Matrix Schrödinger equations and Darboux transformations

This thesis contains the matrix generalisations of some important results known in the theory of the scalar Schrödinger operators. In the first part we discuss the one-dimensional matrix Schrodinger equations in complex domain. The main results here are the local criteria for the Schrödinger operators to have trivial monodromy and a matrix generalisation of the well-known Duistermaat-Grünbaum theorem giving the description of such operators in terms of Darboux transformations. In the second part we consider D-integrable matrix Schrodinger operators in many dimensions. The local criteria on singularities of such operators are found and new examples are constructed. In the last chapter we discuss the soliton solutions of the matrix KdV equations and study the interaction of two solitons.