Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations

We construct Bäcklund transformations (BT) and several classes of exact cnoidal wave and travelling wave solutions for both the Sawada–Kotera (SK) and Kaup–Kupershmidt (KK) equations. AKNS-like system has been constructed for a unified equation of the SK and KK equations. The BT are derived from the Riccati form of inverse method. We generalise the leading order analysis method to include elliptic functions and obtain cnoidal wave solutions for both the SK and KK equations. As reduced cases, we obtain several classes of travelling wave solutions for these equations. Conclusions and some features and comments are given.