Derivation of the Camassa-Holm equations for elastic waves

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.

arXiv comment15 pages

References in corpus (3)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Variational derivation of the Camassa-Holm shallow water equation

Cited by in corpus (6)

• The Generalized Fractional Benjamin-Bona-Mahony Equation: Analytical and Numerical Results
• On the Stability of Solitary Wave Solutions for a Generalized Fractional Benjamin-Bona-Mahony Equation
• The Cauchy problem for fractional Camassa-Holm equation in Besov space
• The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
• On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
• Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations