Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation

TL;DR

This paper analyzes the spectral transverse instability of periodic traveling waves in the generalized KP equation, deriving a geometric index to predict instability and calculating it for specific elliptic solutions.

Contribution

It introduces a geometric orientation index for spectral instability analysis applicable to a broad class of periodic waves in the generalized KP framework.

Findings

Derived a sufficient condition for transverse instability using the orientation index.

Calculated the index explicitly for elliptic solutions of KdV and mKdV equations.

Applicable to waves with minimal smoothness and convexity assumptions.

Abstract

In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.