Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations

TL;DR

This paper numerically investigates the stability of line solitons in two-dimensional Zakharov-Kuznetsov equations, confirming theoretical predictions and revealing velocity-dependent stability and instability behaviors.

Contribution

It provides the first detailed numerical analysis of line soliton stability across different nonlinear regimes of the Zakharov-Kuznetsov equations.

Findings

Stable line solitons below a critical speed in the subcritical case.

Instability and lump formation at higher velocities.

Velocity-dependent stability in critical and supercritical cases.

Abstract

We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov-Kuznetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the $L^2$-critical and supercritical cases but subcritical for the 1D generalized Korteweg-de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time.