Numerical study of elliptic-hyperbolic Davey-Stewartson system: Dromions simulation and blow-up

This paper is devoted to the numerical approximation of the elliptic-hyperbolic form of the Davey-Stewartson equations. A well-suited finite differences scheme that preserves the energy is derived. This scheme is tested to compute the famous dromion 1-1 and dromion 2-2 solutions. The accuracy of Crank-Nicolson scheme is discussed and it is shown that it induces a phase error. The qualitative behaviour of the solutions is then studied; in particular the influence of the initial datum and of the various parameters is pointed out. Finally, numerical experiments show the existence of blow-up solutions.