Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

IF 2.7 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2024-10-01 DOI:10.1016/j.physd.2024.134389

Deng-Shan Wang , Guo-Fu Yu , Dinghao Zhu

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Abstract

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schrödinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non-vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schrödinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.

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在具有阶梯状初始条件的聚焦非线性薛定谔方程背景波上运动的孤子

这项工作涉及在离散谱存在的情况下，具有阶梯状初始条件的聚焦非线性薛定谔方程的长期渐近行为。具有非消失边界的精确阶跃初值问题已经解决，而具有孤子出现的类阶跃初值问题仍未解决。我们研究了这一问题，并通过对与谱函数相关的离散谱的所有可能位置进行分类，探索了在聚焦非线性薛定谔方程的背景波上移动的孤子。结果表明，复平面上的离散谱存在五种区域，分别称为哑区、陷波区、陷波/唤醒区、透射/唤醒区和透射区。利用黎曼-希尔伯特问题的 Deift-Zhou 非线性最陡渐近法，提出了每种情况下解的长期渐近线和孤子的位置。数值模拟与理论分析非常吻合。

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来源期刊

Physica D: Nonlinear Phenomena 物理-物理：数学物理

CiteScore

7.30

自引率

7.50%

发文量

213

审稿时长

65 days

期刊介绍： Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.