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Algebraic-geometric principles of superposition of finite-zone solutions of integrable nonlinear equations. (English. Russian original) Zbl 0612.35117
Russ. Math. Surv. 41, No. 2, 1-49 (1986); translation from Usp. Mat. Nauk 41, No. 2(248), 3-42 (1986).
The paper surveys the progress in the Weierstrass’ theory of reducing the Abelian integrals to the elliptic integrals and the Riemann’s theta- functions to the Jacobi’s theta functions of lower genera. The theory is then used to simplify the solutions of the nonlinear equations obtained by the finite-zone integration method. Among them there are the Korteweg- de Vries, the sine-Gordon, the Kovalevskaya’s heavy top, and the Landau- Lifshitz equations.
Reviewer: S.M.Zverev

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
35L70 Second-order nonlinear hyperbolic equations
35C05 Solutions to PDEs in closed form
33E05 Elliptic functions and integrals
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