Periodically modulated solitary waves of the CH-KP-I equation

TL;DR

This paper proves the existence of periodically modulated solitary waves in the CH-KP-I equation, showing they bifurcate from line solitary waves via a dynamical systems approach and the Lyapunov-Iooss theorem.

Contribution

It demonstrates the existence of new steady solutions with mixed spatial properties for the CH-KP-I equation using bifurcation analysis.

Findings

Existence of steady, partially periodic solitary waves in the CH-KP-I equation.

Bifurcation from line solitary waves via a dimension-breaking bifurcation.

Application of the Lyapunov-Iooss theorem to prove solutions.

Abstract

We consider the CH-KP-I equation. For this equation we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave solutions, i.e. solitary wave solutions to the Camassa-Holm equation, in a dimension-breaking bifurcation. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying the Lyapunov-Iooss theorem.