A Nekhoroshev Type Theorem of Higher Dimensional Nonlinear Schrödinger Equations

Abstract

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 $C_s \varepsilon$.