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This article is cited in 2 scientific papers (total in 2 papers)
Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
Yan Rybalko, Dmitry Shepelsky B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
Abstract:
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schrödinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\to A$ as $x\to +\infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the “shifted step function” $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x<R$ and $q_{R,A}(x)=A$ for $x>R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors.
Key words and phrases:
nonlocal nonlinear Schrödinger equation, Riemann–Hilbert problem, long-time asymptotics, nonlinear steepest descent method.
Received: 18.09.2020
Citation:
Yan Rybalko, Dmitry Shepelsky, “Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data”, Zh. Mat. Fiz. Anal. Geom., 16:4 (2020), 418–453
Linking options:
https://www.mathnet.ru/eng/jmag765 https://www.mathnet.ru/eng/jmag/v16/i4/p418
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Abstract page: | 112 | Full-text PDF : | 44 | References: | 15 |
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