A simple criterion for transverse linear instability of nonlinear waves

We prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain Omega x R subset of R-n x R. For stationary solutions depending upon x is an element of Omega only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon (x, y) is an element of Omega x R. Starting with a formulation of the PDE as a dynamical system in the transverse direction y, we give sufficient conditions for transverse linear instability. We apply the general result to the Davey-Stewartson equations, which arise as modulation equations for three-dimensional water waves. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS.