Second Painleve Hierarchies: Derivation and Basic Properties

Abstract

In this thesis we studied the derivation of the Modified Korteweg-de Vries hierarchy from Korteweg-de Vries hierarchy. Then we derive the Flaschka-Newell second Painlev´e hierarchy from the Modified Korteweg-de Vries hierarchy. Moreover we study B¨acklund transformations, special solutions, and Lax pair representation of Flaschka-Newell second Painlev´e hierarchy. In addition we studied the derivation of Jimbo-Miwa second Painlev´e hierarchy as a reduction of a (2 + 1)-dimensional dispersive water wave non-isospectral hierarchy, and we derived some basic properties of this hierarchy such as B¨acklund transformations, special solutions, and we studied Lax pair representation.