Algebraic approach to group classification of differential equations

Abstract:
The complexity of group classification of differential equations led to the development of a great variety of specialized classification techniques, which are conventionally partitioned into two main approaches. The first approach is based on the direct compatibility analysis and integration of the corresponding determining equations up to a relevant equivalence relation. We discuss the other approach, which is of algebraic nature. Any version of the algebraic method involves, in some way, the classification of algebras of vector fields up to certain equivalence induced by point transformations. The key question is what set of vector fields should be classified and what kind of equivalence should be used. Depending on this and completeness of solution, one can talk about partial preliminary group classification, complete preliminary group classification and complete group classification. Within the framework of group classification, an important role is played by the notion of normalized classes of differential equations. Thus, for a weakly normalized class, complete preliminary group classification and complete group classification coincide. If the class is semi-normalized, the group classification up to equivalence generated by the associated equivalence group coincides with the group classification up to general point equivalence. As normalized classes are both semi-normalized and weakly normalized, it is especially convenient to carry out group classification in such classes by the algebraic method. This is why the normalization property can be used as a criterion for selecting classes of differential equations to be classified or for splitting of such classes into subclasses which are appropriate for group classification. To illustrate the approach developed, we discuss the group classification problems for the classes of nonlinear wave equations and generalized nonlinear Schrödinger equations.