General N-soliton solutions to the two types of nonlocal Gerdjikov-Ivanov equations via Riemann-Hilbert problem

Abstract

This paper mainly makes use of the Riemann-Hilbert approach to solve the two types of nonlocal Gerdjikov-Ivanov equations derived by different nonlocal group reductions. The Riemann-Hilbert problem of the general Gerdjikov-Ivanov equation is constructed and the relations between the Riemann-Hilbert problems of the nonlocal Gerdjikov-Ivanov equations and the above Riemann-Hilbert problem are discussed in two parts. The general N-soliton solutions of the nonlocal Gerdjikov-Ivanov equations are acquired by solving the Riemann-Hilbert problems of the nonlocal equations under the reflectionless case and the matrix forms of the soliton solutions are given. In particular, the dynamics of the solutions are explored and the images of the general one-soliton solutions and two-soliton solutions of two types of nonlocal Gerdjikov-Ivanov equations are shown with the appropriate parameters.