Starting from a 4 × 4 matrix Sylvester equation, the matrix mKdV system as an unreduced equation is worked out and the explicit expression of its solution is presented by applying the Cauchy matrix method. Then, two kinds of reduction conditions are given, under which the complex three-component mKdV(CTC-mKdV) equation and the real three-component mKdV(RTC-mKdV) equation can be obtained, and finally, the explicit expressions of soliton solution and Jordan block solution for CTC-mKdV equation and RTC-mKdV equation are presented, respectively. Specially, the generated conditions of one-soliton solutions, two-soliton solutions, double-pole solutions, symmetry broken solutions and soliton molecule are presented, and their dynamic behaviors were analyzed.

中文翻译：

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三分量 mKdV 方程的柯西矩阵结构和解


从 4 × 4 矩阵 Sylvester 方程开始，计算出矩阵 mKdV 系统作为未约化方程，并通过应用柯西矩阵法表示其解的显式表达式。然后，给出了两种还原条件，得到复三分量mKdV（CTC-mKdV）方程和实三分量mKdV（RTC-mKdV）方程，最后分别给出了CTC-mKdV方程和RTC-mKdV方程的孤子解和Jordan块解的显式表达式。具体来说，介绍了单孤子解、双孤子解、双极解、对称破解和孤子分子的生成条件，并分析了它们的动力学行为。