On a class of nonlocal nonlinear Schrödinger equations and wave collapse

A similar type of nonlocal nonlinear Schrödinger (NLS) system arises in both water waves and nonlinear optics. The nonlocality is due to a coupling between the first harmonic and a mean term. These systems are termed nonlinear Schrödinger with mean or NLSM systems. They were first derived in water waves by Benney-Roskes and later by Davey-Stewartson. Subsequently similar equations were derived and found to be fundamental systems in quadratically nonlinear optical media. Wave collapse can occur in these systems. The collapse structure and the role of the ground state in the collapse process are studied. There are similarities to the well-known collapse mechanism associated with classical NLS system. Numerical simulations show that NLSM collapse occurs with a quasi selfsimilar profile that is a modulation of the corresponding ground-state. Further, it is found that NLSM collapse can be arrested by adding small nonlinear saturation.