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

Abstract

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.