# Lie Groups and PDEs

A friend of mine recently explained to me a little bit about using Lie groups and symmetries to obtain solutions of PDEs. I was interested and wanted to learn a bit more about it. He's been using Olver's "Applications of Lie Groups to Differential Equations" but I found it a bit out of my reach.

I've taken a PDE course that followed Fritz John's "Partial Differential Equations" pretty closely, and a basic differential geometry course (curves and surfaces). I also have limited knowledge of group theory, but he said he didn't have any when he started learning the theory.

So my question is: should I study PDEs or group theory a bit more before attempting to tackle Olver's book, or should I try an easier text first?

Thanks for the help.

• Does Oliver's introduction say what the prerequisites are? Might be a good place to start. – BSteinhurst May 4 '11 at 0:07
• Or just start reading Olver's book and take detours to learn what you need as you go along. – Deane Yang May 4 '11 at 1:17
• Speaking as an outsider, I know that Olver's graduate text (now in a 1993 second edition) is highly regarded and is also virtually the only textbook covering this kind of material. It's clearly essential to start with specific examples as motivation for the use of general Lie group methods, which I think is the way Olver proceeds. This is very much in the spirit of Lie's original program, generalizing Galois theory from polynomial equations to differential equations. – Jim Humphreys May 4 '11 at 13:08
• @Hans this question was asked 7 years ago. I don't think it is necessary at this point to criticize the question as being unsuitable. – KConrad Jan 5 '18 at 0:31
• There also is a more elementary (and very short) introduction to the subject by Olver himself: www-users.math.umn.edu/~olver/s_/sxs.pdf that could be a good place to start – just-someone Nov 17 '18 at 1:42

I found a solid background in PDE, together with some physics, to be a useful entry point to Olver's nice book. There's the 'Lectures on Partial Differential Equations' by V.I.Arnold which is fun to read alongside, if not before. Any solid book on mathematical methods in classical mechanics and quantum mechanics should prove useful as well. Finally, I agree with Deane- the most efficient path is to start reading the book, and learn the material you need as you proceed.

Just to advocate the relevance of groups in PDEs.

• On the one hand, when a PDE admits a group of symmetries (often translations, rotations, but also Galilean transformation or conformal transformation, ...) you may look for special solutions that behave well under some subgroup (they are invariant or equi-variant). This leads to PDEs in smaller dimensions, or even to ODEs. One aspect of this approach leads to special functions, orthogonal polynomials, harmonic analysis and so on.
• On the other hand, the conservation laws play an important role in PDEs, for instance when we look for a priori estimates in order to prove the existence of solutions to either boundary-value problems or Cauchy problems. By Noether's Theorem, there is a correspondance between the symmetries of the PDE and its conservation laws. Notice that this is not specific to PDEs; it happens already in ODEs.
• ...and, in nonlinear equations it frequently occurs that blow-up solutions have near the blow-up point additional symmetries, which can be evidenced via rescaling i.e. groups of transformation. As a consequence, one obtains rigidity results which can lead by contradiction to spectacular results of global existence – Piero D'Ancona May 4 '11 at 12:30
• @Piero: i'd be interested in hearing more about that. Could you give some references? Thanks. – Michael Bächtold May 4 '11 at 15:08
• @PieroD'Ancona I just would like to second Michael's comment -- what you say is very interesting and it would be great if you could provide some references. Thanks in advance. – mathphysicist Sep 9 '18 at 12:20

Try Lawrence Dresner's book on the subject, Applications of Lie's Theory of Ordinary and Partial Differential Equations. It lacks Olver's rigor but gets you up and running on the applications.

For something more elementary than Olver's book see e.g. Hydon's Symmetry methods for differential equations. A beginner's guide or Arrigo's Symmetry Analysis of Differential Equations: An Introduction and, at a more advanced level, e.g. Stephani's Differential equations. Their solution using symmetries.

Here are lecture notes, which build (partly) on the book by Olver and should be an easier starting point: Lie Transformation Groups: An Introduction to Symmetry Group Analysis of Differential Equations