Solutions of matrix soliton equations: some results on Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations

Some recent results concerning nonlinear non-Abelian KdV and mKdV equations are presented. Operator equations are studied in references [2]-[7] where structural properties of KdV type equations are investigated. Now, in particular, on the basis of results, the special finite dimensional case of matrix soliton equations is addressed to: solutions of matrix KdV and mKdV equations are constructed. Baecklund transformations, which connect different third order nonlinear evolution equations [8], represent a key tool in this study. Explicit solution formulae [11, 3] are applied to obtain some 2x2 and 3x3 Matrix mKdV solutions, which seems to be new, are presented.