# resonance in pde

(I migrated the question from overflow b/c I think it belongs here instead.) About a year back I was in Professor Henry Mckean's office and he told me alot of interesting things. One thing he told me was that if a pde has resonances then often the resonances dissapear when the same pde is taken on a natural compactification of the space. Unfortunately at 85 years old he is not in great health anymore and I can't ask him directly what he meant.

One example he gave me is that the transformed pde of burgers equation into the heat equation has resonances but when the same pde is taken in $\mathbb{RP^1}=\mathbb{S^1}$, a compactification of $\mathbb{R}$, the resonances dissapear. He told me that this makes sense because the cole hopf transformation is nonlinear transformation so the resulting pde should have resonances. But projective space is a nonlinear space so the 2 effects cancel.

I really want a reference so I can know what I am talking about. I have a working knowledge of pde I, II and took these courses(sobolev spaces, well posedness results by fixed fixed point or compactness, direct method of calculus of variations, mountain pass lemma etc). For whatever its worth I have taken courses in differential geometry, lie theory and have a strong background in analysis. In short I know every term in what he is saying but I don't know what he means by what he is saying if that makes sense.

For whatever its worth I have been reading about pde resonances at http://www.cims.nyu.edu/~pgermain/Space_time_resonances.pdf Professor Pierre Germain and Professor Shatah are supposed to be authorities in the subject.

• Greetings from a fellow Couranter :) Jun 20, 2015 at 15:36
• Alas, I'm not, was there around 2000. Love seeing the reference to McKean's office and Shatah. Jun 20, 2015 at 20:03