The Riemann–Hilbert Problem and Inverse Scattering

Abstract

The connection between the Riemann–Hilbert factorization on self-intersecting contours and a class of singular integral equations is studied with a pair of decomposing algebras. This provides an effective way of treating the inverse scattering problem for first-order systems. We also show that the matrix functions with positive definite real parts on the real axis and Schwarz reflection invariant elsewhere only have zero partial indices. In particular, this implies the solvability for the inverse scattering problem with skew Schwarz reflection invariant system coefficients $J(z)$ and $q( \cdot ,z)$. This includes, for instance, the system associated with the generalized sine-Gordon equation.