Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity

TL;DR

This paper derives the Camassa-Holm shallow water equation for flows with non-zero vorticity using variational methods and small-parameter expansions, providing a unified approach for irrotational and rotational flows.

Contribution

It introduces a variational derivation of the Camassa-Holm equation applicable to flows with non-zero vorticity, extending previous irrotational models.

Findings

Derivation of the Camassa-Holm equation for non-zero vorticity flows.

Unified framework for irrotational and rotational shallow water waves.

Application of variational methods to water wave modeling.

Abstract

We describe the physical hypotheses underlying the derivation of an approximate model of water waves. For unidirectional surface shallow water waves moving over an irrotational flow as well as over a non-zero vorticity flow, we derive the Camassa-Holm equation by an interplay of variational methods and small-parameter expansions.