| 荀伟康,田守富.一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解*[J].数学年刊A辑,2022,43(3):313~346 |
| 一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解* |
| A New Matrix Modified Korteweg-de Vries Equation:Riemann-Hilbert Approach and Exact Solutions |
| Received:June 12, 2020 Revised:January 14, 2022 |
| DOI：10.16205/j.cnki.cama.2022.0021 |
| 中文关键词: 矩阵型修正 Korteweg-de Vries 方程, Riemann-Hilbert 方法, 精确解, 多孤子解, 孤子分类 |
| 英文关键词:Matrix modified Korteweg-de Vries equation, Riemann-Hilbert approach, Exact solutions, Multi-soliton solutions, Soliton classification |
| 基金项目:国家自然科学基金资助项目(No.11975306),江苏省科学基金资助项目(No.BK20181351),江苏省"六大人才高峰"高层次人才项目(No.JY-059),中央高校基本科研业务费(No.2019ZDPY07, No.2019QNA35)和江苏省研究生科研创新计划项目(No.KYCX21\_2153) |
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| 中文摘要: |
| 在本文中, 一类新的矩阵型修正 Korteweg-de Vries (简记为mmKdV) 方程被首次通过Riemann-Hilbert 方法研究, 而且, 这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正 Korteweg-de Vries 方程.从方程对应的 Lax 对的谱分析入手, 作者成功地建立了方程对应的 Riemann-Hilbert 问题.在无反射势的特殊条件下, mmKdV 方程的精确解可由 Riemann-Hilbert 问题的解给出.而且, 基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类, 从而得到一些有趣的解的现象, 比如呼吸孤子、钟形孤子等. |
| 英文摘要: |
| A new matrix modified Korteweg-de Vries (mmKdV for short) equation with a p×q complex-valued potential matrix function is first studied via Riemann-Hilbert approach,which can be reduced to the well-known coupled modified Korteweg-de Vries equations by selecting special potential matrix. Starting from the special analysis for the Lax pair of this equation, the authors successfully establish a Riemann-Hilbert problem of the equation. By introducing the special conditions of irregularity and reflectionless case, some interesting exact solutions, including the N-soliton solution formula, of the mmKdV equation are derived through solving the corresponding Riemann-Hilbert problem. Moreover, due to the special symmetry of special potential matrices and the N-soliton solution formula, the authors make further efforts to classify the original exact solutions to obtain some other interesting solutions which are all displayed graphically. It is interesting that the local structures and dynamic behaviors of soliton solutions, breather-type solutions and bell-type soliton solutions are all analyzed via taking different types of potential matrices. |
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