An estimate for the Morse index of a Stokes wave and the Khuri-Martin-Wu conjecture

Abstract

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler-Lagrange equation of a certain functional.

This allows one to define the Morse index of a Stokes wave. B. Buffoni, E.N. Dancer, and J.F. Toland (2000) proved, using earlier results by P.I. Plotnikov (1991), that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one.

I will present a quantitative version of this result and explain its connection to the proof of a conjecture by N.N. Khuri, A. Martin and T.T. Wu (2001) on the number of bound states in two dimensions.