Long time stability of plane wave solutions to Schrödinger equation on Torus. Issue 8 (24th May 2022) - Mi, Lufang; Sun, Yingte; Wang, Peizhen
ABSTRACT: We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing ( λ = 1 ) or focusing ( λ = − 1 ) case, 1 i u t = − Δ u + λ | u | 2 u, x ∈ T d, t ∈ R . More precisely, in a Gevrey space 2 G σ := u :   |   ∥ u ∥ σ 2 = ∑ a ∈ Z d e 2 σ | a | | u a | 2 < ∞ for some positive constant σ, we show that solution with the initial datum in the 4 ϵ -neighborhood of the plane wave solution still stays in the C ϵ -neighborhood ( C >4 ) of the plane wave solution for a subexponential long time | t | ≤ ϵ − ζ | ln ⁡ ϵ | ϱ, where ζ = min { 1 4, σ − σ ′ }, σ > σ ′ > 0 and 0 < ϱ < 1 / 6 .