Algebro-geometric solutions of the kadomtsev-petviashvili equation

Abstract

I. M. Krichever suggested a method to solve nonlinear partial di erential equations in the form of the Zaharov-Shabat Equation [L {u100000} @y;A {u100000} @t] = 0 where L and A are di erential operators including derivatives only with respect to the x variable in 1976 [1]. 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 -function, a -function de- ned 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) 3 4 uyy = @ @x ut {u100000} 1 4 (6uux + uxxx) which is an example of the Zaharov-Shabat equation. Following the expository paper of B. A. Dubrovin [2], we will present the construction of such solutions to the KP equation given as u(x; y; t) = 2 @2 @x2 log (xU +yV +tW +z0)+c. It was observed that this construction allows one to investigate the su cient conditions on arbitrary vectors U, V , and W that make the above function u(x; y; t) a solution to the KP equation. We explain the answer to this question for Riemann surfaces of small genera, and mention the result for more general Riemann surfaces which are both given in [2].