Abstract
By requiring B=P^sT^dA and substituting u=A+B into the b-family equation and Novikov equation, we can obtain Alice-Bob peakon systems, where P^s and T^d are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively. The nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived from the Alice-Bob b-family equations by choosing different parameters. Some new types of interesting solutions are solved including explicit one-peakons, two-peakons, and N-peakons solutions.
Highlights
By requiring B = PsTdA and substituting u = A + B into the b-family equation and Novikov equation, we can obtain AliceBob peakon systems, where Ps and Td are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively
The nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived from the AliceBob b-family equations by choosing different parameters
Which was proposed firstly by Degasperis and Procesi. This equation can be considered as a model for shallow water wave and satisfied the asymptotic integrability to third order [17]; in the complete integrability of the system it was proved because of the existence of a Lax pair and a bi-Hamiltonian structure based on a third order spectral problem [18]
Summary
By requiring B = PsTdA and substituting u = A + B into the b-family equation and Novikov equation, we can obtain AliceBob peakon systems, where Ps and Td are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively. The most interesting feature of the CH equation is that it admits peaked peakon solutions [1, 7]. Every member of the b-family equation has peakon solutions for each b.
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