Peakon solutions of Alice-Bob \(b\)-family equation and Novikov equation (Q2425062)

Summary: By requiring \(B = \widehat{P}_s \widehat{T}_d A\) and substituting \(u = A + B\) into the \(b\)-family equation and Novikov equation, we can obtain Alice-Bob peakon systems, where \(\widehat{P}_s\) and \(\widehat{T}_d\) 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.