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Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

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Abstract

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation

$$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0 $$

with step-like initial data \(q(x,0)=0\) for \(x \leqslant 0\) and \(q(x,0)=A\mathrm {e}^{-2iBx}\) for \(x>0\), where \(A>0\) and \(B\in \mathbb R\) are constants. We show that there are three regions in the half-plane \(\{(x,t) | -\infty <x<\infty , t>0\}\), on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for \(x>-4tB\), a plane wave region: \(x<-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )\), an elliptic region: \(-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )<x<-4tB\). Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Jian Xu

  2. School of Mathematical Sciences, Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, People’s Republic of China

    Engui Fan

  3. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, People’s Republic of China

    Yong Chen

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  1. Jian Xu
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  2. Engui Fan
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  3. Yong Chen
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Correspondence to Engui Fan.

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Xu, J., Fan, E. & Chen, Y. Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value. Math Phys Anal Geom 16, 253–288 (2013). https://doi.org/10.1007/s11040-013-9132-3

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