Nonlocal symmetry and exact solutions of the (2+1)-dimensional Gerdjikov–Ivanov equation

Gerdjikov–Ivanov (GI) equation was widely used in quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics and other fields. Based on the algorithm of constructing high-dimensional equations using the conservation laws of low-dimensional equations proposed by Lou et al, this paper we establish a new (2+1)-dimensional GI equation by using the conservation law of (1+1)-dimensional GI equations and related algorithms at first. Then we establish a closed system corresponding to the nonlocal symmetry by introducing some new equations and variables. Furthermore, we successfully obtain some exact solutions for the high-dimensional GI equations, and give some corresponding images of the solutions to analysis the dynamic behavior, including soliton solution and hyperbolic function solution.