Integrability, Darboux transformation and exact solutions for nonlocal couplings of AKNS equations

Abstract

In this paper, we investigate an integrable nonlocal couplings of Ablowitz–Kaup–Newell–Segur (NC-AKNS) equations for the first time. With the help of Lax pair, the NC-AKNS equations are constructed by symmetry reduction method. The construction of nonlocal equations shows that the symmetry reduction method can not only construct a single nonlocal equation, but also can construct a set of equations. In order to study the exact solutions of these nonlocal equations, the method of Darboux transformation is used in this paper. Different from the local equation, the Darboux transformation of the nonlocal equation needs to establish the relation between the spectral parameters. Under some restrictions, 1-fold Darboux transformation is given and two types of soliton solutions are constructed. In order to study the properties of the solutions, the corresponding graphs are given.