Long‐time asymptotics to the defocusing generalized nonlinear Schrödinger equation with the decaying initial value problem

Abstract

In this paper, the main work is to study the long‐time asymptotics of the defocusing generalized nonlinear Schrödinger equation with the decaying initial value. The Riemann‐Hilbert method and the nonlinear steepest descent method by Deift‐Zhou have made great contributions to obtain it. Starting from the Lax pair of the defocusing generalized nonlinear Schrödinger equation, the associated oscillatory Riemann‐Hilbert problem can be obtained. Then, via the stationary point, the steepest decent contours, and the trigonometric decomposition of jump matrix, we get the solvable Riemann‐Hilbert problem from the associated oscillatory Riemann‐Hilbert problem. Based on the decaying initial value in Schwartz space, the Weber equation, and the standard parabolic cylinder function, the expression of the solution for the generalized nonlinear Schrödinger equation can be given.