The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Abstract

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.

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

“…Proof. The proof of (2.2) is the same as Lemma 2.1 in [11], which generalizes the result given in Lemma 1.1 in [9].…”

“…where the last inequality is based on Lemma 7.5 in [11] with p = 2, and we finish the proof of (2.25).…”

“…Recently, Cong-Liu-Shi-Yuan [11] generalized Bourgain's result from θ = 1/2 to 0 < θ < 1, i.e the actions of the tori satisfying 1 2 e −r|n| θ ≤ I n ≤ 2e −r|n| θ , θ ∈ (0, 1) and r > 0.…”

“…We drop the index s of H s , N s , R s , Φ s and shorten the index s + 1 as +. We desire to eliminate the terms R 0 , R 1 in (2.5) by the coordinate transformation Φ, which is obtained as the time-1 map X t F | t=1 of a Hamiltonian vector field 11) and the homological equations become…”

“…where the last inequality is based on (7.37) in [11] and C(θ) is a positive constant depending only on θ. In view of (2.22), one has (2.34)…”