Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two

Abstract

We investigate numerically the two-dimensional travelling waves of the nonlinear Schrodinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy–momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross–Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and “one-dimensional spreading” phenomena.