Transverse spectral stability of small periodic traveling waves for the KP equation

TL;DR

This paper investigates the spectral stability of small periodic traveling waves in the KP equation, revealing instability in KP-I and stability in KP-II under certain perturbations, advancing understanding of wave behavior in these systems.

Contribution

It provides the first detailed spectral stability analysis of small periodic waves in the KP equation, distinguishing stability properties between KP-I and KP-II.

Findings

KP-I waves are spectrally unstable to both periodic and localized perturbations.

KP-II waves are spectrally stable to periodic perturbations with long transverse wavelengths.

Abstract

The Kadomtsev-Petviashvili (KP) equation possesses a four-parameter family of one-dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two-dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one-dimensional traveling wave, or non-periodic (localized or bounded). We focus on the so-called KP-I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP-II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations which are periodic in the direction of propagation, and have long wavelengths in the transverse direction.