On small-amplitude asymmetric water waves

TL;DR

This paper extends existing methods to analyze the existence of small-amplitude asymmetric water waves, providing necessary and sufficient conditions, and applies these to specific equations to determine when such waves do or do not exist.

Contribution

It generalizes a method for proving asymmetric solutions, offering a unified approach applicable to various water wave models and establishing criteria for their existence.

Findings

Nonexistence of small-amplitude waves in certain equations

Conditions for the existence of asymmetric solutions

Potential for asymmetric waves in finite depth models

Abstract

We generalize the method used by M{\ae}hlen & Seth [17] used to prove the existence of small-amplitude asymmetric solutions to the capillary-gravity Whiham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetry solutions for this class of equations. The result is then applied to an infinite depth capillary-gravity Whitham equation and an infinite depth capillary-gravity Babenko equation to show the nonexistence of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary-gravity Babenko equation.