Transverse instability of the CH-KP-I equation

TL;DR

This paper demonstrates the transverse instability of line solitary waves in the two-dimensional CH-KP-I equation, extending stability analysis to a more complex nonlinear PDE with periodic transverse perturbations.

Contribution

It establishes the linear and nonlinear transverse instability of solitary waves in the CH-KP-I equation, adapting existing frameworks to handle high nonlinearity.

Findings

Proves linear instability of solitary waves.

Shows nonlinear effects are dominated by linear instability.

Extends instability results to a high nonlinearity setting.

Abstract

The Camassa-Holm-Kadomtsev-Petviashvili-I equation (CH-KP-I) is a two dimensional generalization of the Camassa-Holm equation (CH). In this paper, we prove transverse instability of the line solitary waves under periodic transverse perturbations. The proof is based on the framework of the paper written by Rousset and Tzvetkov. Due to the high nonlinearity, our proof requires necessary modification. Specifically, we first establish the linear instability of the line solitary waves. Then through an approximation procedure, we prove that the linear effect actually dominates the nonlinear behavior.