Lie Groups and PDEs - MathOverflow

Lie Groups and PDEs

2011-05-03T23:41:36Z

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.

Answer by Nilima Nigam for Lie Groups and PDEs

2011-05-04T01:55:09Z

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.

Answer by Denis Serre for Lie Groups and PDEs

2011-05-04T11:58:17Z

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.