Taiwanese Journal of Mathematics

A Nekhoroshev Type Theorem of Higher Dimensional Nonlinear Schrödinger Equations

Shidi Zhou and Jiansheng Geng

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In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS (nonlinear Schrödinger equations):\[  \mathrm{i} \partial_{t} u - \Delta u + V * u  + \partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad    x \in \mathbb{T}^d, \; t \in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an $\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0 = u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm $\|u(t)\|_s$ remains bounded by $C_s \varepsilon$.

Article information

Taiwanese J. Math. Volume 21, Number 5 (2017), 1115-1132.

Received: 8 October 2016
Revised: 2 January 2017
Accepted: 8 January 2017
First available in Project Euclid: 1 August 2017

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Primary: 37K55: Perturbations, KAM for infinite-dimensional systems 35B10: Periodic solutions

KAM theory Hamiltonian systems Schrödinger equation Birkhoff normal form


Zhou, Shidi; Geng, Jiansheng. A Nekhoroshev Type Theorem of Higher Dimensional Nonlinear Schrödinger Equations. Taiwanese J. Math. 21 (2017), no. 5, 1115--1132. doi:10.11650/tjm/7951. https://projecteuclid.org/euclid.twjm/1501599185

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