The Cauchy matrix structure and solutions of the three-component mKdV equations

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. •We generalize Cauchy matrix method to derive the matrix mKdV equation.•Two kinds of reductions yield the complex and real three-component mKdV equations.•Soliton reflection, double-pole soliton, symmetry broken soliton, soliton molecule.