Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

Details Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

2010-04-12

arxiv.org/abs/1004.1888v1

Mathematics

Analysis of PDEs

55 pages, 4 figures

Scientific paper

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").

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