Two-component complex modified Korteweg-de Vries equations: new soliton solutions from novel binary Darboux transformation

We derive a $2^N\times 2^N$ Lax pair in the form of block matrices for the $N$-component complex modified Korteweg–de Vries (mKdV) equations and construct a novel binary Darboux transformation with $N=2$. Based on Lax pairs and adjoint Lax pairs, we present a new type of Darboux matrices in which eigenvalues could be equal to adjoint eigenvalues. As an illustration, by taking the zero seed solutions, we construct new soliton solutions using the binary Darboux transformation for $2$-component complex mKdV equations with a Lax pair of $4\times 4$ matrix spectral problems. New two- and three-soliton solutions are provided explicitly by choosing appropriate parameters. Furthermore, dynamics and interactions of two- and three-soliton solutions are also explored graphically.