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 Camassa–Holm and Degasperis–Procesi equations, respectively), 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 Painlev´e III equation, while for all other choices of b the resulting ordinary differential equation is not of Painlev´e type.