Kadomtsev-Petviashvili denkleminin cebirsel-geometrik çözümleri

In 1976, I. M. Krichever suggested a method to solve nonlinear partial differential equations in the form of the Zaharov-Shabat Equation. The method uses so called Baker-Akhiezer functions on Riemann surfaces and provides periodic and conditionally periodic solutions to such nonlinear equations that can be expressed in terms of the so called Riemann theta function, a theta function defined on some n dimensional complex space where the Riemann matrix of the function corresponds to a Riemann surface. In this thesis, we will mainly consider the Kadomtsev-Petviashvili equation (or KP equation) which is an example of the Zaharov-Shabat equation. Following the expository paper "Theta Functions and Nonlinear Equations" by B. A. Dubrovin, we will present the construction of a family of solutions to the KP equation that can be given in terms of the Riemann theta function. Another topic Dubrovin deals with in the article is the effectivization of these solutions, meaning the constructed solutions are substituted into the KP equation and some conditions are found. Finally, we will present those results obtained by effectivization.