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Stability and instability of standing waves for the generalized Davey- Stewartson system. (English) Zbl 0827.35122

Summary: We study the stability and instability properties of standing waves for the equation \[ iu_t + \Delta u + a |u |^{p - 1} u + E_1 \bigl( |u |^2 \bigr) u = 0 \] in \(\mathbb{R}^2\) or \(\mathbb{R}^3\), which derives from the generalized Davey-Stewartson system in the elliptic-elliptic case. We show that if \(n = 2\) and \(a(p - 3) < 0\), then the standing waves generated by the set of minimizers for the associated variational problem are stable. We also show that if \(n = 3\), \(a > 0\) and \(1 + 4/3 < p < 5\) or \(a < 0\) and \(1 < p < 3\), then the standing waves are strongly unstable. We employ the concentration compactness principle due to Lions and the compactness lemma due to Lieb to solve the associated minimization problem.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35A15 Variational methods applied to PDEs
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