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 $H_0 = -\Delta + V$, $\lambda = \pm1$, 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 nonresonant ($e_0 > 2e_1$) 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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