The Laplace method, algebraic curves, and nonlinear equations (Q1070121)

An algebro-geometric generalization of the Laplace method [\textit{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.