Long-time asymptotics and the bright N-soliton solutions of the Kundu-Eckhaus equation via the Riemann-Hilbert approach

The long-time asymptotics and bright N-soliton solutions of the Kundu-Eckhaus equation are studied by Riemann-Hilbert approach. Firstly, the initial value problem of the defocusing Kundu-Eckhaus equation is considered and its long-time asymptotics is derived based on the nonlinear steepest descent method of Deift-Zhou. Then the linear spectral problem of the focusing Kundu-Eckhaus equation is investigated via Riemann-Hilbert formulation and the bright N-soliton solutions of this equation are obtained explicitly.