Superharmonic instability for regularized long-wave models

TL;DR

This paper investigates the spectral stability of periodic traveling waves in regularized long-wave models, revealing a novel instability phenomenon where spectrum extends into infinity off the imaginary axis, supported by analytical and numerical analysis.

Contribution

It introduces the first detailed spectral analysis of regularized long-wave models, uncovering a new instability phenomenon not observed in classical models.

Findings

Spectrum can extend into infinity off the imaginary axis.

Spectral behavior varies with wave parameters.

Numerical experiments confirm analytical predictions.

Abstract

We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney--Luke, and Benjamin--Bona--Mahony equations. Of particular interest is a striking new instability phenomenon -- spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg--de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We carry out asymptotic spectral analysis to short wavelength perturbations, distinguishing whether the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments to corroborate our analytical findings.