Numerical study of the semiclassical limit of the Davey–Stewartson II equations

Abstract

We present the first detailed numerical study of the semiclassical limit of the Davey–Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing initial data with a single hump. The formal limit of these equations for vanishing semiclassical parameter ϵ, the semiclassical equations, is numerically integrated up to the formation of a shock. The use of parallelized algorithms allows one to determine the critical time tc and the critical solution for these 2 + 1-dimensional shocks. It is shown that the solutions generically break in isolated points similarly to the case of the 1 + 1-dimensional cubic nonlinear Schrödinger equation, i.e., cubic singularities in the defocusing case and square root singularities in the focusing case. For small values of ϵ, the full Davey–Stewartson II equations are integrated for the same initial data up to the critical time tc. The scaling in ϵ of the difference between these solutions is found to be the same as in the 1 + 1 dimensional case, proportional to ϵ2/7 for the defocusing case and proportional to ϵ2/5 in the focusing case. We document the Davey–Stewartson II solutions for small ϵ for times much larger than the critical time tc. It is shown that zones of rapid modulated oscillations are formed near the shocks of the solutions to the semiclassical equations. For smaller ϵ, the oscillatory zones become smaller and more sharply delimited to lens-shaped regions. Rapid oscillations are also found in the focusing case for initial data where the singularities of the solution to the semiclassical equations do not coincide. If these singularities do coincide, which happens when the initial data are symmetric with respect to an interchange of the spatial coordinates, no such zone is observed. Instead the initial hump develops into a blow-up of the L∞ norm of the solution. We study the dependence of the blow-up time on the semiclassical parameter ϵ.