Walter Strauss

Brown University

Modulational instability of Stokes waves

This is an exposition of joint work Huy Q. Nguyen. We prove that the classical small-amplitude irrotational steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. However, a rigorous proof has never been found except in the case of finite depth. We provide a completely different and self-contained proof of both the finite and infinite depth cases. Our linearization retains the physical variables and is compatible with energy estimates for the nonlinear problem.