Questions about *classical* gauge field theory (Abelian and Non-Abelian)

[petergreat]

I know little about these issues, so my questions may be dumb.
1. Do exact solutions exist for non-Abelian gauge theory without matter fields?
2. Do exact solutions exist for Abelian/Non-Abelian theory with matter fields (scalar or spinor)?
2. Are solutions well-behaved? I'm asking because no one can prove it for Navier-Stokes equation. Another reason is that Non-linear waves can develop singularities, and gluons fields are non-linear waves.

 

[tom.stoer]

regarding 1: yes, e.g. instantons and merons are exact solutions of the theory; but due to the non-linearity they are of limited relevance as they can't be used to construct all solutions

regarding 2: afaik no!

regarding 3: afaik no proof exists

general remark: classical solutions are of limited relevance; all the hard stuff in QCD is purely QFT; even instantons and merons are used in rare cases only; so it's really the QFT that needs to be solved and it seems that classical solutions do not really help. Maybe this is slightly different in lattice gauge theories ...