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The Riemann problem and matrix-valued potentials with a convergent Baker-Akhiezer function. (English) Zbl 1178.30048
Theor. Math. Phys. 144, No. 3, 1264-1278 (2005); translation from Teor. Mat. Fiz. 144, No. 3, 453-471 (2005).
Summary: 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 \(\mathbb C _{xt}^{2}\) without singularities on \(\mathbb R _{xt} ^{2}\) . 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 \(\in \mathbb C\), is a potential with a convergent Baker-Akhiezer function for the corresponding matrix-valued differential operator of the first order.

MSC:
30E25 Boundary value problems in the complex plane
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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