论文标题
新的矩阵修改的Korteweg-de Vries方程：Riemann-Hilbert方法和精确解决方案
A new matrix modified Korteweg-de Vries equation: Riemann-Hilbert approach and exact solutions
论文作者
论文摘要
首先通过Riemann-Hilbert方法研究了一个新的矩阵修改后的Korteweg-de Vries（MMKDV）方程，该方程式具有$ p \ times q $ Q $复合物值的潜在矩阵函数，可以将其简化为众所周知的耦合修改后的Korteweg-De Vries方程，以选择特殊的特殊潜在矩阵。从对该方程的宽松对的特殊分析开始，我们成功地建立了该方程式的Riemann-Hilbert问题。通过引入不规则性和反射案例的特殊条件，通过解决相应的Riemann-Hilbert问题来得出了MMKDV方程的一些有趣的精确解决方案，包括$ n $ soliton解决方案公式。此外，由于特殊潜在矩阵和$ n $ soliton解决方案公式的特殊对称性，我们将进一步努力对原始精确解决方案进行分类，以获取其他一些有趣的解决方案，这些解决方案都以图形方式显示。有趣的是，通过采用不同类型的潜在矩阵，分析了孤子溶液，呼吸型溶液和贝尔型孤子溶液的局部结构和动态行为。
A new matrix modified Korteweg-de Vries (mmKdV) equation with a $p\times q$ 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, we 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, we 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.
