Derivation of the Camassa-Holm equations for elastic waves

TL;DR

This paper formally derives the Camassa-Holm and fractional Camassa-Holm equations as models for small-amplitude long waves in nonlocally and nonlinearly elastic media, extending their application beyond shallow water waves.

Contribution

It provides a novel derivation of the Camassa-Holm equations for elastic media, including fractional variants, using asymptotic expansions and integral-type constitutive relations.

Findings

Derivation of the Camassa-Holm equation from the improved Boussinesq equation.

Extension to fractional Camassa-Holm equations with fractional kernel functions.

Application of standard asymptotic techniques to elastic wave equations.

Abstract

In this paper we provide a formal derivation of both the Camassa-Holm equation and the fractional Camassa-Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic medium. We first show that the equation of motion for the nonlocally and nonlinearly elastic medium reduces to the improved Boussinesq equation for a particular choice of the kernel function appearing in the integral-type constitutive relation. We then derive the Camassa-Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters tend to zero independently. Our approach follows mainly the standard techniques used widely in the literature to derive the Camassa-Holm equation for shallow water waves. The case where the Fourier transform of the kernel function has fractional powers is also considered and the fractional Camassa-Holm equation is derived using the asymptotic expansion technique.