On quasiperiodic boundary condition problem

JOURNAL OF MATHEMATICAL PHYSICS 46, 013503 (2005)

On quasiperiodic boundary condition problem Y. Charles Lia) Department of Mathematics, University of Missouri, Columbia, Missouri 65211 (Received 28 April 2004; accepted 29 September 2004; published online 3 January 2005)

The paper raises the question of posing the quasiperiodic boundary condition in the Cauchy problem of partial differential equations. Using the one-dimensional cubic nonlinear Schrödinger as a simple example, we illustrated the various types of questions including global well-posedness, spectra of linear operators, and foliations. © 2005 American Institute of Physics. [DOI: 10.1063/1.1832754]

I. INTRODUCTION

The quasiperiodic boundary condition problem can be posed for a variety of partial differential equations (PDE) including, e.g., parabolic and hyperbolic equations. Questions that can be asked include local and global well-posedness, dynamics in phase spaces, and asymptotics, etc. Here we take a simple PDE–one-dimensional cubic nonlinear Schrödinger equation (NLS), to study its phase space foliations. Typical fluid flows are defined on unbounded domain with nondecaying boundary conditions. For example, the Poiseuille flow or the boundary layer flow has nondecaying boundary conditions along the longitudinal direction. In fact, turbulence develops along this longitudinal direction. In many cases, turbulent fluid flows contain both temporal and spatial randomness. Temporal randomness is often caused by temporal chaotic motions. Spatial randomness is often caused by vortex (energy) cascade or inverse cascade. In such cases, periodic boundary conditions put too much constraint. Quasiperiodic or more general boundary conditions are more relevant. The one-dimensional (1D) cubic NLS under periodic boundary conditions is well understood. It is globally well-posed. Under quasiperiodic boundary conditions, global well-posedness is not known. Under periodic boundary conditions, Stokes wave solution has a finite number of unstable eigenvalues. On the other hand, under quasiperiodic boundary conditions, it has infinitely many unstable eigenvalues dense on an interval. There is no spectral gap. But explicit expressions of the foliation in phase space can be obtained via a Darboux transformation.

II. FORMULATION OF THE PROBLEM

Consider the 1D cubic nonlinear Schrödinger equation 共2.1兲

iqt = qxx + 2兩q兩2q,

where q is a complex-valued function of two real variables 共t , x兲 , i = 冑−1. We pose a quasiperiodic boundary condition with two base frequencies ␤1 and ␤2, ␤1 / ␤2 is irrational. That is, q = q共t, ␪1, ␪2兲,

␪1 = ␤1x,

␪2 = ␤2x,

and q is periodic in both ␪1 and ␪2 with period 2␲. Thus a)

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J. Math. Phys. 46, 013503 (2005)

Y. Charles Li

兺

q = q共t, ␪兲 =

qk共t兲eik·␪,

冉冊 冉冊 ␪1 , ␪2

␪=

k僆Z2

k1 . k2

k=

It seems that the more natural norm is

兺

2 = 储q储共s兲

共1 + 兩k兩2兲s兩qk兩2 ,

k僆Z2

rather than 2 = 储q储关s兴

兺

关1 + 共k · ␤兲2兴s兩qk兩2,

␤=

k僆Z2

冉 冊

␤1 . ␤2

In terms of Fourier transforms, (2.1) can be rewritten as i

dqk ⳵H =− , dt ⳵ qk 共2.2兲

dqk ⳵ H = i , dt ⳵ qk where H=

兺

k僆Z2

1 2a a→+⬁

= lim

k僆Z2

冕

冏

兺 ˆ兺

共k · ␤兲2兩qk兩2 −

qkˆqk−kˆ

k僆Z2

冏

2

共2.3兲

a

关兩qx兩2 − 兩q兩4兴dx.

共2.4兲

−a

Using (2.4), the NLS (2.1) can be rewritten as iqt = −

␦H , ␦q 共2.5兲

␦H . iqt = ␦q Obviously, I=

兺

k僆Z2

1 a→+⬁ 2a

兩qk兩2 = lim

冕

a

兩q兩2 dx

−a

is an invariant. III. WELL-POSEDNESS

Explicitly (2.2) can be written as iq˙k = − 共k · ␤兲2qk + 2

兺

qkˆq˜k+kˆqk−k˜ .

共3.1兲

˜ 僆Z2 kˆ,k

The method of variation of parameters leads to the integral equation

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013503-3

J. Math. Phys. 46, 013503 (2005)

On quasiperiodic boundary condition problem

2

qk共t兲 = ei共k · ␤兲 tqk共0兲 − 2i

冕

t

ei共k · ␤兲

兺

2共t−␶兲

0

qkˆq˜k+kˆqk−k˜ d␶ .

共3.2兲

˜ 僆Z2 kˆ,k

Notice that (3.1) bears more resemblance to two-dimensional (2D), rather than 1D, NLS under the periodic boundary condition. Local well-posedness can be easily established,1 since the nonlinear term is still locally Lipschitz. Theorem 3.1 (Local well-posedness): For any q0 僆 H共s兲, s 艌 2, there exists a unique solution q共t兲 僆 C0共关0 , ␶兴 , H共s兲兲 where ␶ = ␶共储q0储共s兲兲, to the Cauchy problem of (3.1) with initial condition q共0兲 = q0. For any fixed t 僆 关0 , ␶兴, q共t兲 is C⬁ in q0. The interesting open problem is whether or not (3.1) has global well-posedness. On the one hand, it resembles 2D NLS under periodic boundary condition, therefore, it may not have global well-posedness. In fact, the first term in the Hamiltonian (2.3) is weaker than 兺k僆Z2兩k兩2兩qk兩2 of the 2D NLS periodic case. Thus the Hamiltonian cannot bound the H共1兲 norm. On the other hand, it is still an integrable system, therefore, an infinite sequence of invariants is at one’s disposal. IV. THE SPECTRUM OF A LINEAR NLS OPERATOR

Setting ⳵x = 0 in (2.1), one gets an ODE defined on the invariant complex plane iqt = 2兩q兩2q with all periodic solutions (the so-called Stokes waves) q = ce−i关2c

2t+␥兴

共4.1兲

.

Linearize the NLS in the manner q = 共c + qˆ兲e−i关2c

2t+␥兴

,

one has iqˆt = qˆxx + 2c2共qˆ + qˆ兲. Let qˆ =

兺

qˆk共t兲eik·␪ ,

共4.2兲

k僆Z2

one gets i Let

冉 冊冉

d qˆk dt qˆ −k

=

2c2 − 共k · ␤兲2

2c2

− 2c2

共k · ␤兲2 − 2c2

冉 冊 冉冊 qˆk

qˆ−k

= e␭t

A , B

冊冉 冊 qˆk

qˆ−k

.

共4.3兲

共4.4兲

where ␭, A, and B are complex constants, then ␭ = ± 共k · ␤兲冑共2c兲2 − 共k · ␤兲2 .

共4.5兲

Lemma 4.1: The set 兵k · ␤其k僆Z2 is dense in R. Proof: This proof is furnished by Banks.2 For any real number z, let 关z兴 denote the greatest integer less than or equal to z, and let 兵z其 = z − 关z兴 be the fractional part of z; then 0 艋 兵z其 艋 1. For any irrational number a, it is known that the fractional parts 兵na其n僆Z are uniformly distributed over

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J. Math. Phys. 46, 013503 (2005)

Y. Charles Li

the unit interval [0,1]. For any fixed b 僆 R, given any ⑀ ⬎ 0, let k2 be chosen such that 兩兵k2共␤2/␤1兲其 − 兵b/␤1其兩 ⬍ ⑀/␤1 , and choose k1 = 关b / ␤1兴 − 关k2共␤2 / ␤1兲兴, then 兩k1 + k2共␤2/␤1兲 − b/␤1兩 = 兩k1 + 关k2共␤2/␤1兲兴 − 关b/␤1兴 + 兵k2共␤2/␤1兲其 − 兵b/␤1其兩 = 兩兵k2共␤2/␤1兲其 − 兵b/␤1其兩 ⬍ ⑀/␤1 . Multiplying by ␤1, one obtains 兩k · ␤兩 ⬍ ⑀. This proves the lemma. Theorem 4.2: The spectrum of the linear NLS operator in H共s兲, s 艌 0 is

䊏

␴ = ␴ p 艛 ␴c = 关− 2c2,2c2兴 艛 iR where ␴ p is given by (4.5) and is everywhere dense in ␴. Proof: The maximum of the function z2共共2c兲2 − z2兲,

z僆R

is 4c2. By Lemma 4.1 and the fact that the spectrum ␴ is a closed set, we have that 关− 2c2,2c2兴 艛 iR 傺 ␴ . In terms of the Fourier transform (4.2), the linear NLS operator has the representation given by (4.3),

Lk = − i

冉

2c2 − 共k · ␤兲2

2c2

− 2c2

共k · ␤兲2 − 2c2

冊

.

If ␭ 苸 关−2c2 , 2c2兴 艛 iR, then there is an absolute constant C such that 储共Lk − ␭兲−1储 艋 C,

∀k

and this is true even for some k, 共k · ␤兲2 might be equal to 共2c兲2. Thus such ␭ belongs to the resolvent set, and

␴ = 关− 2c2,2c2兴 艛 iR. Let ␭ 僆 ␴ / ␴ p where ␴ p is the point spectrum given by (4.5), then there is a sequence ␭ j 僆 ␴ p such that ␭ j → ␭, and 储共Lk j − ␭兲−1储 艌 1/兩␭ j − ␭兩 → + ⬁; 䊏 thus ␭ 僆 ␴c is the continuous spectrum. This proves the theorem. Remark 4.3: For NLS under periodic boundary condition, the spectrum of the linear NLS operator consists of only discrete point spectrum given by ␭ = ± k␤冑共2c兲2 − 共k␤兲2 , where k 僆 Z, and ␤ is a positive constant. For any fixed c ⬎ 0, there is a finite number of unstable modes. There are gaps among the unstable, center, and stable spectra. As shown above, under quasiperiodic boundary condition, the point spectrum is dense, and there is also a continuous spectrum. For any fixed c ⬎ 0, there are infinitely many unstable modes. There is no gap among the unstable, center, and stable spectra.

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013503-5

J. Math. Phys. 46, 013503 (2005)

On quasiperiodic boundary condition problem

V. FOLIATIONS

Although there is no spectral gap in this quasiperiodic setting, foliations can still be established via explicit expressions. The tool used is the so-called Darboux transformation. The NLS (2.1) has the Lax pair,

␺x = U␺,

␺t = V␺ ,

where

U=i

V=i

冉

冉 冊 ␭

q

q −␭

2␭2 − 兩q兩2

,

2␭q − iqx

2␭q + iqx − 2␭2 + 兩q兩2

冊

.

Theorem 5.1: Let q共t , x兲 be a solution, and let ␾ be an eigenfunction of the Lax pair at ␭ = ␯ for any ␯ 僆 C. Use ␾ to define a matrix, G=⌫

冉

␭−␯

0

0

␭−␯

冊

⌫−1 ,

where ⌫=

冉

冊

␾1 − ␾2 . ␾2 ␾1

We define Q and ⌿ by Q = q + 2共␯ − ␯兲

␾ 1␾ 2 , 兩 ␾ 1兩 2 + 兩 ␾ 2兩 2

⌿ = G␺ ,

共5.1兲

where ␺ solves the Lax pair at (q , ␭). Then ⌿ solves the Lax pair at (Q , ␭) and Q solves the NLS. This is a well-known theorem in the integrable theory, see, e.g., Ref. 3. The transformation (5.1) is called a Darboux transformation. For example, let q = aei␪共t兲,

␪共t兲 = − 关2a2t + ␥兴,

where a is the amplitude and ␥ is the phase. The eigenfunctions of the Lax pair are

␾± =

冉

冊

aei␪共t兲/2 e±i2␭␤t±i␤x, 共± ␤ − ␭兲e−i␪共t兲/2

␭ = 冑␤2 − a2 .

In order to have temporal growth, ␤2 ⬍ a2. For ␤ = ␤1, ␭ = ␯ = i␴, let

␾ = c +␾ + + c −␾ − , where c± are two arbitrary complex constants. For ␤ = ␤2, ␭ = ␯ˆ = i␴ˆ , let

␾ˆ = cˆ+␾+ + cˆ−␾− , where cˆ± are two arbitrary complex constants. By iterating the Darboux transformation (5.1) at ␯ and ␯ˆ , one gets

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013503-6

J. Math. Phys. 46, 013503 (2005)

Y. Charles Li

Q = q + 2共␯ − ␯兲

ˆ ⌽ ˆ ⌽ ␾ 1␾ 2 1 2 ˆ ˆ 兲 + 2共 ␯ ␯ , − ˆ 兩2 兩 ␾ 1兩 2 + 兩 ␾ 2兩 2 ˆ 兩2 + 兩⌽ 兩⌽ 1 2

where ˆ = ⌽ 1

1 ˆ 1 + 共 ␯ − ␯ 兲 ␾ 1␾ 2␾ ˆ 2其, 兵关共␯ˆ − ␯兲兩␾1兩2 + 共␯ˆ − ␯兲兩␾2兩2兴␾ 兩 ␾ 1兩 2 + 兩 ␾ 2兩 2

ˆ = ⌽ 2

1 ˆ 1 + 关共␯ˆ − ␯兲兩␾1兩2 + 共␯ˆ − ␯兲兩␾2兩2兴␾ ˆ 2其. 兵共␯ − ␯兲␾1␾2␾ 兩 ␾ 1兩 2 + 兩 ␾ 2兩 2

Explicitly, one has ˜ + q sin ␽ ˆ Q=Q 0

兿2

冒兿

共5.2兲

,

1

where ˜ = q关1 + sin ␽ sech ␶ cos X兴−1关cos 2␽ − i sin 2␽ tanh ␶ − sin ␽ sech ␶ cosX兴, Q 0 0 0 0

兿1 =

冋

ˆ 兲2共1 + sin ␽ sech ␶ cos X兲2 + 1 共sin 2␽ 兲2共sech ␶兲2共1 − cos 2X兲 共sin ␽ 0 0 0 8

册

ˆ sech ␶ˆ cos Xˆ兲 − 1 sin 2␽ sin 2␽ ˆ sech ␶ sech ␶ˆ 共1 + sin ␽ sech ␶ cos X兲 ⫻共1 + sin ␽ 0 0 0 0 2 ⫻sin X sin Xˆ + 共sin ␽0兲2关1 + 2 sin ␽0 sech ␶ cos X + 关共cos X兲2 − 共cos ␽0兲2兴共sech ␶兲2兴 ˆ sech ␶ˆ cos Xˆ兲 − 2 sin ␽ ˆ sin ␽ 关cos ␽ ˆ cos ␽ tanh ␶ˆ tanh ␶ ⫻共1 + sin ␽ 0 0 0 0 0 ˆ + sech ␶ˆ cos Xˆ兲兴共1 + sin ␽ sech ␶ cos X兲, + 共sin ␽0 + sech ␶ cos X兲共sin ␽ 0 0

兿2 =

冋

ˆ 兲2共1 + sin ␽ sech ␶ cos X兲2 + 1 共sin 2␽ 兲2共sech ␶兲2共1 − cos 2X兲 − 2共sin ␽ 0 0 0 4

册

ˆ + sech ␶ˆ cos Xˆ + i cos ␽ ˆ tanh ␶ˆ 兲 + 2共sin ␽ 兲2共− cos ␽ tanh ␶ + i sin ␽ ⫻共sin ␽ 0 0 0 0 0 ˆ + sech ␶ˆ cos Xˆ − i cos ␽ ˆ tanh ␶ˆ 兲 + 2 sin ␽ 共sin ␽ + sech ␶ cos X + i sech ␶ cos X兲2共sin␽ 0 0 0 0 ˆ 共1 + sin ␽ sech ␶ cos X兲共1 + sin ␽ ˆ sech ␶ˆ cos Xˆ兲 + i cos ␽0 tanh ␶兲关2sin ␽ 0 0 0 ˆ sech ␶ sech ␶ˆ sin X sin Xˆ兴, − sin 2␽0 cos ␽ 0 and ˆ

␤1 + ␯ = aei␽0,

␤2 + ␯ˆ = aei␽0 ,

c+/c− = e␳+i␽,

cˆ+/cˆ− = e␳ˆ +i␽ ,

␶ = 4 ␴ ␤ 1t − ␳ ,

␶ˆ = 4␴ˆ ␤2t − ␳ˆ ,

ˆ

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013503-7

J. Math. Phys. 46, 013503 (2005)

On quasiperiodic boundary condition problem

X = 2␤1x + ␽ − ␽0 + ␲/2,

ˆ −␽ ˆ + ␲/2. Xˆ = 2␤2x + ␽ 0

The foliation here is with respect to the two linear unstable modes (2␤1,0) and (0,2␤2) in (4.5). The temporal growth condition ␤21 ⬍ a2 or ␤22 ⬍ a2 is in agreement with (4.5). Thus (5.2) represents a class of solutions with quasiperiodic boundary condition. For fixed a, ␤1, and ␤2, the parameters ˆ . As t → ± ⬁, e.g., ␤ , ␤ , ␴, and ␴ˆ are all positive, are ␥, ␳, ␳ˆ , ␽, and ␽ 1 2 ˆ

Q → qe⫿i2共␽0+␽0兲 . VI. CONCLUSION AND DISCUSSION

From the presentation in this paper, one can see that the first interesting question on such quasiperiodic boundary condition problem is the global well-posedness. In terms of Fourier transforms, one can see that the integrable NLS resembles the 2D more than the 1D periodic problem. I tend to believe that it may have finite-time blow-up solutions, which will be truly interesting. Also linearization in the quasiperiodic case often leads to a linear operator with continuous spectrum and with no spectral gap. Therefore, the phase space foliation is a challenging and interesting problem. In this paper, through Darboux transformation, such foliation can still be established. 1

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. Vol. 44 (Springer Verlag, New York, 1983). V. Banks (private communication). 3 Y. Li, Chaos in Partial Differential Equations (International Press, Somerville, MA, 2004). 2

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