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

Abstract

Abstract. We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of asymptotic wave models describing the unidirectional propagation of small-but-finite amplitude long waves. When the kernel function is the well-known exponential kernel, the asymptotic description is provided by the Korteweg-de Vries (KdV) equation, the Benjamin-Bona-Mahony (BBM) equation, or the Camassa-Holm (CH) equation. When the Fourier transform of the kernel function has fractional powers, it turns out that fractional forms of these equations describe unidirectional propagation of the waves. We then compare the exact solutions of the KdV equation and the BBM equation with the numerical solutions of the nonlocal model. We observe that the solution of the nonlocal model is well approximated by associated solutions of the KdV equation and the BBM equation over the time interval considered.

Nonlinear Wave Modulation in Nanorods Based on Nonlocal Elasticity Theory by Using Multiple-Scale Formalism

“…Furthermore, Akgun and Demiray [33] studied the modulation of non-linear axial and transverse waves in a thin elastic tube filled with fluid and obtained nonlinear Schrödinger equation which corresponds to two nonlinear equations related to the axial and transverse motions of the tube material. Erbay, Erbay, and Erkip [34] studied a unidirectional wave motion in a nonlocally and nonlinearly elastic medium. Duruk, Erbay, and Erkip [35] investigated the blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations.…”

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

“…In a recent study [11], the CH equation has been also derived as an appropriate model for the unidirectional propagation of long elastic waves in an infinite, nonlocally and nonlinearly elastic medium (see also [12]). The constitutive behavior of the nonlocally and nonlinearly elastic medium is described by a convolution integral (we refer the reader to [9,10] for a detailed description of the nonlocally and nonlinearly elastic medium) and in the case of quadratic nonlinearity the one-dimensional equation of motion reduces to the nonlocal equation given in (2).…”