Modulational Instability in Equations of KdV Type

TL;DR

This paper reviews recent mathematical progress on the instability of slowly modulated nonlinear wave trains in dispersive media described by KdV-type equations, building on Whitham's modulation theory from the 1970s.

Contribution

It provides a summary of recent advances in understanding the instability phenomena of modulated wave trains in KdV-type equations.

Findings

Analysis of modulational instability in KdV equations

Rigorous justification of Whitham's modulation predictions

Identification of conditions leading to wave train instability

Abstract

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.