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

Abstract. In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62-94.] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.

The existence of two-dimensional Kolmogorov–Arnold–Moser invariant tori is proved for cubic complex Ginzburg–Landau equation of higher spatial dimension. As a consequence, the equation possesses a Cantorian branch of nontraveling-wave quasiperiodic solutions of two-dimensional frequency vector.

No abstract