Spectral properties of the Frechet derivatives of stratified steady Stokes waves

TL;DR

This paper investigates the spectral characteristics of Frechet derivatives of stratified steady water waves, demonstrating conditions under which subharmonic waves are absent and analyzing eigenvalue implications for wave periodicity.

Contribution

It provides new spectral analysis results for the Frechet derivatives of stratified steady Stokes waves, linking eigenvalues to wave periodicity and stability.

Findings

First eigenvalue is always negative.

If the second eigenvalue is positive, no multi-periodic waves exist nearby.

Spectral properties determine wave stability and periodicity constraints.

Abstract

We consider stratified steady water waves in a two dimensional channel. Our main subject is spectral properties of the Frechet derivatives of steady water Stokes waves. One of main results is the absence of subharmonic water waves in a neighborhood of a Stokes wave. The main assumption is formulated in terms of the eigenvalues of the Frechet derivative evaluated at this wave and considered in the class of periodic solutions of the same period. The first eigenvalue is always negative. We show that if the second eigenvalue is positive then there are no waves with multiple periods in a neighborhood of the Stokes wave.