Variational derivation of two-component Camassa-Holm shallow water system

TL;DR

This paper derives the two-component Camassa-Holm system using a variational approach, showing its connection to shallow water wave equations and highlighting its integrability and approximation properties.

Contribution

It introduces a variational derivation of the two-component Camassa-Holm system, linking it to shallow water wave dynamics and demonstrating its integrability.

Findings

Derivation of the two-component Camassa-Holm system via variational methods

Connection of the system to shallow water wave equations

Identification of the system as an integrable model

Abstract

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa-Holm system (1). We show that the two-component Camassa-Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.