CHAPTER V - Invariant Solutions

Abstract

This chapter presents the study of the operation of the admitted group on the set of solutions of a differential equation that begins by the consideration of fixed points of that operation, which are invariant solutions. These are solutions that transform into themselves under transformations from the admitted group. The chapter discusses the general theory of invariants and invariant manifolds. The theory is related to the fact that the determination of the invariants for many parameters groups requires the analysis of not one vector field but a family of vector fields. The description of nonsingular invariant manifolds of the group Gr(f) is given in the following theorem concerning representation of nonsingular invariant manifolds Ψ satisfying the condition: dim Ψ <n. If the differential equation E admits the group Gr(f), then it also admits any of its subgroups H. In accordance with this, the concept of an invariant solution is defined in such a way that it also covers all subgroups H ⊂ Gr (h). The solution u Є SE is called an invariant H-solution of the equation E if the manifold U, corresponding to it, is an invariant manifold of the group H.