Similarity reductions of peakon equations: the $b$-family

The $b$-family is a one-parameter family of Hamiltonian partial differential equations of nonevolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases $b=2$ and $b=3$ (the respective Camassa–Holm and Degasperis–Procesi equations), the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter $b$ it is nonintegrable. After a discussion of traveling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the $b$-family and show that when $b=2$ or $b=3$, this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of $b$ the resulting ordinary differential equation is not of Painlevé type.