Long-time asymptotic for the derivative nonlinear Schr\"odinger equation with decaying initial value

TL;DR

This paper develops a new Riemann-Hilbert problem approach to analyze the long-time behavior of solutions to the derivative nonlinear Schrödinger equation with decaying initial data, advancing asymptotic analysis techniques.

Contribution

It introduces a novel Riemann-Hilbert formalism for the DNLS initial value problem, enabling long-time asymptotic analysis via nonlinear steepest descent methods.

Findings

New Riemann-Hilbert problem formulation for DNLS

Application of Deift-Zhou method to derive asymptotics

Enhanced understanding of long-time behavior of DNLS solutions

Abstract

We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the solution of some associated Riemann-Hilbert problem. This new Riemann-Hilbert problem for the DNLS equation will lead us to use nonlinear steepest-descent/stationary phase method or Deift-Zhou method to derive the long-time asymptotic for the DNLS equation on the line.