The Riemann - Hilbert problem associated with the quantum nonlinear Schrödinger equation

We consider the dynamical correlation functions of the quantum nonlinear Schrodinger equation. In a previous paper we found that the dynamical correlation functions can be described by the vacuum expectation value of an operator-valued Fredholm determinant. In this paper we show that a Riemann - Hilbert problem can be associated with this Fredholm determinant. This Riemann - Hilbert problem formulation permits us to write down completely integrable equations for the Fredholm determinant and to perform an asymptotic analysis for the correlation function.