{"title":"耦合修正非线性薛定谔方程的孤子分子和呼吸正子解","authors":"Tao Xu , Jinyan Zhu","doi":"10.1016/j.wavemoti.2024.103347","DOIUrl":null,"url":null,"abstract":"<div><p>The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general <span><math><mrow><mi>M</mi><mo>−</mo></mrow></math></span>SM-<span><math><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mi>M</mi><mo>)</mo></mrow></math></span>-soliton solutions (<span><math><mrow><mn>0</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mi>N</mi></mrow></math></span>), including <span><math><mi>M</mi></math></span> SMs and <span><math><mrow><mi>N</mi><mo>−</mo><mi>M</mi></mrow></math></span> solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., <span><math><mi>N</mi></math></span>-SM solutions (<span><math><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></math></span>) and <span><math><mi>N</mi></math></span>-soliton solutions (<span><math><mrow><mi>M</mi><mo>=</mo><mn>0</mn></mrow></math></span>). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.</p></div>","PeriodicalId":49367,"journal":{"name":"Wave Motion","volume":"129 ","pages":"Article 103347"},"PeriodicalIF":2.1000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation\",\"authors\":\"Tao Xu , Jinyan Zhu\",\"doi\":\"10.1016/j.wavemoti.2024.103347\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general <span><math><mrow><mi>M</mi><mo>−</mo></mrow></math></span>SM-<span><math><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mi>M</mi><mo>)</mo></mrow></math></span>-soliton solutions (<span><math><mrow><mn>0</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mi>N</mi></mrow></math></span>), including <span><math><mi>M</mi></math></span> SMs and <span><math><mrow><mi>N</mi><mo>−</mo><mi>M</mi></mrow></math></span> solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., <span><math><mi>N</mi></math></span>-SM solutions (<span><math><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></math></span>) and <span><math><mi>N</mi></math></span>-soliton solutions (<span><math><mrow><mi>M</mi><mo>=</mo><mn>0</mn></mrow></math></span>). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.</p></div>\",\"PeriodicalId\":49367,\"journal\":{\"name\":\"Wave Motion\",\"volume\":\"129 \",\"pages\":\"Article 103347\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-05-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Wave Motion\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0165212524000775\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ACOUSTICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Wave Motion","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165212524000775","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ACOUSTICS","Score":null,"Total":0}
引用次数: 0
摘要
达尔布变换(DT)方法研究了耦合修正非线性薛定谔方程,该方程可视为非线性薛定谔方程和导数非线性薛定谔方程的组合。在矢量修正导数非线性薛定谔方程谱问题的基础上,成功地构造了相关的拉克斯对和紧凑行列式的 DT,确保了耦合修正非线性薛定谔方程的可整性。根据 DT 方法和极限技术,系统地讨论了两大类解,即孤子分子解(SMs)和呼吸正子解(B-P)。从零平面波背景开始,通过接收 DT 方法巧妙地推导出了一般的 M-SM-(N-M)-孤子解(0≤M≤N),包括 M SM 和 N-M 孤子。特别是,从上述一般解可以还原出两种退化情况,即 N-SM 解(M=N)和 N 孤子解(M=0)。从非零平面波背景出发,通过 DT 和极限技术构造出高阶 B-P 解。有趣的是,B-P 解的中心区域表现出流氓波的模式,因此建议用它们来解释流氓波的产生机制。最后,详细讨论了这些接收解的相应动力学。
Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation
The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general SM--soliton solutions (), including SMs and solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., -SM solutions () and -soliton solutions (). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.
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.