The Riemann problem and matrix-valued potentials with a convergent Baker-Akhiezer function

摘要

We obtain a simple sufficient condition for the sole-ability of the Riemann factorization problem for matrix-valued functions on a circle. This condition is based on the symmetry principle. As an application, we consider nonlinear evolution equations that can be obtained by a unitary reduction from the zero-curvature equations connecting a linear function of the spectral parameter z and a. polynomial of z. We consider solutions obtained by dressing the zero solution with a. function holomorphic at infinity. We show that all such solutions are meromorphic functions on C-xt(2) t without singularities on R-xt(2). This class of solutions contains all generic finite-gap solutions and many rapidly decreasing solutions but is not exhausted by them. Any solution of this class, regarded as a. function of x for almost every fixed t is an element of C, is a potential with a convergent Baker-Akhiezer function for the corresponding matrix-valued differential operator of the first order.