Landau Damping in Nonlinear SCHRÖDINGER Equations

By using the Wigner transform, the modulational instability analysis for a wide class of nonlinear Schrödinger equations describing different physical situations is carried out in phase space. In this framework, a kinetic-like description similar to the one based on the Vlasov equation which is used for describing the collective longitudinal dynamics of charged-particle bunches in accelerating machines is provided. In particular, the modulational instability (MI) corresponds to the usual coherent instability of the particle bunch. The main result od this analysis is the prediction of the phenomenon of Landau damping (LD) which seems to be in competition with the MI. This approach provides stability charts fully similar to the ones describing charged-particle beams in accelerating machines. Recent investigations on MI and LD in nonlinear Schrödinger equations including memory-effect terms are reviewed and new results are put forward.