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A special addition theorem for theta-functions and nonlinear equations. (English. Russian original) Zbl 0920.14022

Sib. Math. J. 39, No. 5, 973-976 (1998); translation from Sib. Mat. Zh. 39, No. 5, 1127-1130 (1998).
In the article under review, the authors derive some consequences from the Mumford theorem characterizing the Jacobi varieties of hyperelliptic algebraic curves in terms of theta-constants. These formulas, treated as special addition theorems for the corresponding theta-functions, are applied to effectivization of theta-functional formulas for finite gap solutions to the Korteweg-de Vries and sine-Gordon equations.

MSC:

14K25 Theta functions and abelian varieties
35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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