The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker-Akhiezer Function |
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Authors: | A. V. Domrin |
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Affiliation: | (1) Moscow State University, Moscow, Russia |
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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 ℂ xt2 without singularities on ℝxt2. 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.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 453–471, September, 2005. |
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Keywords: | Riemann factorization problem zero-curvature conditions |
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