Lectures on the Theory of Group Properties of Differential Equations

Description

Ovsyannikov was Ibragimov's dissertation advisor.

pp. 32-4 (PDF pp. 44-6), "1.5.3 Second-order ordinary differential equations," contain a very concise proof that 2nd order ODEs admit no more than an 8 parameter Lie group of symmetries.

These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models.

These are well-written lectures on invariants, invariant partial differential equations, invariant solutions of partial differential equations, reduction of differential equations by means of invariants, and so on, with many examples though not with essentially new theoretical contributions except perhaps the study and the use of the smallest invariant manifold containing a given manifold, its invariance defect and the reduction of solutions of an invariant differential equation by means of invariant solutions. At the end a few problems are formulated. Reviewed by H. Freudenthal