On the first bifurcation of solitary waves

TL;DR

This paper investigates the bifurcation phenomena of solitary water waves with vorticity in a finite-depth channel, establishing the existence and simplicity of the first bifurcation point from laminar flow to extreme waves.

Contribution

It proves the existence of a bifurcation point on solitary wave branches and shows that the first bifurcation occurs at a simple eigenvalue, advancing understanding of wave stability.

Findings

Existence of a bifurcation point on solitary wave branches

First bifurcation occurs at a simple eigenvalue

Bifurcation point always exists on the wave branch

Abstract

We consider solitary water waves on the vorticity flow in a two-dimensional channel of finite depth. The main object of study is a branch of solitary waves starting from a laminar flow and then approaching an extreme wave. We prove that there always exists a bifurcation point on such branches. Moreover, the crossing number of the first bifurcation point is 1, i.e. the bifurcation occurs at a simple eigenvalue.