Stable Directions for Degenerate Excited States of Nonlinear Schr\"odinger Equations

TL;DR

This paper studies nonlinear Schrödinger equations with specific potentials, constructing stable directions for excited states and analyzing their behavior in resonant and non-resonant cases.

Contribution

It introduces the existence of nonlinear excited states and identifies stable directions in phase space for these states in both resonant and non-resonant scenarios.

Findings

Existence of two branches of nonlinear excited states.

Construction of finite-codimension stable regions in phase space.

Analysis applicable to both resonant and non-resonant cases.

Abstract

We consider nonlinear Schr\"{o}dinger equations, $i\partial_t \psi = H_0 \psi + \lambda |\psi|^2\psi$ in $\mathbb{R}^3 \times [0,\infty)$, where $H_0 = -\Delta + V$, $\lambda=\pm 1$, the potential $V$ is radial and spatially decaying, and the linear Hamiltonian $H_0$ has only two eigenvalues $e_0 < e_1 <0$, where $e_0$ is simple, and $e_1$ has multiplicity three. We show that there exist two branches of small "nonlinear excited state" standing-wave solutions, and in both the resonant ($e_0 < 2e_1$) and non-resonant ($e_0 > 2e_1$) cases, we construct certain finite-codimension regions of the phase space consisting of solutions converging to these excited states at time infinity ("stable directions").