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.

Soliton-shape-preserving and soliton-complex interactions for a (1+1)-dimensional nonlinear dispersive-wave system in shallow water

“…Relevant topics can be found in [46][47][48][49][50][51][52][53][54][55][56]. Different from [32], substituting the seeds q = r = 0 into System (4), we have derived the multi-soliton solutions with the determinant representations for System (2), i.e., expressions (23) and (24) {selecting v = w = 0 into System (26), we have obtained expressions (30) and (31), which are different from the results in [28,29]}.…”

Painlevé property, Lax pair and Darboux transformation of the variable-coefficient modified Kortweg-de Vries model in fluid-filled elastic tubes

“…More on relevant issues can also be found in Refs. [46][47][48][49][50]. Furthermore, the Darboux transformation [Eqs.…”

Extended double Wronskian solutions to the Whitham–Broer–Kaup equations in shallow water

“…Relevant issues are seen in [39][40][41][42][43][44]. Firstly, we have transformed (1) into (3), the generalized AKNS system, of which bilinear forms (5) have been obtained by using rational transformations (4).…”