A class of equations with peakon and pulson solutions (with an Appendix by Harry Braden and John Byatt-Smith)

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 $b\ne 1$.

arXiv commentContribution to volume of Journal of Nonlinear Mathematical Physics in honour of Francesco Calogero

Cited by in corpus (8)

• A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations
• A general family of multi-peakon equations and their properties
• Stability of smooth solitary waves in the b-Camassa-Holm equation
• The inverse spectral problem for the discrete cubic string
• Stability of stationary solutions for nonintegrable peakon equations
• Fireball sheath instability
• Dynamics of conservative peakons in a system of Popowicz
• Similarity reductions of peakon equations: the $b$-family