A new Riemann–Hilbert problem in a model of stimulated Raman Scattering

Abstract

The Riemann–Hilbert problem proposed in [2] for the integrable stimulated Raman scattering (SRS) model was shown to be solvable under an additional condition: the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities. In the general case, it can be shown that a spectral singularity occurs at k = 0. On the other hand, the initial boundary value (IBV) problem for the SRS equations is known to be well posed: using PDE techniques, this has been established in [3]. Therefore, it seems natural to try to find a new RH problem that is solvable in the presence of arbitrary spectral singularities. The formulation of such a RH problem is the main aim of the paper. Then the solution of the nonlinear initial boundary value problem for the SRS equations is expressed in terms of the solution of a linear problem which is the Riemann–Hilbert problem for a sectionally analytic matrix function.