Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2 x 2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C (Q(1), Q(2)) is defined. To each class C(Q(1), Q(2)) a Riemann surface Sigma is associated, so that the factorization of the elements of C(Q(1), Q(2)) is reduced to solving a scalar Riemann-Hilbert problem on Sigma. For the solution of this problem, a notion of Sigma-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems. (C) 2008 Elsevier Inc. All rights reserved.