High order nonlocal symmetries and exact solutions of variable coefficient KdV equation

Abstract

In this paper, nonlocal symmetries and exact solutions of variable coefficient Korteweg-de Vries (KdV) equation are studied for the first time. Using pseudo-potential, high order nonlocal symmetries of time-dependent coefficient KdV equation are obtained. In order to construct new exact analytic solutions, new variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give symmetry reduction and some exact analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves are discussed in detail, and the corresponding images are given to illustrate their dynamic behavior.