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Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.jde.2020.08.003
Yan Rybalko , Dmitry Shepelsky

Abstract We study the Cauchy problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation i q t ( x , t ) + q x x ( x , t ) + 2 q 2 ( x , t ) q ¯ ( − x , t ) = 0 with a step-like initial data: q ( x , 0 ) = q 0 ( x ) , where q 0 ( x ) = o ( 1 ) as x → − ∞ and q 0 ( x ) = A + o ( 1 ) as x → ∞ , 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 − ∞ x ∞ , 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.

中文翻译:

具有阶跃初始数据的非局部非线性薛定谔方程的长时间渐近

摘要 我们研究可积非局部非线性薛定谔 (NNLS) 方程 iqt ( x , t ) + qxx ( x , t ) + 2 q 2 ( x , t ) q ¯ ( − x , t ) = 0 的柯西问题,其中 a阶跃式初始数据: q ( x , 0 ) = q 0 ( x ) ,其中 q 0 ( x ) = o ( 1 ) as x → − ∞ 和 q 0 ( x ) = A + o ( 1 ) as x → ∞ ,任意正常数 A > 0 。主要目的是研究这个问题的解决方案的长期行为。我们表明渐近线在半平面的四分之一平面中具有性质不同的形式 − ∞ x ∞ , t > 0 : (i) 对于 x 0 ,解接近缓慢衰减的 Zakharov-Manakov 型调制波; (ii) 对于 x > 0 ,解接近“调制常数”。
更新日期:2021-01-01
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