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

TL;DR

This paper analyzes the long-time behavior of solutions to a derivative nonlinear Schrödinger equation with step-like initial data, revealing four distinct asymptotic regions with different wave behaviors using Riemann-Hilbert problem techniques.

Contribution

It provides the first detailed asymptotic description of the GI-type DNLS equation with step-like initial conditions, identifying four qualitatively different regions in the long-time limit.

Findings

Identified four asymptotic regions with distinct wave behaviors

Derived explicit asymptotic formulas for each region

Used Riemann-Hilbert analysis to rigorously justify results

Abstract

We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the derivative nonlinear Schr\"odinger (DNLS) equation: $$iq_t+q_{xx}-iq^2\bar{q}_x+\frac{1}{2}|q|^4{q}=0.$$ with steplike initial data: $q(x,0)=0$ for $x\le 0$ and $q(x,0)=Ae^{-2iBx}$ for $x>0$,where $A>0$ and $B\in \R$ are constants.The paper aims at studying the long-time asymptotics of the solution to this problem.We show that there are four regions in the half-plane $-\infty<x<\infty,t>0$,where the asymptotics has qualitatively different forms:a slowly decaying self-similar wave of Zakharov-Manakov type for $x>-4tB$, a plane wave region:$x<-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})$, an elliptic region:$-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})<x<-4tB$. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem.