Proof of modulational instability of Stokes waves in deep water

Abstract: It is proven that small-amplitude 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.

It has never been proven rigorously except in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases. Our linearization retains the physical variables and is compatible with energy estimates for the nonlinear problem.

This is joint work with Walter Strauss (Brown University).

analysis of PDEs

Audience: researchers in the topic

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