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**Stable directions for degenerate excited states of nonlinear Schrödinger equations.**
*(English)*
Zbl 1230.35129

This paper concerns nonlinear Schrödinger equations with a (smooth and spatially decaying) potential
\[
\text{i} \partial_t \psi = H_0 \psi \pm |\psi|^2 \psi = 0, \quad H_0 \psi = - \Delta \psi + V(x) \psi,
\]
such that the operator \(H_0\) has exactly two eigenvalues \(e_0<e_1<0\), in the case where \(e_0\) is simple and \(e_1\) has multiplicity three. The purpose is to extend several results in [T.-P. Tsai and H.-T. Yau, Commun. Partial Differ. Equations 27, No. 11-12, 2363–2402 (2002; Zbl 1021.35113)] and [T.-P. Tsai and H.-T. Yau, Int. Math. Res. Not. 2002, No. 31, 1629–1673 (2002; Zbl 1011.35120)], devoted to the case where both \(e_0\) and \(e_1\) are simple eigenvalues.

First, the authors prove a specific result of existence of two branches of small excited states, solutions \(Q_1\) of \( H_0 Q_1 \pm |Q_1|^2 Q_1 = EQ_1, \) for \(E\) close to \(e_1\), such that the corresponding standing solutions of the Schrödinger model \(\psi(x,t) = \text{e}^{-\text{i} Et} Q_1(x)\) are unstable.

Second, the authors extend the main result in [T.-P. Tsai and H.-T. Yau, Int. Math. Res. Not. 2002, No. 31, 1629–1673 (2002; Zbl 1011.35120)], proving that in spite of their instability, there exist solutions converging to these excited states asymptotically in large time, under standard assumptions.

The paper also contains many references of previous works related to stability and asymptotic stability of ground states (solutions corresponding to \(E\) close to \(e_0\)).

First, the authors prove a specific result of existence of two branches of small excited states, solutions \(Q_1\) of \( H_0 Q_1 \pm |Q_1|^2 Q_1 = EQ_1, \) for \(E\) close to \(e_1\), such that the corresponding standing solutions of the Schrödinger model \(\psi(x,t) = \text{e}^{-\text{i} Et} Q_1(x)\) are unstable.

Second, the authors extend the main result in [T.-P. Tsai and H.-T. Yau, Int. Math. Res. Not. 2002, No. 31, 1629–1673 (2002; Zbl 1011.35120)], proving that in spite of their instability, there exist solutions converging to these excited states asymptotically in large time, under standard assumptions.

The paper also contains many references of previous works related to stability and asymptotic stability of ground states (solutions corresponding to \(E\) close to \(e_0\)).

Reviewer: Yvan Martel (Palaiseau)