The laplace method, algebraic curves, and nonlinear equations
Copyright policy )The laplace method, algebraic curves, and nonlinear equations
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.
linear Schrödinger equation, differential-difference schemes, Partial differential equations of mathematical physics and other areas of application, Linear ordinary differential equations and systems, Partial functional-differential equations, Laplace method, Kadomtsev-Petviashvili, finite-gap potential, Solutions to PDEs in closed form
linear Schrödinger equation, differential-difference schemes, Partial differential equations of mathematical physics and other areas of application, Linear ordinary differential equations and systems, Partial functional-differential equations, Laplace method, Kadomtsev-Petviashvili, finite-gap potential, Solutions to PDEs in closed form
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