Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study

TL;DR

This numerical study examines the transverse stability of cnoidal waves in KP equations, revealing amplitude-dependent stability in KP-I and universal stability in KP-II, aligning with recent analytical findings.

Contribution

It provides the first comprehensive numerical analysis of transverse stability of cnoidal waves in KP equations, confirming analytical predictions and exploring amplitude effects.

Findings

KP-I cnoidal waves are stable at small amplitudes

Stability threshold exists for KP-I waves

KP-II cnoidal waves are stable for all amplitudes

Abstract

We numerically investigate transverse stability and instability of so-called cnoidal waves, i.e., periodic traveling wave solutions of the Korteweg-de Vries equation, under the time-evolution of the Kadomtsev-Petviashvili equation. In particular, we find that in KP-I small amplitude cnoidal waves are stable (at least for spatially localized perturbations) and only become unstable above a certain threshold. In contrast to that, KP-II is found to be stable for all amplitudes, or, equivalently, wave speeds. This is in accordance with recent analytical results for solitary waves given in \cite{RT1, RT2}.