On negative eigenvalues of the spectral problem for water waves of highest amplitude

TL;DR

This paper investigates the spectral properties of steady water waves with extreme form, revealing the existence of arbitrarily large negative eigenvalues and providing their asymptotic behavior.

Contribution

It demonstrates that the spectral problem for highest amplitude water waves has infinitely many large negative eigenvalues and derives their asymptotics, advancing understanding of water wave stability.

Findings

Spectrum contains arbitrarily large negative eigenvalues.

Negative eigenvalues are simple.

Asymptotic behavior of these eigenvalues is characterized.

Abstract

We consider a spectral problem associated with steady water waves of extreme form on the free surface of a rotational flow. It is proved that the spectrum of this problem contains arbitrary large negative eigenvalues and they are simple. Moreover, the asymptotics of such eigenvalues is obtained.