Linear versus nonlinear dissipation for critical NLS equation

The focusing nonlinear Schrِdinger equation at critical dimension is considered in the presence of a weak damping, either linear or involving a power of the wave intensity. Special attention is paid to the case of a diffusive damping term, that describes Landau damping for a low-frequency Alfvén wave subject to filamentation instability. When compared with the nonlinear damping classically used to model multi-photon absorption in nonlinear optics, significant differences concerning the wave energy dissipation are numerically observed and interpreted by means of an asymptotic ODE model resulting from a modulation analysis of the solution near collapse. It is in particular shown that whereas for a nonlinear damping, dissipation is almost totally concentrated in the collapse events, it remains sizeable while the wave defocuses when the damping is linear.