Long-time asymptotics for the nonlocal nonlinear Schrodinger equation with step-like initial data

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:270

Article

Rybalko, Yan^1,2  Shepelsky, Dmitry^1,2 

[1] B Verkin Inst Low Temp Phys & Engn, 47 Nauky Ave, UA-61103 Kharkiv, Ukraine

[2] V Karazin Kharkiv Natl Univ, 4 Svobody Sq, UA-61022 Kharkiv, Ukraine

关键词: Nonlocal integrable equations;    Riemann-Hilbert problem;    Large time asymptotics;    Cauchy problem with step-like initial values;    Nonlinear steepest descent method;   

DOI  :  10.1016/j.jde.2020.08.003

来源: Elsevier

【 摘 要 】

We study the Cauchy problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation iq(t)(x, t) + q(xx)(x, t) + 2q(2)(x, t)(q) over bar(-x, t) = 0 with a step-like initial data: q(x , 0) = q(0)(x), where q(0)(x) = o(1) as x -> -infinity and q(0)(x) = A + o(1) as x -> infinity, with an arbitrary positive constant A > 0. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane -infinity < x < infinity, t > 0: (i) for x < 0, the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for x > 0, the solution approaches the modulated constant. The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. (C) 2020 Elsevier Inc. All rights reserved.

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