Algebro-Geometric Solutions for the Kadomtsev-Petviashvili Hierarchy

Based on the idea of symmetric constraint, we apply the Gesztesy-Holden’s method to derive explicit representations of the Baker-Ahkiezer function ${\psi }_1$ of the KP hierarchy, from which we provide theta function representations of algebro-geometric solutions for the whole Kadomtsev-Petviashvili (KP) hierarchy. This provides a approach to obtain some special subclasses of algebro-geometric solutions for the KP hierarchy and other high dimensional hierarchy of equations.