Lax pair and Darboux transformation for multi-component modified Korteweg–de Vries equations

Abstract

In this paper, through generalizing the 2 × 2 matrix Ablowitz–Kaup–Newell–Segur linear eigenvalue problem to the 2N × 2N case, a new Lax pair associated with the multi-component modified Korteweg–de Vries equations is derived in the form of block matrices. Furthermore, the Darboux transformation is applied to this integrable multi-component system, and the n-times iterative potential formula is presented by applying the Darboux transformation successively. This formula enables us to construct a series of explicit solutions of multi-component modified Korteweg–de Vries equations. In illustration, starting from the zero background, we construct the multi-soliton solutions by performing the symbolic computation.