Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

Abstract

We consider nonlinear Schrödinger 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").

55 pages, 4 figures