Davey–Stewartson and Related Systems

Abstract

In this chapter we first recall the derivation of Davey–Stewartson systems in the context of water waves. Actually the Davey–Stewartson systems are singular limits of more general, “universal” systems, the Benney–Roskes/Zakharov–Rubenchik systems that describe the interactions of short and long waves. Then we survey the rigorous results obtained by PDE and Inverse Scattering techniques for the Davey–Stewartson and the related Ishimori systems. We also comment on the three-dimensional Davey–Stewartson systems and on the “hyperbolic” nonlinear Schrödinger equations that are not integrable but have physical relevance and lead to many open questions. We also present various numerical simulations aiming to illustrate the aforementioned results and to propose conjectures, in particular in the non-integrable case.