Standing waves in nonlinear Schrödinger equations

Stefan Le Coz (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For the physical applications as well as for the mathematical properties of the equation, a fundamental issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present various methods developed to study the existence and stability of standing waves. We prove the existence of standing waves by using a variational approach. When stability holds, it is obtained by proving a coercivity property for a linearized operator. Another approach based on variational and compactness arguments is also presented. When instability holds, we show by a method combining a Virial identity and variational arguments that the standing waves are unstable by blow-up.

Domaines: Equations aux dérivées partielles [math.AP]

Stefan Le Coz. Standing waves in nonlinear Schrödinger equations. Analytical and Numerical Aspects of Partial Differential Equations, de Gruyter, pp.151-192, 2008. ⟨hal-00731236⟩