具有非零边界条件的非均质离散非线性薛定谔方程的黎曼-希尔伯特方法

IF 2.1 3区 物理与天体物理 Q2 ACOUSTICS
Ya-Hui Liu, Rui Guo, Jian-Wen Zhang
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引用次数: 0

摘要

本文系统地研究了黎曼-希尔伯特(Riemann-Hilbert,RH)方法,并得到了具有非零边界条件(NZBCs)的非均相离散非线性薛定谔(NLS)方程的孤子解。我们从谱问题出发,引入均匀化变量κ以避免双值函数和黎曼曲面的复杂性,推导出特征函数和散射系数的解析性、渐近性和对称性,进而成功地构建了 RH 问题和势的重构公式。在无反射条件下,结合散射系数和特征函数的时间演化,我们得到了由系数变化引起传播方向不同的各种一阶孤子解。基于解析解和特殊参数值的选择,我们得到了两个孤子解的碰撞机制。此外,RH 问题的重要优势在于可以进一步用于研究孤子解析和解的长期渐近行为。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Riemann–Hilbert approach for the inhomogeneous discrete nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable κ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.

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来源期刊
Wave Motion
Wave Motion 物理-力学
CiteScore
4.10
自引率
8.30%
发文量
118
审稿时长
3 months
期刊介绍: Wave Motion is devoted to the cross fertilization of ideas, and to stimulating interaction between workers in various research areas in which wave propagation phenomena play a dominant role. The description and analysis of wave propagation phenomena provides a unifying thread connecting diverse areas of engineering and the physical sciences such as acoustics, optics, geophysics, seismology, electromagnetic theory, solid and fluid mechanics. The journal publishes papers on analytical, numerical and experimental methods. Papers that address fundamentally new topics in wave phenomena or develop wave propagation methods for solving direct and inverse problems are of interest to the journal.
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