Long time stability of plane wave solutions to Schrödinger equation on Torus

ABSTRACT

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing ($\lambda =1$) or focusing ($\lambda =-1$) case, $\mathrm{i}{u}_t=-\mathrm{\Delta }u+\lambda |u{|}^{2}u,x\in {\mathbb{T}}^d,t\in \mathbb{R}.$ More precisely, in a Gevrey space $G_{\sigma }:=\left\{u:|\parallel u{\parallel }_{\sigma }^2=\sum _{a\in {\mathbb{Z}}^d}{e}^{2\sigma \sqrt{|a|}}|u_a{|}^2<\mathrm{\infty }\right\}$ for some positive constant σ, we show that solution with the initial datum in the $4ϵ$-neighborhood of the plane wave solution still stays in the $Cϵ$-neighborhood ( C>4 ) of the plane wave solution for a subexponential long time $|t|\le {ϵ}^{-\zeta |\ln ϵ{|}^{\varrho }},$ where $\zeta =min\{\frac{1}{4},\sigma -{\sigma }^{\prime }\},\sigma >{\sigma }^{\prime }>0$ and $0<\varrho <1/6$.

Keywords:

• Nekhoroshev type estimate
• tame property
• hamiltonian partial differential equation
• plane wave solutions
• stability

2000 Mathematics Subject Classifications:

• Primary 37K55
• 37J40
• Secondary 35B35
• 35Q35

Acknowledgments

The authors are very grateful to Professor Hongzi Cong for his helpful suggestions and the helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by Shandong Provincial Natural Science Foundation No. ZR2019MA062 and Binzhou University (BZXYL1402). The second author is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 19KJB110025) and School Foundation of Yangzhou University (Grant No. 2019CXJ009).