Long-time asymptotics for a modified nonlinear Schrödinger equation via the Dbar steepest descent method

The long-time asymptotics for a modified nonlinear Schrödinger (mNLS) equation with Schwartz initial value is studied. By analysing the Lax pair of the mNLS equation, the relationship between the solution of the initial value problem and the solution of a Riemann–Hilbert (RH) problem is established. Then the Dbar steepest descent method is applied. The essence of this method is to make continuous extensions of the scattering data, and then construct a mixed RH-Dbar problem, which can be decomposed into a pure RH problem and a pure Dbar problem. The results are divided into two cases according to the different positions of stationary phase points. In our final asymptotic expression, the leading order is from the pure RH problem, which is divided into a local solvable RH model and a small norm RH problem, and the error order is from the pure Dbar problem.