Real-analytic solutions of the nonlinear Schrödinger equation

We establish that the Riemann problem on the factorization of formal matrix-valued Laurent series subject to unitary symmetry has a solution. As an application, we show that any local real-analytic solution (in $ x$ and $ t$) of the focusing nonlinear Schrödinger equation has a real-analytic extension to some strip parallel to the $ x$-axis and that in each such strip there exists a solution that cannot be extended further.