N-(soliton, breather) interactions for general multi-component third-fifth-order mKdV equations via Riemann–Hilbert method

In this paper, we mainly focus on the Riemann–Hilbert approach solving the general n-component third-fifth-order mKdV (n-(3,5)-mKdV) equations containing the n-component mKdV equation, fifth-order mKdV equation, and their combination. Starting from the spectral analysis of the (n+1)-order matrix Lax pair, we give the corresponding (n+1)-order matrixed-type Riemann–Hilbert problem. By solving the Riemann–Hilbert problem, N-soliton solutions of the n-(3,5)-mKdV equations can be found. Particularly, for the case of reflectionless and some types of spectral parameters of the Lax pair, we analyze the interactions of some kinds of solutions of the 3-component mKdV equation, fifth-order mKdV equation, and third-fifth-order mKdV equations, including breather solutions, W-shaped solutions, anti-bright and bright solutions. Some elastic collisions between them are also presented. In addition, the multi-pole solutions are obtained for the n-(3,5)-mKdV equations through the L’Hospital’s rule. These results will be useful to further understand the related wave phenomena in the multi-component physical systems. •The (n+1)-order matrixed-type Riemann–Hilbert problem is presented.•N-soliton solutions are found for the n-(3,5)-mKdV equations.•The multi-pole solutions are obtained for the n-(3,5)-mKdV equations through the L’Hospital’s rule.