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

TL;DR

This paper derives new generalized Camassa-Holm equations from Boussinesq-type equations using asymptotic expansion, introducing power-type nonlinearities and fractional dispersive terms, expanding the mathematical modeling of shallow water waves.

Contribution

The paper introduces novel generalized Camassa-Holm equations with power-type nonlinearities and fractional dispersive effects, not previously documented in literature.

Findings

Derived generalized equations reduce to classical CH in specific limits.

New equations incorporate fractional dispersive terms.

Comparison shows these equations are unique in literature.

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.