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Invariants-preserving integration of the modified Camassa–Holm equation

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Abstract

McLachlan and Zhang (J Diff Equ 246:3241–3259, 2009) proposed a family of generalized versions of the well-known Camassa–Holm (CH) equation, which they called “modified Camassa–Holm (mCH) equation.” The mCH family has one striking feature in common: each of them is equipped with two invariants, “momentum” and “energy,” and forms a Hamiltonian equation with respect to the energy. In this paper, we construct three numerical integrators that preserve one or two of the invariants making use of the Hamiltonian structure, and give theoretical analyses of the schemes. We also present several numerical examples, which not only confirm the effectiveness of the schemes but also suggest a new insight that some solutions of the mCH can behave like solitons.

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-0033, Japan

    Yuto Miyatake & Takayasu Matsuo

  2. Cybermedia Center, Osaka University, Machikaneyama 1-32, Toyonaka, Osaka, 560-0043, Japan

    Daisuke Furihata

Authors
  1. Yuto Miyatake
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  2. Takayasu Matsuo
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  3. Daisuke Furihata
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Corresponding author

Correspondence to Yuto Miyatake.

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Miyatake, Y., Matsuo, T. & Furihata, D. Invariants-preserving integration of the modified Camassa–Holm equation. Japan J. Indust. Appl. Math. 28, 351–381 (2011). https://doi.org/10.1007/s13160-011-0043-z

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  • DOI: https://doi.org/10.1007/s13160-011-0043-z

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