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Elliptic solitons and Heun’s equation. (English) Zbl 1072.34102
Kuznetsov, V. B. (ed.), The Kowalevski property. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2885-1/pbk). CRM Proc. Lect. Notes 32, 287-305 (2002).
Summary: We find a new class of algebraic geometric solutions of Heun’s equation with the accessory parameter belonging to a hyperelliptic curve. Dependence of these solutions from the accessory parameter as well as their relation to Heun’s polynomials are studied. Methods for calculating the algebraic genus of the curve, and its branching points, are suggested. Monodromy group is considered. Numerous examples are given.
34M35Singularities, monodromy, local behavior of solutions, normal forms
33E30Functions coming from differential, difference and integral equations
33E10Lamé, Mathieu, and spheroidal wave functions
34L40Particular ordinary differential operators