An estimate for the Morse index of a Stokes wave

TL;DR

This paper provides a quantitative estimate for the Morse index of Stokes waves, which are steady periodic water waves, linking the boundedness of Morse indices to the proximity of singular solutions.

Contribution

It introduces a quantitative version of the relation between Morse index bounds and the distance from singular Stokes waves.

Findings

Establishes a quantitative estimate for the Morse index of Stokes waves.

Shows that bounded Morse indices imply a certain distance from singular solutions.

Provides insights into the stability and bifurcation structure of Stokes waves.

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. It is well known 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. The paper presents a quantitative variant of this result.