The inverse scattering transform for the focusing nonlinear Schrödinger equationwith non-zero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the so-called theta conditio...

In [6], a purely algebraic analogon of Katz’ middle convolution functor (see [10]) is given. In this paper, we find an explicit Riemann-Hilbert correspondence for this functor. This leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the Riemann-Hilbert correspondence.