Residual Symmetry Reduction and Consistent Riccati Expansion of the Generalized Kaup-Kupershmidt Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11405110, 11275129, 11472177, and the Natural Science Foundation of Zhejiang Province of China under Grant Nos. LY18A050001 and LQ13A050002

Abstract

The residual symmetry of the generalized Kaup-Kupershmidt (gKK) equation is obtained from the truncated Painlevé expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Furthermore, the gKK equation is proved to integrable in the sense of owning consistent Riccati expansion and some new Bäcklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.