Romanovskij, R. K.; Sadovnichuk, S. G. 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. Reviewer: I.A.Taimanov (Novosibirsk) Cited in 2 Reviews 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.) Keywords:theta-function; soliton; Jacobi varieties; hyperelliptic algebraic curves; theta-constant PDFBibTeX XMLCite \textit{R. K. Romanovskij} and \textit{S. G. Sadovnichuk}, Sib. Math. J. 39, No. 5, 1127--1130 (1998; Zbl 0920.14022); translation from Sib. Mat. Zh. 39, No. 5, 1127--1130 (1998)