The nonlinear Schroedinger equation appears in several physical or biological models and enjoys various interesting mathematical properties. In particular, this equation admits standing waves, i.e. solutions whose profile remains unchanged under the evolution in time. For the physical applications, it is essential to know the stability properties of waves with respect to perturbations. It is also important to understand the nature of any potential instability. The talk will start by a review of classical results on nonlinear Schroedinger equations and their standing waves. Then we will discuss stability and instability issues. Finally, we will present recent stability/instability results on a nonlinear Schroedinger equation perturbed by a Dirac potential.
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