Abstract:
We obtain a simple sufficient condition for the solvability 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 C2xt without singularities on R2xt. 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∈C, is a potential with a convergent Baker–Akhiezer function for the corresponding matrix-valued differential operator of the first order.
Citation:
A. V. Domrin, “The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker–Akhiezer Function”, TMF, 144:3 (2005), 453–471; Theoret. and Math. Phys., 144:3 (2005), 1264–1278