Darboux Transformation Of The Two-coupled Discrete Modified Korteweg-de Vries Equation And Its Exact Solutions

The aim of the present paper is to study the two-coupled discrete modified Korteweg-de Vries equation and construct its exact solutions with the help of the Darboux transfor-mation. The two-coupled discrete modified Korteweg-de Vries equation possess the Lax representation: We introduce a linear transformation whereΦn is a fundamental matrix of solution to (0.9), Mn is a polynomial matrix of A andλ-1. Let det Mn has 8k different roots, that is from which the transformation matrix Mn can be determined by solving the resulting linear system of equations. Supposeφn be a column vector ofΦn. Under the transformation (0.10), we obtain from (0.9) that It is proved that Ln, Ln and Nn, Nn have the same form, respectively, and the Darboux transformation formulae from old potentials an, bn into new ones are as follows: This means that (0.12) is also a Lax representation of the the two-coupled discrete modified Korteweg-de Vries equation, and the Darboux transformation (0.13) transforms a solution of the two-coupled discrete modified Korteweg-de Vries equation into its a new solution. As an application, we obtain nontrivial exact solutions of the two-coupled discrete modified Korteweg-de Vries equation from its trivial solution by using the Darboux transformation.