Proceedings of the International Conference
SPT 2001
Symmetry and Perturbation Theor Dario Bambusi Giuseppe Gaeta Mariano Cadoni
World Scientific
Proceedings of the International Conference
SPT 2001 Symmetry and Perturbation Theory
Proceedings of the International Conference
SPT 2001 Symmetry and Perturbation Theory Cala Gonone, Sardinia, Italy
6 - 1 3 May 2001
Edited by
Dario Bambusi Universita di Milano, Italy
Giuseppe Gaeta Universita di Milano, Italy University di Roma, Italy
Mariano Cadoni Universita di Cagliari, Italy
V f e World Scientific w l
New Jersey'London'Singapore* NewJersey London'Singapore* Hong Kong
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Preface The third conference on Symmetry and Perturbation Theory (SPT2001) took place in Cala Gonone, a small village on the beautiful eastern coast of Sardinia, on 6-13 May 2001. This followed the conferences of the same title held in Torino 1 in december 1996 and in Roma 2 in december 1998. The conference was attended by over 50 mathematicians, physicists and chemists, and was a nice occasion to have interdisciplinary discussion involving rather different communities; we hope that the reader of these proceedings will find within this volume some remnant of the relaxed and fruitful atmosphere we enjoyed in Cala Gonone, and we trust he/she will find plenty of useful information on the advancement of research in this field, or better said in the different fields at whose crossroads symmetry and perturbation theory sit. In order to respect the interdisciplinary character of the conference, we avoided to separate the papers into specialized sessions, and just collected them in alphabetical order (by author's name). We also give, together with the conference program and the list of participants, the list of papers appeared in the proceedings of previous SPT conferences. In the course of the conference we had a special session devoted to Louis Michel - who died on 30 December 1999 - and his influence on the subject of the conference, organized by his collaborator and friend Boris Zhilinskii. This session has seen, after a speech by Boris on Louis' life and work, the talks of Yuri Gufan, James Montaldi, Dimitrii Sadovskii, and Joshua Zak. On the one hand, it would have been natural to put these talks in a special section of these proceedings; but on the other hand, a cursory look at the table of contents will show to anybody slightly familiar with the work of Louis that it would be very reductive to confine his influence to this special session. The words written by Boris on "Symmetry, Perturbation Theory, and Louis Michel" suitably close this volume stressing the influence of Louis in the field.
Acknowledgements We would like to stress that we asked our authors a serious effort to have the proceedings ready within less than three months from the conference; we would like to thank them again here for having responded positively to this requirement. There are also, well sure, a number of individuals and institutions whose help was crucial for the success of the conference. We would like first of all to thank all those being part of the Scientific Committee of SPT2001 for their constant advice and help. This was made of: Dario Bambusi (Milano), Pascal Chossat (Nice), Giampaolo Cicogna (Pisa), Antonio Degasperis (Roma), Giuseppe Gaeta (Roma and Milano), Jeroen Lamb (London), Giuseppe Marmo (Napoli), Mark Roberts (Warwick and Surrey), Gianfranco Sartori (Padova), Ferdinand Verhulst (Utrecht), Sebastian Walcher (Munich), and Boris Zhilinskii (Dunquerque). A conference gathering different communities is stimulating, but presents a problem of different backgrounds; to overcome this we asked to a number of people to write "tutorial papers" on some selected topic (these are being published elsewhere 3 ). We would like to warmly thank them, and even more those who were in the end unable to attend the conference, for their help. The "Pro-Loco" of the city of Dorgali (in whose territory Cala Gonone lies) was very helpful whenever we had some problems, and when we had no problem as well; we would like to thank the people working there for their most friendly and smiling help. Last but definitely not least, we received financial help which made possible the conference and the publication of these proceedings; this was provided by the Dipartimento di Matematica dell'Universita di Milano and by the Universita di Cagliari; to these Institutions go our warmest thanks. Dario Bambusi, Giuseppe Gaeta, Mariano Cadoni Milano, Roma and Cagliari, July 2001 References 1. D. Bambusi and G. Gaeta eds., "Symmetry and Perturbation Theory", Quaderni GNPM-CNR, Firenze 1997 2. A. Degasperis and G. Gaeta eds., "Symmetry and Perturbation Theory - SPT98", World Scientific, Singapore 1999 3. Special issue of Acta Applicandae Mathematicae, to appear
VII
CONTENTS Preface
v
Acknowledgements
vii
Geometry and Dynamics of Hyperelliptically Separable Systems S. Abenda Multiple Hopf Bifurcation in Problems with 0(2) Symmetry: Kuramoto-Sivashinski Equation F. Amdjadi
1
9
Sternberg-Chen Theorem for Equivariant Hamiltonian Vector Fields . . . . 19 G. R. Belitskii and A. Ya. Kopanskii0, A Functional Analysis Approach to Arnold Diffusion M. Berti
29
The Symplectic Evans Matrix and Solitary Wave Instability T. Bridges and G. Perks
32
Classical Symmetries for a Boussinesq Equation with Nonlinear Dispersion M. S. Bruzon, M. L. Gandarias and J. Ramirez
38
Pseudo-Normal Forms and their Applications A. Delshams and J. Tomds Ldzaro
46
Periodic Orbits of Langmuir's Atom F. Diacu and E. Perez-Chavela
51
Heteroclinic Cycles and Wreath Product Symmetries A. P. S. Dias, B. Dionne and I. Stewart
53
Linearizing Resonant Normal Forms G. Gaeta
58
Symmetry Analysis and Reduction of the Schwarz-Korteweg-De Vries Equation in (2 + 1) Dimensions M. L. Gandarias, M. S. Bruzon and J. Ramirez
66
"For multi-author papers or abstracts, the underlined name corresponds to the author presenting the communication at SPT2001 IX
X
Tori Breakdown in Coupled Map Lattices C. Giberti
76
Evolution of the Universe in Two Higgs-Doublets Standard Models Yu. M. Gufan, O. D. Lalakulich, G. M. Vereshkov and G. Sartori
78
Possible Ground States of D-Wave Condensates in Isotropic Space through Geometric Invariant Theory Yu. M. Gufan, A. V. Popov, G. Sartori, V. Talamini, G. Valente and E. B. Vinberg Parent Phase as a Zero Approximation in Phase Transition Theory Yu. M. Gufan, I. A. Sergienko and M. B. Stryukov Symmetry and Reduction of the 2 + 1 Dimensional Variable Coefficient Burgers Equation F. Gungor
92
106
113
A Two-Dimensional Version of the Camassa-Holm Equation H.-P. Kruse. J. Scheurle and W. Du
120
C°° Symmetries and Equations with Symmetry Algebra SC(2, R) C. Muriel and J. L. Romero
128
Generalizations of Gordon's Theorem N. Nekhoroshev
137
Moving Frames: A Brief Survey P. J. Olver
143
Critical Point Theory and Hamiltonian Dynamics around Critical Elements J. -P. Ortega and T. S. Ratiu
151
Computing Invariant Manifolds of Perturbed Dynamical Systems J. Palacidn and P. Yanguas
159
Periodic Solutions for Resonant Nonlinear PDEs S. Paleari
167
A Symmetric Normal Form for the Fermi Pasta Ulam Chain B. Rink
175
xi
One Dimensional Infinite Symmetries, Boundary Conditions, and Locol Conservation Laws V. Rosenhaus
183
Normal Forms, Geometry, and Dynamics of Atomic and Molecular Systems with Symmetry D. Sadovskii
191
Higher Order Resonance in Two Degrees of Freedom Hamiltonian System J. M. Tuwankotta and F. Verhulst
206
Stability of Hamiltonian Relative Equilibria by Energy Methods C. Wulff, G. Patrick and M. Roberts
214
Topologically Unavoidable Degeneracies in Band Structure of Solids . . . . 222 J. Zak Symmetry, Perturbation Theory, and Louis Michel
231
B. Zhilinskii Conference Program
235
List of Participants
239
List of Tutorial Papers
243
SPT98
245
SPT96
247
G E O M E T R Y A N D D Y N A M I C S OF HYPERELLIPTICALLY SEPARABLE S Y S T E M S SIMONETTA ABENDA Dipartimento di Matematica e C.I.R.A.M., Via Saragozza 8, 40123 Bologna BO , ITALY E-mail:
[email protected] In this paper we focus on the Jacobi-Mumford system and its generalizations.
Many classical integrable systems (like the Euler, Lagrange and Kowalewski tops or the Neumann system) as well as finite dimensional reductions of many integrable PDEs share the property of being algebraically completely integrable systems 4 . This means that they are completely integrable Hamiltonian systems in the usual sense and, moreover, their complexified invariant tori are open subsets of complex Abelian tori on which the complexified flow is linear. To such systems the powerful algebro-geometrical techniques may be applied. However, the requirement that complexified invariant tori are complex Abelian tori is extremely restrictive and does not include most of ArnoldLiouville integrable systems with algebraic first integrals, the simplest example being the geodesic flow on a triaxial ellipsoid in its natural coordinates 3 as well as certain reductions of integrable PDEs 6 , 5 . The geodesic flow on the triaxial ellipsoid and finite dimensional reduction of the Harry-Dym hierarchy are typical examples of hyperelliptically separable systems with deficiency1,2, that is real completely integrable Hamiltonian systems whose generic complexified invariant manifolds are open susbsets of n-dimensional strata of (generalized) hyperelliptic Jacobians (or their coverings). Moreover, we require the existence of coordinates on the (generalized) Jacobian of which n evolve linearly in time and are locally a maximal system of independent coordinates on the stratum. Deficiency is the difference between the dimension of the (generalized) hyperelliptic Jacobian and the dimension of the stratum. In particular, an integrable system is both hyperelliptically separable and algebraically completely integrable if and only if its deficiency is zero. We now present some geometrical and dynamical properties of hyperelliptically separable systems starting with the classical Neumann system (see for instance Moser 10 and references therein) of a point mass on the iV-dimensional unit sphere SN = {q = (gi,...,gjv+i) G B.N+1 : q\-{ t-gjv+i — !}> subject 1
to the quadratic potential U^ = ]£f=i cnqf, where 01 < • • • < OAT+I- The system may be put in Hamiltonian form H(p, q) = \{p\-\ hp;v+i)+W^(q), where p = ( p i , . . . ,Pn+i) is the conjugate vector momentum to q (and we use the canonical Poisson structure). The Neumann system is a completely integrable system in the sense of Arnold-Liouville 7 , that is possesses a sufficient number of indepedent first integrals in involution, which we denote co(p, q) = H(p, q ) , . . . ,cjv-i(p, q), and whose expressions may be obtained from (1) and (4) below. Let /
N+l
E
N+l
QiPi
1=1
L(X) = *(A)
N+1
„2
Ei
-i-T- / A(A) =
\ (1)
N+l
-E t=l
QiPi
-a,i
}
0, JV+l
(2)
A + £(P?-««?). o t=i
with N+l
$(A)=n(A-«i).
(3)
t=l
Then the Neumann system may be put in Lax form ^ L ( A ) = [L(A),i(A)]. Moreover, upon fixing the constants of motion CQ, ... ,c/v, the characteristic equation det(L(A) - fil) = - * ( A ) (co + ci A + • • • + CN^X"'1
- XN) -fi?
= Q, (4)
with J = diag(l, 1), defines a genus N hyperelliptic curve T (for definitions and properties, see Siegel 13 ). An alternative description is the following one. Let us introduce the spheroconic change of coordinates (flj - A i ) - - - ( a , - A n )
Then the Hamiltonian takes the Staeckel form7 1
N
JV
(5)
3
with (ik conjugate momentum to A*, and, upon fixing constants of motion, the equations of motion take the form of Abel-Jacobi differential equations
where R{X) = -#(A)(co + • • • + c^-rX"-1 - XN) and /i 2 = R(X), is again the affine part of the hyperelliptic curve T found in (4). It is easy to check that the N differentials appearing in the left hand side of (6) form a basis of the holomorphic differentials associated to the hyperelliptic curve T (for definitions and properties see Siegel 13 ). Moreover, coordinates (Ai,/ii),... , (AAT,/XJV) are points on the curve T and the complete image of the iV-symmetric product of T, T ^ , through the Abel-Jacobi map
** = E /
V7Wm>
k = l,...,N,
(7)
with (A0, Ho) fixed basepoint, is the Jacobi variety of T, Jac(r). Then comparing (6) and (7), we conclude that the closure of the generic complexified invariant manifold is the complex Abelian torus Jac(r) and that the flow evolves linearly in time on such complex torus, since d<j>i = • • • = d^jv-i = 0, dN = dt. Following Adler and VanMoerbeke 4 , we call the Neumann system algebraically completely integrable or, following Abenda and Fedorov 1 , hyperelliptically separable with deficiency zero. The above construction can be repeated for any hyperelliptically separable system with zero deficiency, as originally shown by Mumford 11 in the odd case (the terms odd and even mean that s is respectively odd or even in fi2 = n*=i(^ ~ e ()i )• Since the Neumann system is "odd", we just briefly recall the Jacobi-Mumford construction in this case. Mumford found expressions of coordinates and translationally invariant vector fields on the ZN+1-dimensional bundle T over the 22V+ 1-dimensional base of odd hyperelliptic curves of genus N, T, parametrized by the coefficients of their affine part, 2AT+1
T :
M2 =
R(X) = J ] (A - e,),
(8)
whose fibers are open subsets of the Jacobi variety Jac(r). Indeed, let U(X) =XN + U1XN~1+--- + UN, V(X) =V1XN-' + --- + VN, W(X) - XN+1 + W0XN + ... + WK.
(9)
4
Then the morphism, n : C 3 j v + 1 -> C 2 J V + 1 , defined as R(X) = n(U(X),V{X), W(X)) = U(X)W(X) + V2(X), associates the coefficients of a convenient hyperelliptic curve (8) to any choice of coefficients in (9) and the preimage, ir~1(R), is an open subset of Jac(r). Finally, Mumford constructed N commuting vector fields D\,..., DN globally defined on C3N+1 and such that they generate the tangent space to 7r -1 (i?) (that is to Jac(r)) at each point. The Jacobi-Mumford system may be put in Lax form
±L(\) at L f A )
_(nA),
=
U(X)\
[L(\),A(\,\*)], L(A')
1 / 0 ,
0\
(10) where P* — (A*,/**) £ T and the corresponding restriction of the flow to Jac(T) is tangent to P* € T C Jac(r). In the case of the Neumann system, comparing (1), (2) and (10), we have L(X) = L(X),
A(X,X*) = A{X),
11
with P* the infinity point . Many generalizations of such construction have been proposed (see for instance Previato 12 and Beauville8 for the case of completely algebraically integrable systems associated to r-gonal curves, Novikov and Veselov15 when fibers are complex tori and Vanhaecke14 when fibers are symmetric products of algebraic curves). We now focus on the case of hyperelliptically separable systems with deficiency and, as before, we show the construction explicitly on an example. The Neumann system admits real integrable generalizations on SN which are hyperelliptically separable with deficiency. Let min{J-l,JV-l} W(0(q) =
_
£
^'-'-^(qjfii+itf),
l>2
initialized by W°^(q) = 1 and W ^ ( q ) = W(q), the Neumann potential, with iij 's coefficients of *(A) $ 2 v-5*— = A" + t l i ^ A " - 1 + • • • + fiw(q). f—f A — a,
Then the generalized Neumann Hamiltonian # ( , ) ( p , q ) = \(p\
+ w ( / ) (q),
+ • • • +PN+I)
i> i
is completely integrable in Arnold-Liouville sense. The equations of the generalized Neumann system may be put in Lax form for any I > 1, |I«
( A )
= [L«>(A),i
