Long-time asymptotic for the derivative nonlinear Schrödinger equation with step-like initial value

Abstract

We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the derivative nonlinear Schrödinger (DNLS) equation: $$i{q}_t+q_{xx}-i{q}^2{\bar{q}}_x+\frac{1}{2}|q{|}^{4}q=0.$$ with steplike initial data: $q(x,0)=0$ for $x\le 0$ and $q(x,0)=A{e}^{-2iBx}$ for $x>0$,where $A>0$ and $B\in \mathbb{R}$ 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.

43 pages. arXiv admin note: text overlap with arXiv:solv-int/9701001 by other authors