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Matveev, V. B.; Smirnov, A. O.

Dubrovin method and the Toda chain. (English. Russian original) Zbl 1537.37083

St. Petersbg. Math. J. 34, No. 6, 1019-1037 (2023); translation from Algebra Anal. 34, No. 6, 170-196 (2022).
Summary: A hierarchy of Lax pairs with \(2\times 2\) matrix coefficients is presented. The compatibility conditions for these pairs include the Toda chain equation, and other differential-difference integrable systems. Various kinds of finite gap solutions for such systems are constructed. Examples of simplest one- and two-phase solutions are given, together with the corresponding spectral curves.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions

Keywords:

Toda chain; Lax pair; theta functions; finite-gap solutions; spectral curves
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References:

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