The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction

Abstract

The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.