Ben-Gurion University, Jacob Blaustein Institute for Desert Research,
Sede-Boker Campus, 84990, Israel
E-mail: georg@bgumail.bgu.ac.il

Expanded Lie group transformations and similarity reductions of differential equations

Abstract:
The generalization of the Lie group method aimed at finding transformations between different equations is developed. In contrast to a B\" acklund transform, which can also relate solutions of two distinct equations but, in general, does not possess group properties, our method is actually the Lie group method extended by including additional variables in the set of the variables to be transformed. The method is applicable to both ordinary and partial differential equations. Introducing the expanded transformations permits enriching the concept of similarity reductions which may yield some new applications. In particular, the classical similarity reductions of the PDEs may be treated as the extended group transformations that are singular in some variables.