Modulational instability in nonlinear nonlocal equations of regularized long wave type

TL;DR

This paper analyzes the stability of periodic traveling waves in nonlocal nonlinear equations of BBM and Boussinesq types, deriving instability criteria and extending previous results for KdV equations.

Contribution

It extends recent stability analyses to nonlocal equations of BBM and Boussinesq types, deriving modulational instability indices based on wave number.

Findings

Small periodic waves in BBM are unstable to long wavelength perturbations above a critical wave number.

Small periodic waves in Boussinesq are stable to square integrable perturbations.

Derived explicit modulational instability indices as functions of wave number.

Abstract

We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin- Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.