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Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy. (English) Zbl 1309.37070
In this work, the authors describe for the first time quasi-periodic solutions of the Kaup-Kupershmidt (KK) equation by using a nontrivial \(3 \times 3\) spectral problem. To prove this result, they solve as a fist step a stationary zero-curve equation related to this \(3 \times 3\) spectral problem. As second step, they introduce the Baker-Akhiezer functions for the KK hierarchy. This allows to decompose the KK equations into a system of solvable Dubrovin-type ODEs which simplifies the problem. Finally, they use the well-known Baker-Akhiezer property and the asymptotic expansion of a related meromorphic function to obtain their explicit Riemann theta function representation. This leads to quasi-periodic solutions of the whole KK hierarchy in terms of Riemann theta functions.

MSC:
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H42 Theta functions and curves; Schottky problem
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