Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse instability of its periodic traveling wave solutions

TL;DR

This paper develops Whitham modulation equations for the Zakharov-Kuznetsov equation to analyze the transverse stability of its periodic traveling waves, revealing their linear instability and providing explicit growth rates validated by numerical and analytical methods.

Contribution

It derives the Whitham modulation equations for the Zakharov-Kuznetsov equation and applies them to determine the transverse stability of periodic traveling waves, a novel analysis in this context.

Findings

All periodic traveling wave solutions are linearly unstable.

Explicit growth rates for the most unstable wave numbers are obtained.

Predictions are validated by numerical eigenvalue analysis and analytical calculations.

Abstract

We derive the Whitham modulation equations for the Zakharov-Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all such solutions are linearly unstable, and we obtain an explicit expression for the growth rate of the most unstable wave numbers. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. Finally, we calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches.