Derivation of the Camassa-Holm equations for elastic waves

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Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2015) Derivation of the Camassa-Holm equations for elastic waves. Physics Letters A, 379 (12-13). pp. 956-961. ISSN 0375-9601 (Print) 1873-2429 (Online)

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Official URL: http://dx.doi.org/10.1016/j.physleta.2015.01.031


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.

Item Type:Article
Uncontrolled Keywords:Camassa-Holm equation; Fractional Camassa-Holm equation; Nonlocal elasticity; Improved Boussinesq equation; Asymptotic expansions
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:27409
Deposited By:Albert Erkip
Deposited On:08 Dec 2015 14:45
Last Modified:23 Aug 2019 10:25

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