Localised coherent solutions of the DSI and DSII equations - a numerical study

We implement several numerical methods for computing the solution of the Cauchy problems for two well-known integrable equations in two space dimensions, known as the Davey–Stewartson I and II equations. Most of our computations confirm earlier theoretical predictions, including a recent result implying the instablity of a class of special solutions of the DSII equation. These solutions, known as lumps, had been previously considered natural candidates as coherent asymptotic structures of the evolution. In contrast to all other cases, when the initial condition has sufficiently large energy no global existence result is known for the DSII equation, in the focusing regime. Our preliminary computations indicate in this case the possibility that the solution blows up, hence that no global existence result can hold.