A simple criterion for transverse linear instability of nonlinear waves

Abstract

We prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain Ω×R⊂Rn×R. For stationary solutions depending upon x∈Ω only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon (x,y)∈Ω×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.