈ l and g ∈ l0 . The operation ∆ := ∂ + ∂ − = ∂ − ∂ + is called (one dimensional) discrete Laplacian. References: [Atkinson (1964); Gel’fond (1971); Samarskii (2001); Teschl (2000)].

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Index

Abel’s summation by parts formula, 186 average kinetic energy, 123 average potential energy, 123

expansion wave, 80 Floquet condition, 16 Floquet transform, 16 FPU α-model, 7 FPU β-model, 7 Fréchet derivative, 179 Frenkel-Kontorova model, 22 functional of class C 1 , 179 functional of class C 2 , 180

band function, 17 Bloch condition, 16 Bloch eigenfunction, 17 Bloch wave, 17 chain of coupled nonlinear oscillators, 18 compression wave, 80 critical point, 180 critical value, 180

gap breather, 76 gap solution, 76 Gateaux derivative, 179 Gelfand transform, 16 geometrically distinct solutions, 63 gradient, 179

diatomic lattice, 8 difference operation, 185 difference operation in divergence form, 186 discrete ϕ4 -equation, 23 discrete breather, 29 discrete Laplacian, 7, 186 discrete nonlinear Klein-Gordon equation, 22 discrete nonlinear wave equation, 22 discrete wave equation, 7 dispersion relation, 17 DNKG equation, 22 DNW equation, 22

harmonic lattice, 12 homoclinic, 74 lattice with impurities, 8 left difference, 185 left shift, 185 Lennard-Jones potential, 8 linking geometry, 182 linking theorems, 181 monoatomic lattice, 8 mountain pass geometry, 181 mountain pass theorem, 180 mountain pass value, 181

energy inequality, 133 193

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multiatomic lattice, 8 multibump solution, 62 Nehari functional, 108 Nehari manifold, 108 Palais-Smale condition, 180 periodic ground wave, 106 periodic travelling wave, 78 plane wave, 18 profile, 77 profile function, 77 profile function for relative displacements, 78 quasimomentum, 16 relative displacement, 5 right difference, 185 right shift, 185 second Fréchet derivative, 180 second Gateaux derivative, 179 Sobolev embedding theorem, 173 Sobolev spaces, 172 solitary ground wave, 111 solitary travelling wave, 78 spectral band, 17 spectral gap, 17 speed of sound, 80 speed of wave, 77 subadditivity inequality, 132 subsonic periodic travelling wave, 88 Toda lattice, 8 travelling transition layer, 167 travelling wave, 77 weak derivative, 172

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Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices

Travelling waves and periodic oscillations in Fermi-Pasta-Ulam lattices

Travelling Waves and Periodic Oscillatio

Almost periodic oscillations and waves

Almost Periodic Oscillations and Waves

Magnetization Oscillations and Waves

Magnetization Oscillations and Waves

Waves and Oscillations, Second Edition

Almost periodic oscilation and waves

Quasi-periodic oscillations for wave equations under periodic forcing

Travelling Light

Waves and Oscillations: A Prelude to Quantum Mechanics

Waves and Oscillations: A Prelude to Quantum Mechanics

Waves and Oscillations: A Prelude to Quantum Mechanics

Waves and oscillations: A prelude to quantum mechanics

Oscillations in Ionized Gases

Oscillations in Neural Systems

Oscillations in Chemical Reactions

Nonlinear oscillations

Lattices and ordered sets

Lattices and Ordered Sets

Lattices and ordered sets

Continuous lattices and domains

Quarks, gluons and lattices

Continuous Lattices and Domains

Lattices and Ordered Sets

Neutrino Oscillations

Nonlinear Oscillations

Travelling players in Shakespeare's England

Nonlinear Oscillations in Mechanical Engineering

Oscillations in Planar Dynamic Systems

Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices