Inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary conditions

Abstract

In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. Based on the Lax pair of the Gerdjikov–Ivanov equation, we derive its Jost solutions with nonzero boundary. Further asymptotic behaviors, analyticity and the symmetries of the Jost solutions and the spectral matrix are in detail derived. The formula of N-soliton solutions is obtained via transforming the problem of nonzero boundary into the corresponding matrix Riemann–Hilbert problem. As examples of N-soliton formula, for $$N=1$$ and $$N=2$$ , respectively, different kinds of soliton solutions and breather solutions are explicitly presented according to different distributions of the spectrum. The dynamical features of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions.