Stability of KAM tori for nonlinear Schrödinger equation

Abstract

It is proved that the KAM tori (thus quasi-periodic solutions) are long time stable for nonlinear Schrödinger equation.Consider a function W (x, y; ξ ) : D(s, r) × Π → C is analytic about the variable (x, y) ∈ D(s, r) and C 1 -smooth about the parameter ξ ∈ Π with the following formwhere |α| = ∑ n j=1 |α j |. In this paper, always by | · | denotes 1-norm for complex vectors in C n or C N . Definition 2.3. Introduce the complex T 0 -neighborhoods D(s, r 1 , r 2 ) = (x, y, q,q) ∈ P p | ||Im x|| ≤ s, ||y|| ≤ r 2 1 , ||q|| p + ||q|| p ≤ r 2 . Let r 1 = r 2 = r. Consider a function W (x, y, q,q; ξ ) : D(s, r, r) × Π → C is analytic about the variable (x, y, q,q) ∈ D(s, r, r) and C 1 -smooth about the parameter ξ ∈ Π with the following form W (x, y, q,q; ξ ) = ∑ α∈N n ,β ,γ∈N N W αβ γ (x; ξ )y α q βqγ . Define the modulus ⌊W ⌉ D(s,r)×Π (q,q) of W (x, y, q,q; ξ ) by ⌊W ⌉ D(s,r)×Π (q,q) := ∑ β ,γ∈N N

Long time stability of KAM tori for nonlinear wave equation

“…Finally The following lemma estimates p-tame norm of the solution of homological equation during KAM iterative procedure and normal form iterative procedure, which can be parallel proved following the proof of Theorem 3.4 in [9]: For each α ∈ N n , β ∈ NZ, k ∈ Z n , j ≥ 1 and some fixed constant τ > 0, assume the following inequality holds U αβ (k; ξ) + ∂ ξ j U αβ (k; ξ) ≤ |k| + 1 τ V αβ (k; ξ) + ∂ ξ j V αβ (k; ξ) ,…”

“…In Section 3, we construct a normal form of order 2, which satisfies p-tame property, around the KAM tori based on the standard KAM method (see Theorem 3.1) and a partial normal form of order M + 2 in the δ-neighborhood of the KAM tori (see Theorem 3.2). Since the iterative procedure is parallel to [9], we only prove the measure estimate in detail. Finally, due to the partial normal form of order M + 2 and p-tame property, we show that the KAM tori are stable in a long time (see Theorem 3.3).…”

“…Finally define the p-tame norm of the Hamiltonian vector field X W as follows, instead of a bounded map form 2 b,p to 2 b,p as in [9].…”

“…The following lemma compares p-tame norm with the usual weighted norm for Hamiltonian vector field, which can be parallel proved following the proof of Theorem 3.5 in [9]: has p-tame property on the domain D(s, r, r) × Π for some 0 < s, r ≤ 1. Let X t U be the phase flow generalized by the Hamiltonian vector field X U .…”

KAMTheorem with Normal Frequencies of FiniteLimit‐Pointsfor Some Shallow Water Equations

“…These quasi-periodic solutions are of course global and of recurrent property. As done in [20], those solutions whose initial data are close to any quasi-periodic solution are almost global, that is, assuming u 0 .t; x/ with initial datum u 0 .0; x/ is a quasi-periodic solution for BBM or gPC equation subject to the typical boundary conditions, then any solution u.t; x/ with initial value satisfying ku.0; x/ u 0 .0; x/k p < with any 0 < ( 1 obeys that the solution u.t; x/ exists for time jtj < L 1…”