KAM Theory For Nonlinear Schr(?)dinger Equation With The Mixed Dispersion

In quantum mechanics, the Schrodinger equation is not only a fun-damental equation, but also a basic assumption. Many famous mathe-maticians focus on the existence of periodic or quasi-periodic solutions of Schrodinger equation for a long time. KAM theory is one of the main methods. In last century, the celebrated KAM theory which is the land-mark of the development of Hamiltonian systems was constructed by Kolmogorov, Arnold and Moser. It explains the dynamics fundamental question. It also gives the reasonable explanation of the stability of solar system.Theorem. Let the integrable Hamiltonian H1(I) be real analytic and nondegenerate, and let the perturbed Hamiltonian H(I,θ) = H1(I)+∈H2(I,θ) be of class Cr (r> 2n). Then, for sufficiently small∈, the perturbed system possesses smooth invariant n tori with linear flow for allω∈Ω(n,γ):={ω∈Rn||ω·k|≥γ|k|-n, (?)k∈Zn\{0}}, i. e. restricted to the invariant n-tori, the vector field is analytically conjugate toφ=ω.In the 1980's, the classical KAM theory was generalized to infinite dimensional setting by Kuksin, Wayne and Poschel. It's applied to the nonlinear partial equation successfully.Later, Poschel restated it and got a result which is called Cantor manifold theorem.In some neighborhood of the origin in Hα,ρ, we consider a Hamil-tonian H=Λ+Q+R, where R denotes a higher order perturbation of an integrable normal formΛ+Q. Under coordinates q= (q1,..., qn) in Hα,ρ, whereand withthe normal form is composed of terms whereα,βare constant vectors and A, B are constant matrices. Then there exists a n-dimensional invariant manifold E={q= 0} which is completely filled with the invariant tori We call T(Ⅰ) an elliptic rotational torus with frequenciesω(Ⅰ). Our aim is to prove that there exist a family of n tori and a Lipschitz continuous embedding where C (?) Pn is a Cantor set, such that the restriction ofψto each torus T(I) is an embedding of an elliptic rotational n torus for the Hamiltian H. The image S of T[C] is called a Cantor manifold of an elliptic rotational n tori which is given by the embeddingψ:T[C]→ε. Moreover, The embeddingψis close to the inclusion mapψ0:E→Hα,ρ, and Cantor manifoldεis tangent to E at the origin.We make the following assumptions. (H1). Nondegeneracy. The normal formΛ+Q is nondegenerate in the sense that det A≠0,≠0,+≠0, (?)(k,l)∈Zn×Z∞, where 1≤|l|≤2,ω=α+AI,Ω=β+BI. (H2). Spectral Asymptotic. There exist d≥1 andδ< d - 1 such thatβj=jd+…+O(jδ), where the dots denote the terms of order less than d in j. (H3). Regularity. where A(Hα,ρ, Hα,ρ) denotes the class of all maps from some neighbor-hood of the origin in Hα,ρinto Hα,ρ, which are real analytic in the real and imaginary parts of the complex coordinate q. Theorem. (The Cantor manifold theorem). Suppose the Hamiltonian H =Λ+Q+R satisfies assumptions(H1) - (H3), and whereThen there exists a Cantor manifoldεof real analytic, elliptic diophan-tine n tori given by a Lipschitz continuous embeddingψ:T[C]→ε, where C has full density at the origin, andψis close to the inclusion mapψ0 with Moreover,εis tangent to E at the origin.This paper obtains the existence of quasi-periodic solutions of the nonlinear Schrodinger equation with mixed dispersion under the Dirich-let boundary condition. i. e.,Theorem. Consider (1.2). Then for any given d∈Z+ and the index set J={j1<...< jd}, there exists an Cantor manifoldεJ of real anal-ysis, linearly stability, Diophantine d-tori for the system (1.2) given by a Lipschitz continuous embeddingψ:Tj[C]→εJ which is the high pertur-bation of the inclusion mapψ0:TJ[C]→P. Moreover, the Diophiantine tori carry quasi-periodic motions.