The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with non-zero boundary conditions

In this paper, we investigate the focusing Kundu-Eckhaus equation with non-zero boundary conditions. An appropriate two-sheeted Riemann surface is introduced to map the spectral parameter k into a single-valued parameter z. Starting from the Lax pair of Kundu-Eckhaus equation, two kinds of Jost solutions are constructed. Further, their asymptotic, analyticity, symmetries as well as spectral matrix are analyzed in detail. It is shown that the solution of the Kundu-Eckhaus equation with non-zero boundary conditions can be characterized with a matrix Riemann-Hilbert problem. Then a formula of N-soliton solutions is derived by solving the Riemann-Hilbert problem. As applications of the N-soliton formula, the first-order explicit soliton solutions with different dynamical features are obtained and analyzed.