Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations; pp. 256–262

• Published in: Proceedings of the Estonian Academy of Sciences
• Main Authors: Hüsnü Ata Erbay, Saadet Erbay, Albert Erkip
• Format: Article
• Language: English
• Published: Estonian Academy Publishers 2015-08-01
• Subjects: nonlocal elasticity; korteweg–de vries equation; benjamin–bona–mahony equation; camassa–holm equation; fractional camassa–holm equation.
• Online Access: https://kirj.ee/public/proceedings_pdf/2015/issue_3/Proc-2015-3-256-262.pdf
• ISSN: 1736-6046; 1736-7530

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 Kortewegâde Vries (KdV) equation, the BenjaminâBonaâMahony (BBM) equation, or the CamassaâHolm (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.