Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems

Abstract

A method for explicit Wiener–Hopf factorization of 2 × 2 matrix-valued functions is presented together with an abstract definition of a class of functions, C(Q₁, Q₂), to which it applies. The method involves the reduction of the original factorization problem to certain nonlinear scalar Riemann–Hilbert problems, which are easier to solve. The class C(Q₁, Q₂) contains a wide range of classes dealt with in the literature, including the well-known Daniele–Khrapkov class. The structure of the factors in the factorization of any element of the class C(Q₁, Q₂) is studied and a relation between the two columns of the factors, which gives one of the columns in terms of the other through a linear transformation, is established. This greatly simplifies the complete determination of the factors and gives relevant information on the nature of the factorization. Two examples suggested by applications are completely worked out.