A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

DOI

10.2991/jnmp.2005.12.s1.31How to use a DOI?

Abstract

We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b = 2 and g is the peakon kernel (i.e. g(x) = exp(-|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b = 3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However, for b = 2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g. The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b = 1.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Darryl D. Holm
AU  - Andrew N.W. Hone
PY  - 2005
DA  - 2005/01/01
TI  - A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)
JO  - Journal of Nonlinear Mathematical Physics
SP  - 380
EP  - 394
VL  - 12
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.s1.31
DO  - 10.2991/jnmp.2005.12.s1.31
ID  - Holm2005
ER  -