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Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations. (English. Russian original) Zbl 0978.58012
Funct. Anal. Appl. 34, No. 3, 159-171 (2000); translation from Funkts. Anal. Prilozh. 34, No. 3, 1-16 (2000).
This is an excellent paper. The authors deal with the development of a method for the integration of nonlinear partial differential equations on the basis of uniformization of the Jacobi varieties of algebraic curves with the help of \(p\)-functions of several variables, the theory of which was founded by Klein. The study in that direction [see V. M. Buchstaber, V. Z. Enolskii and D. V. Leikin, Rev. Math. Math. Phys. 10, No. 2, 3–120 (1997; Zbl 0911.14019)] is motivated by the fact that under such uniformization the most important nonlinear equations of mathematical physics appear natural and explicitly. By way of application, the authors obtain the system of nonlinear partial differential equations integrable in trigonal \(p\)-functions. This system in particular contains the Boussinesq equations.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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