The Riemann Problem and Matrix-Valued Potentials with a~Convergent Baker--Akhiezer Function

A. V. Domrin

Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 3, pp. 453-471

Résumé

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 $\mathbb{C}^2_{xt}$ without singularities on $\mathbb{R}^2_{xt}$. 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 $t\in\mathbb{C}$, is a potential with a convergent Baker–Akhiezer function for the corresponding matrix-valued differential operator of the first order.

Détail

Keywords: Riemann factorization problem, zero-curvature conditions.