Holm, Darryl D., Hone, Andrew N.W. (2005) A class of equations with peakon and pulson solutions (with an Appendix by Harry Braden and John ByattSmith). Journal of Nonlinear Mathematical Physics, 12 (Sup.1). pp. 380394. ISSN 14029251. (KAR id:41497)
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Official URL http://staff.www.ltu.se/~norbert/home_journal/elec... 
Abstract
We consider a family of integrodifferential 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 CamassaHolm equation results, while the DegasperisProcesi equation is obtained from the peakon kernel with b=3. Although these two cases are integrable, generically the corresponding integroPDE is nonintegrable. 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 skewsymmetric antiderivative of g. The nonlocal bracket reduces to a noncanonical Poisson bracket for the peakon dynamical system, for any value of b?1.
Item Type:  Article 

Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Q Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations Q Science > QA Mathematics (inc Computing science) > QA801 Analytic mechanics 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Andrew Hone 
Date Deposited:  21 Jun 2014 22:46 UTC 
Last Modified:  16 Feb 2021 12:54 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/41497 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
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