Elliptic curves and soliton matrix differential equations

A systematic exposition is given of recent results of the general algebraic theory of soliton equations with an elliptic sheaf which generalize a number of integrable two-dimensional matrix differential equations, starting with the equations of principal chiral fields. Local conservation laws, Bäcklund-Darboux transformations, algebra-geometric solutions, quantum integrals, and Hamiltonian structure are studied.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 22, pp. 205–265, 1984.