Theoretical and Mathematical Physics

, Volume 144, Issue 3, pp 1264–1278 | Cite as

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

  • A. V. Domrin


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 ℂ xt 2 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 ∈ ℂ, is a potential with a convergent Baker-Akhiezer function for the corresponding matrix-valued differential operator of the first order.


Riemann factorization problem zero-curvature conditions 


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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. V. Domrin
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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