Application of Lie group analysis of PDE (beyond calculation of exact solutions)

Question

I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide successful applications of Lie symmetry group method for the study of a PDE, other than the calculation of exact solutions? Thank you very much.

P.S., I am not quite sure whether this question is “about research-level mathematics”. I posted it in math.stackexchange two days ago, and there is no answer until now. But I really need some help.

Answer 1

Not an expert but these are sources I have looked at when I had essentially this question. All these sources contain approximate methods (such as approximate symmetries/groups):

• Y.N. Grigoriev, N.H. Ibragimov, V.F. Kovalev & S.V. Meleshko (2010). Symmetries of Integro-Differential Equations. Springer.
• G. Bluman, G. & S.C. Anco (2002). Symmetry and integration methods for differential equations. Springer.
• B.J. Cantwell (2002). Introduction to symmetry analysis. Cambridge University Press.

Cantwell is extensively applying symmetry methods to fluid dynamics. Grigoriev et al discuss applications of plasma physics and mechanics.

Answer 2

The book by Olle Stormark, Lie's Structural Approach to PDE Systems, Cambridge 2000 provides an introduction to a number of topics on the relationship between Lie groups and characteristics of PDEs, integrability, and the equivalence problem.

Note Stormark's comment at the very end of the book:

Its purpose certainly is not to replace reading the original papers of Lie, Cartan and Vessiot, but quite the opposite, to encourage it and hopefully make it more fruitful.

The bibliography provides a good starting point for this.

J.F. Pommaret's Partial Differential Equations and Group Theory New Perspectives for Applications Kluwer 1994 provides some applications of Lie pseudogroups to control theory and continuum physics.

Answer 3

You might consider 'Applications of Lie Groups to Differential Equations' by Olver. I have this book and I quote:

Besides the basic theory of manifolds, Lie groups and algebras, transformation groups and differential forms, the book delves into the more theoretical subjects of prolongation theory and differential equations, the Cauchy-Kovalevskaya theorem, characteristics and integrability of differential equations, extended jet spaces over manifolds, quotient manifolds, adjoint and co-adjoint representations of Lie groups, the calculus of variations and the inverse problem of characterizing those systems which are Euler-Lagrange equations of some variational problem, differential operators, higher Euler operators and the variational complex, and the general theory of Poisson structures, both for finite-dimensional Hamiltonian systems as well as systems of evolution equations, all of which have direct bearing on the symmetry analysis of differential equations. It is hoped that after reading this book, the reader will, with a minimum of difficulty, be able to readily apply these important group-theoretic methods to the systems of differential equations he or she is interested in, and make new and interesting deductions concerning them.