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The Laplace method, algebraic curves, and nonlinear equations. (English. Russian original) Zbl 0583.35086
Funct. Anal. Appl. 18, 210-223 (1984); translation from Funkts. Anal. Prilozh. 18, No. 3, 43-56 (1984).
An algebro-geometric generalization of the Laplace method [E. Goursat, ”Cours d’analyse mathématique”. Tome II (1949; Zbl 0034.341)] is developed. It allows to find integrable equations of the form $$y''=(u(x)+ax+b)y$$ where a,b are constants and u(x) a periodic function tending to a finite-gap potential as $$| x| \to \infty$$. The construction is based on the concept of a Laplace type differential. A multiparameter generalization of the latter results in new classes of exact solutions to the Kadomtsev-Petviashvili equation.
Reviewer: S.Duzhin

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35C05 Solutions to PDEs in closed form 34A30 Linear ordinary differential equations and systems, general 35R10 Functional partial differential equations
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