Similarity reductions of peakon equations: the -family
arXiv:2201.00265 · doi:10.1134/S0040577922080104
Abstract
The -family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary 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 (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 it is non-integrable. After a discussion of travelling 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 -family, and show that when or this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of the resulting ordinary differential equation is not of Painlevé type.
Some additional minor typos corrected; additional remark about scaling similarity reduction for b=0 added