Transverse Spectral Instabilities in Konopelchenko-Dubrovsky Equation

TL;DR

This paper investigates the transverse spectral stability of small-amplitude periodic traveling waves in the (2+1)-dimensional Konopelchenko-Dubrovsky equation, revealing conditions for stability and instability under various perturbations.

Contribution

It provides a detailed analysis of the transverse spectral stability of these waves, including stability criteria for different types of perturbations and implications for related equations.

Findings

Long-wavelength perturbations cause transverse instability.

Short-wavelength or mean-zero periodic perturbations preserve stability.

Results apply to special cases like KP-II and mKP-II equations.

Abstract

We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two-dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean-zero periodic or square-integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation - namely, KP-II and mKP-II equations.