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Elliptic solitons and Gröbner bases. (English) Zbl 1070.37050
Summary: We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite Ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New integrable potentials and corresponding solutions of the Sawada-Kotera, Kaup-Kupershmidt, Boussinesq equations and others are found.

MSC:
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
33E05 Elliptic functions and integrals
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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