Title: On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system,
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.