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.


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