On quasiperiodic boundary condition problem

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1 JOURNAL OF MATHEMATICAL PHYSICS 46, (005) On quasiperiodic boundary condition problem Y. Charles Li a) Department of Mathematics, University of Missouri, Columbia, Missouri 65 (Received 8 April 004; accepted 9 September 004; published online 3 January 005) 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. 005 American Institute of Physics. [DOI: 0.063/.83754] 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 (D) 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 D cubic nonlinear Schrödinger equation iq t = q xx +q q,. q is a complex-valued function of two real variables t,x, i=. We pose a quasiperiodic boundary condition with two base frequencies and, / is irrational. That is, q = qt,,, = x, = x, and q is periodic in both and with period. Thus a) Electronic mail: cli@math.missouri.edu /005/46()/03503/7/$.50 46, American Institute of Physics

2 Y. Charles Li J. Math. Phys. 46, (005) It seems that the more natural norm is rather than q = qt, = q k te ik, = k kz, = k k. q s = +k s q k, kz q s = +k s q k, = kz. In terms of Fourier transforms, (.) can be rewritten as i dq k dt i dq k dt = H q k, = H q k, H = k q k kz kz = lim a a +a a q x q 4 dx. q kˆq kˆz k kˆ..3.4 Using (.4), the NLS (.) can be rewritten as Obviously, is an invariant. iq t = H q, iq t = H q. I = q k = lim kz a + a a a q dx.5 III. WELL-POSEDNESS Explicitly (.) can be written as iq k = k q k + kˆ,k Z q kˆq k +kˆq k k. 3. The method of variation of parameters leads to the integral equation

3 On quasiperiodic boundary condition problem J. Math. Phys. 46, (005) q k t = e ik t q k 0 i0 t e ik t kˆ,k Z q kˆq k +kˆq k k d. Notice that (3.) bears more resemblance to two-dimensional (D), rather than D, NLS under the periodic boundary condition. Local well-posedness can be easily established, since the nonlinear term is still locally Lipschitz. Theorem 3. (Local well-posedness): For any q 0 H s, s, there exists a unique solution qtc 0 0,,H s =q 0 s, to the Cauchy problem of (3.) with initial condition q0=q 0. For any fixed t0,, qt is C in q 0. The interesting open problem is whether or not (3.) has global well-posedness. On the one hand, it resembles D NLS under periodic boundary condition, therefore, it may not have global well-posedness. In fact, the first term in the Hamiltonian (.3) is weaker than kz k q k of the D NLS periodic case. Thus the Hamiltonian cannot bound the H 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 (.), one gets an ODE defined on the invariant complex plane iq t =q q with all periodic solutions (the so-called Stokes waves) 3. Linearize the NLS in the manner one has Let q = ce ic t+. q = c + qˆe ic t+, iqˆ t = qˆ xx +c qˆ + qˆ. qˆ = kz qˆ kte ik, one gets Let i d dt qˆ k qˆ k = c k c qˆ k qˆ k t = e B, A, A, and B are complex constants, then c qˆ k k c qˆ k =±k c k. Lemma 4.: The set k kz is dense in R. Proof: This proof is furnished by Banks. For any real number z, let z denote the greatest integer less than or equal to z, and let z=z z be the fractional part of z; then 0z. For any irrational number a, it is known that the fractional parts na nz are uniformly distributed over 4.5

4 Y. Charles Li J. Math. Phys. 46, (005) the unit interval [0,]. For any fixed br, given any 0, let k be chosen such that and choose k =b/ k /, then k / b/ /, k + k / b/ = k + k / b/ + k / b/ = k / b/ /. Multiplying by, one obtains k. This proves the lemma. Theorem 4.: The spectrum of the linear NLS operator in H s, s0 is = p c = c,c ir p is given by (4.5) and is every dense in. Proof: The maximum of the function z c z, z R is 4c. By Lemma 4. and the fact that the spectrum is a closed set, we have that c,c ir. In terms of the Fourier transform (4.), the linear NLS operator has the representation given by (4.3), L k = i c k c c k c. If c,c ir, then there is an absolute constant C such that L k C, and this is true even for some k, k might be equal to c. Thus such belongs to the resolvent set, and k = c,c ir. Let / p p is the point spectrum given by (4.5), then there is a sequence j p such that j, and L kj / j + ; thus 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 c k, kz, and is a positive constant. For any fixed c0, 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 c0, there are infinitely many unstable modes. There is no gap among the unstable, center, and stable spectra.

5 On quasiperiodic boundary condition problem J. Math. Phys. 46, (005) 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 (.) has the Lax pair, x = U, U = i q t = V, q, V = i q q + iq x q iq x + q. Theorem 5.: Let qt,x be a solution, and let be an eigenfunction of the Lax pair at = for any C. Use to define a matrix, 0 G = 0, We define Q and by =. Q = q + +, = G, 5. 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.) is called a Darboux transformation. For example, let q = ae it, t = a t +, a is the amplitude and is the phase. The eigenfunctions of the Lax pair are ± = ae it/ ± e it/e ±it±ix, = a. In order to have temporal growth, a. For =, ==i, let = c c, c ± are two arbitrary complex constants. For =, =ˆ =iˆ, let ˆ = ĉ ĉ, ĉ ± are two arbitrary complex constants. By iterating the Darboux transformation (5.) at and ˆ, one gets

6 Y. Charles Li J. Math. Phys. 46, (005) Q = q + + +ˆ ˆ ˆ ˆ ˆ + ˆ, ˆ = + ˆ + ˆ ˆ + ˆ, ˆ = Explicitly, one has + ˆ + ˆ + ˆ ˆ. Q = Q + q sin ˆ 0, 5. Q = q + sin 0 sech cos X cos 0 i sin 0 tanh sin 0 sech cosx, =sin ˆ 0 + sin 0 sech cos X + 8 sin 0 sech cos X + sin ˆ 0 sech ˆ cos Xˆ sin 0 sin ˆ 0 sech sech ˆ + sin 0 sech cos X sin X sin Xˆ + sin 0 +sin 0 sech cos X + cos X cos 0 sech + sin ˆ 0 sech ˆ cos Xˆ sin ˆ 0 sin 0 cos ˆ 0 cos 0 tanh ˆ tanh + sin 0 + sech cos Xsin ˆ 0 + sech ˆ cos Xˆ + sin 0 sech cos X, = sin ˆ 0 + sin 0 sech cos X + 4 sin 0 sech cos X sin ˆ 0 + sech ˆ cos Xˆ + i cos ˆ 0tanh ˆ +sin 0 cos 0 tanh + i sin 0 + i sech cos X sinˆ 0 + sech ˆ cos Xˆ i cos ˆ 0 tanh ˆ + sin 0 sin 0 + sech cos X + i cos 0 tanh sin ˆ 0 + sin 0 sech cos X + sin ˆ 0 sech ˆ cos Xˆ sin 0 cos ˆ 0 sech sech ˆ sin X sin Xˆ, and + = ae i 0, + ˆ = ae iˆ 0, c + /c = e +i, ĉ + /ĉ = e ˆ+iˆ, =4 t, ˆ =4ˆ t ˆ,

7 On quasiperiodic boundary condition problem J. Math. Phys. 46, (005) X = x /, Xˆ = x + ˆ ˆ 0 + /. The foliation here is with respect to the two linear unstable modes (,0) and (0, ) in (4.5). The temporal growth condition a or a is in agreement with (4.5). Thus (5.) represents a class of solutions with quasiperiodic boundary condition. For fixed a,, and, the parameters are,, ˆ,, and ˆ.Ast ±, e.g.,,,, and ˆ are all positive, Q qe i 0 +ˆ 0. 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 D more than the D 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. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. Vol. 44 (Springer Verlag, New York, 983). V. Banks (private communication). 3 Y. Li, Chaos in Partial Differential Equations (International Press, Somerville, MA, 004).

Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger. equations

J. Math. Anal. Appl. 315 (2006) 642 655 www.elsevier.com/locate/jmaa Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations Y. Charles Li Department of