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Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

We consider nonlinear Schrödinger equations, $i\partial_t ψ= H_0 ψ+ λ|ψ|^2ψ$ in $\mathbb{R}^3 \times [0,\infty)$, where $H_0 = -Δ+ V$, $λ=\pm 1$, 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").