Wave collapse in a class of nonlocal nonlinear Schrِdinger equations

Wave collapse is investigated in nonlocal nonlinear Schrödinger (NLS) systems, where a nonlocal potential is coupled to an underlying mean term. Such systems, here referred to as NLS-Mean (NLSM) systems, are also known as Benney–Roskes or Davey–Stewartson type and they arise in studies of shallow water waves and nonlinear optics. The role of the ground-state in global-existence theory is elucidated. The ground-state is computed using a fixed-point method. The critical-powers for collapse predicted by the Virial Theorem, global-existence theory, and by direct numerical simulations of the NLSM are found to be in good agreement with each other for a wide range of parameters. The ground-state profile in the water-wave case is found to be generically narrower along the direction of propagation, whereas in the optics case it is generically wider along the axis of linear polarization. In addition, numerical simulations show that NLSM collapse occurs with a quasi self-similar profile that is a modulation of the corresponding astigmatic ground-state, which is in the same spirit as in NLS collapse. It is also found that NLSM collapse can be arrested by small nonlinear saturation.