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On the Equivalence of Eulerian and Lagrangian Variables for the Two-Component Camassa–Holm System

In: Current Research in Nonlinear Analysis

Author

Listed:
  • Markus Grasmair

    (NTNU Norwegian University of Science and Technology)

  • Katrin Grunert

    (NTNU Norwegian University of Science and Technology)

  • Helge Holden

    (NTNU Norwegian University of Science and Technology)

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.

Suggested Citation

  • Markus Grasmair & Katrin Grunert & Helge Holden, 2018. "On the Equivalence of Eulerian and Lagrangian Variables for the Two-Component Camassa–Holm System," Springer Optimization and Its Applications, in: Themistocles M. Rassias (ed.), Current Research in Nonlinear Analysis, pages 157-201, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-89800-1_7
    DOI: 10.1007/978-3-319-89800-1_7
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