The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity

TL;DR

This paper analyzes the Benjamin-Feir modulational instability in a broad class of KdV-like equations with general dispersion and monomial nonlinearities, revealing a universal spectral pattern associated with instability.

Contribution

It provides a rigorous spectral characterization of small-amplitude waves in KdV-like equations, demonstrating the universal formation of a figure-eight spectral pattern during modulational instability.

Findings

Spectrum near the origin forms a closed figure-eight pattern when unstable

Complete spectral characterization for small-amplitude traveling waves

Applicable to a wide class of dispersive nonlinear wave equations

Abstract

Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For small-amplitude traveling wave solutions, we provide a complete characterization of the spectrum near the origin of the linear operator obtained from linearizing about periodic traveling waves. We prove rigorously that, when the modulational instability is present, the spectrum connected to the origin consists of curves that invariably form a closed figure-eight pattern.