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

Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2015) Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations. In: IUTAM Symposium on the Complexity of Nonlinear Waves, Talinn, Estonia

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Official URL: http://dx.doi.org/10.3176/proc.2015.3.08

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.

Item Type:Papers in Conference Proceedings
Additional Information:Document Type:Article
Uncontrolled Keywords:Nonlocal elasticity; Korteweg-de Vries equation; Benjamin-Bona-Mahony equation; Camassa-Holm equation; Fractional Camassa-Holm equation
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:27209
Deposited By:Albert Erkip
Deposited On:05 Oct 2015 16:06
Last Modified:05 Oct 2015 16:06

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