Similarity reductions of peakon equations: integrable cubic equations

We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa–Holm (mCH) equation and Novikov’s equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov’s equation to a negative mKdV flow and a negative Sawada–Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second order and second degree, while for Novikov’s equation the reduction is a particular case of Painlevé V. Furthermore, we show that each of these two different Painlevé-type equations is related to the particular cases of Painlevé III that arise from analogous similarity reductions of the Camassa–Holm and the Degasperis–Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlevé III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.