Groups Transformations and Critical Asymptotics Applications to Non-Linear Differential and Partial Derivative Equations

Abstract

This work is connected to different fields of physics and mathematics. A first and obvious reason is that non-linear differential and partial derivative equations are found in many — if not all — fields of physics. But, more important, as a methodology the use of transformations groups is closely connected to other methodologies which in the recent years have played an important role in theoretical physics. The first deals with the concept of self-similarity and dimensional analysis as exposed in the classical book by Sedov (1). In fact the self-similar group technique is a straightforward continuation of the dimensional analysis. A second methodology which has found important applications in statistical mechanics and field theory is the so-called renormalisation technique (2). We will deal with quite similar ideas in our force and time renormalisation concept. Finally we will see that these transformations are also very useful to optimize numerical analysis method.