Construction of soliton solutions of the matrix Korteweg-de Vries and modified Korteweg-de Vries equations

Nonlinear evolution equations known also as non-commutative soliton equations are considered. In particular, the matrix modified Korteweg-deVries (mKdV) equation is studied. Based on Backlund transformations which allow to ¨ reveal algebraic properties as well as construct solutions of nonlinear evolution equations, a method to construct explicit solutions is presented. The produced solutions can be considered as soliton solutions; they exhibit the typical behaviour of solitons. Indeed, they are shown to represent a generalisation of the corresponding scalar solutions. Finally, perspectives concerning the construction of soliton solutions admitted by further matrix equations are given.