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.