# On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system

## TL;DR

This paper investigates the relationship between Eulerian and Lagrangian variables in the two-component Camassa-Holm system, establishing criteria for their convergence and methods to approximate solutions across wave breaking.

## Contribution

It provides a rigorous analysis of the equivalence of convergence in Eulerian and Lagrangian coordinates and introduces a way to approximate solutions past wave breaking.

## Key findings

- Convergence in Eulerian coordinates is equivalent to convergence in Lagrangian coordinates.
- Criteria for convergence are identified.
- Global conservative solutions can be approximated by smooth solutions without wave breaking.

## Abstract

The Camassa-Holm equation and its two-component Camassa-Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave breaking, it is common to reformulate the original equation given in Eulerian coordinates, into a system of ordinary differential equations in Lagrangian coordinates. It is of considerable interest to study the stability of solutions and how this is manifested in Eulerian and Lagrangian variables. We identify criteria of convergence, such that convergence in Eulerian coordinates is equivalent to convergence in Lagrangian coordinates. In addition, we show how one can approximate global conservative solutions of the scalar Camassa-Holm equation by smooth solutions of the two-component Camassa-Holm system that do not experience wave breaking.