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 that 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.

Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations

“…We underline that dispersive wave propagation is due to the internal structure of the medium but not due to the existence of the boundaries. This feature of the model differentiates the asymptotic derivation of the CH equation for elastic waves in [9] from previous work in the literature. For a derivation based on the dispersive wave propagation resulting from the existence of the boundaries, we refer the reader to [14,15] where CH-type equations are derived asymptotically for elastic waves.…”

“…A few of the most commonly used ones are listed in [6]. In [9], for two particular forms of the kernel function β and the quadratic nonlinearity g(u) = u 2 , various asymptotic equations describing unidirectional wave propagation of small-but-finite amplitude long waves have been derived. Therefore, in the remaining part of this study, we consider only the quadratic nonlinearity and focus on the two particular kernel functions: an exponential kernel and a fractional-type kernel function.…”

“…The second type of the kernel function considered in [9] is the fractional-type kernel function defined by β (ξ ) = (1 +…”

“…In this section we briefly review the unidirectional wave equations derived in [9] for small-but-finite amplitude long waves propagating in the one-dimensional nonlocal media governed by (4) and (5). From a wave propagation point of view, both (4) and (5) are dispersive wave equations.…”

“…In subsequent papers [7,8], global existence and nonexistence of solutions of the initial-value problems posed for various generalizations of the model were investigated. In a very recent study [9], using an asymptotic expansion technique, the asymptotic equations governing unidirectional wave propagation of small-butfinite amplitude long waves in the nonlinear nonlocal elastic medium were derived. The asymptotic equations derived include the Korteweg-de Vries (KdV) equation [10], the Benjamin-Bona-Mahony (BBM) equation [11], and the Camassa-Holm (CH) equation [12] as well as fractional generalizations of these equations.…”