Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

TL;DR

This paper establishes a general criterion for the transverse nonlinear instability of one-dimensional solitary waves in Hamiltonian PDEs, applicable to various equations like KP-I, NLS, Boussinesq, and Zakharov-Kuznetsov.

Contribution

It introduces a unified theoretical framework for analyzing transverse instability in Hamiltonian PDEs based on the sign condition of the linear parts.

Findings

Proves transverse nonlinear instability under specified conditions.

Applicable to multiple important Hamiltonian PDEs.

Provides a basis for further research in wave stability analysis.

Abstract

We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation "have the same sign". Our result applies to the generalized KP-I equation, the Nonlinear Schr\"odinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other contexts.