{"title":"On the two nonzero boundary problems of the AB system with multiple poles","authors":"","doi":"10.1016/j.chaos.2024.115560","DOIUrl":null,"url":null,"abstract":"<div><div>The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of <span><math><mi>N</mi></math></span>-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":null,"pages":null},"PeriodicalIF":5.3000,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos Solitons & Fractals","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0960077924011123","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of -order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.
黎曼-希尔伯特方法用于研究具有两个非零边界条件的 AB 系统。为了重构势,讨论了拉克斯对的谱分析和相应的黎曼-希尔伯特问题。对于这两个非零边界问题,我们分别考虑了截面解析函数具有双极和三极的情况。导出的散射数据具有多个零点或多对零点,由此可以得到相应极点处特征函数之间的关系。通过求解代数方程,我们可以得出不同情况下的 N 阶解公式。作为应用,我们得到了无反射情况下不同类型的孤子和呼吸子,并揭示了这些解的动态分析。
期刊介绍:
Chaos, Solitons & Fractals strives to establish itself as a premier journal in the interdisciplinary realm of Nonlinear Science, Non-equilibrium, and Complex Phenomena. It welcomes submissions covering a broad spectrum of topics within this field, including dynamics, non-equilibrium processes in physics, chemistry, and geophysics, complex matter and networks, mathematical models, computational biology, applications to quantum and mesoscopic phenomena, fluctuations and random processes, self-organization, and social phenomena.