Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2004-12-10
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Contribution to volume of Journal of Nonlinear Mathematical Physics in honour of Francesco Calogero
Scientific paper
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 $b\neq 1$.
Holm Darryl D.
Hone Andrew N. W.
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