Elliptic solitons and Gröbner bases

Abstract

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.

Lamé polynomials, hyperelliptic reductions and Lamé band structure

“…A rigorous investigation must be more careful. For example, if the ideal generated by Article submitted to Royal Society the three equations contains a polynomial involving only B (and g 2 , g 3 ), then the Hermite-Krichever Ansatz will not apply for arbitrary values of B (Brezhnev 2000). In practice, this problem seems not to arise.…”

“…A rigorous investigation must be more careful. For example, if the ideal generated by the three equations contained a polynomial involving only B (and g 2 , g 3 ), then a solution of the Hermite-Krichever form would exist for very few values of B (Brezhnev 2004). In practice, this problem does not arise: except at a finite number of values of B (at most), the Ansatz can be employed (Gesztesy & Weikard 1998).…”

“…For example, to obtain a single polynomial equation for B (involving g 2 , g 3 of course), one may eliminate y 0 from a −1 = 0, b −1 = 0 by computing their resultants against the third equation; and then eliminate x 0 . Alternatively, a Gröbner basis calculation may be performed (Brezhnev 2000).…”