## 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.

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Does Oliver's introduction say what the prerequisites are? Might be a good place to start. – BSteinhurst May 4 2011 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 2011 at 1:17
Mathematical physics by Hassani has a chapter or two that is a kind of introduction to the material in Olver's book. I think this can be a good entry point (or it is possible that you decide Hassani's version is enough for now, but I doubt it because it is very interesting). – timur May 4 2011 at 2:43
According to Olver, his book is pretty self contained with regards to what it uses of Lie theory and manifolds. I think I'm going to follow Deane's suggestion and supplement it with an easier book, for examples, and such. – cako May 4 2011 at 4:09
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 2011 at 13:08
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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.

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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.
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...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 2011 at 12:30
@Piero: i'd be interested in hearing more about that. Could you give some references? Thanks. – Michael May 4 2011 at 15:08