The long-time asymptotic of the derivative nonlinear Schr$\ddot{o}$dinger equation with step-like initial value

TL;DR

This paper analyzes the long-time behavior of solutions to the derivative nonlinear Schrödinger equation with step-like initial conditions using the Deift-Zhou method, revealing asymptotic solutions involving Theta, parabolic cylinder, and Airy functions.

Contribution

It introduces a $g$-function mechanism to handle exponential growth in the Riemann-Hilbert problem, providing detailed asymptotics for the DNLS equation with step-like initial data.

Findings

Asymptotic solution expressed by Theta function on genus 3 Riemann surface

Subleading terms involve parabolic cylinder and Airy functions

Effective handling of exponential growth via $g$-function mechanism

Abstract

Consideration in this present paper is the long-time asymptotic of solutions to the derivative nonlinear Schr$\ddot{o}$dinger equation with the step-like initial value \begin{eqnarray} q(x,0)=q_{0}(x)=\begin{cases} \begin{split} A_{1}e^{i\phi}e^{2iBx}, \quad\quad x<0,\\ A_{2}e^{-2iBx}, \quad\quad~~ x>0. \end{split}\nonumber \end{cases} \end{eqnarray} by Deift-Zhou method. The step-like initial problem described by a matrix Riemann-Hilbert problem. A crucial ingredient used in this paper is to introduce $g$-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as $t\rightarrow\infty$. It is shown that the leading order term of the asymptotic solution of the DNLS equation expressed by the Theta function $\Theta$ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.