Transverse instability of plane wave soliton solutions of the Novikov-Veselov equation

TL;DR

This paper investigates the transverse stability of plane wave soliton solutions in the Novikov-Veselov equation, demonstrating through numerical simulations that these solutions are unstable under transverse perturbations.

Contribution

It introduces a hybrid semi-implicit spectral numerical scheme and applies it to analyze the transverse stability of solitons in the NV equation, revealing their instability.

Findings

Plane wave solitons are unstable to transverse perturbations.

A new hybrid numerical scheme was developed for nonlinear PDEs.

Numerical simulations confirmed the instability of solitons under transverse disturbances.

Abstract

The Novikov-Veselov (NV) equation is a dispersive (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. This paper considers the stability of plane wave soliton solutions of the NV equation to transverse perturbations. To investigate the behavior of the perturbations, a hybrid semi-implicit/spectral numerical scheme was developed, applicable to other nonlinear PDE systems. Numerical simulations of the evolution of transversely perturbed plane wave solutions are presented. In particular, it is established that plane wave soliton solutions are not stable for transverse perturbations.