Introduction to nonlocal and generalized lie symmetries of partial differential equations.

Abstract

A symmetry of a differential equation is a transformation that takes a solution of the differential equation to another solution. These transformations are useful in both pure and applied study of partial differential equations. Of particular interest are symmetries which form Lie groups under the operation of function composition. Lie theory guarantees that the structure of the symmetry group is determined by a set of vector fields, called infinitesimal generators. An interesting generalization is to allow the coefficients of these vector fields to depend on the partial derivatives of the dependent variable in the PDE. This thesis presents an intuitive introduction to the theory of generalized symmetries. Included are accessible arguments leading to well-known results, as well as details not well-documented elsewhere. The thesis concludes with a summary of some recent research regarding the theory of nonlocal symmetries, or symmetries generated by infinitesimal generators whose coefficients are allowed to depend on anti-derivatives of the dependent variable.