## 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