Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

DOI

10.1080/14029251.2016.1199493How to use a DOI?

Keywords

Generalized Camassa-Holm equation; modified Camassa-Holm equation; fractional Camassa-Holm equation; improved Boussinesq equation; asymptotic expansions

Abstract

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.

Copyright

© 2016 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - H. A. Erbay
AU  - S. Erbay
AU  - A. Erkip
PY  - 2021
DA  - 2021/01/06
TI  - Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 314
EP  - 322
VL  - 23
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2016.1199493
DO  - 10.1080/14029251.2016.1199493
ID  - Erbay2021
ER  -