Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

TL;DR

This paper derives and analyzes Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity, classifying solutions and examining stability properties.

Contribution

It introduces explicit periodic solutions, derives modulation equations via an averaged variational principle, and studies their hyperbolicity and stability.

Findings

Periodic solutions classified in terms of Jacobi elliptic functions.

Loss of hyperbolicity correlates with modulational instability.

Numerical verification of instability onset and additional short-wave instabilities.

Abstract

A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink, and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic traveling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic speeds is shown to lead to modulational instability of periodic traveling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic traveling waves and initial value problems that also reveal additional short-wave instabilities.