When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates are just plain easy to detect).

Now, is it always possible to find a change of variables that transforms a PDE into a product of independent functions $f_i(x_i)$, with $$\frac{\partial f_i}{\partial x_j} = \delta_{ij}$$ How is this change of variables found? And, in case it doesn't always exist, what are the simmetries the expression of the PDE must have so that we can prove its existence?

Bonus question: what relationship does it have with the Hamilton-Jacobi equations?

• this question is far deeper than it appears. See ima.umn.edu/~miller/sepofvariablestalk.pdf for a sense of why this question must be narrowed to obtain a good answer. I think if you limit it to Hamilton-Jacobi in a reasonable context the linked items shows you what's what. – James S. Cook Jun 20 '14 at 15:28
• See this answer for links to the general Lie-theoretic approach. – Bill Dubuque Jun 20 '14 at 15:30
• This paper by (my colleagues) Waksjö & Rauch-Wojciechowski might also be of interest: dx.doi.org/10.1023/B:MPAG.0000007238.37788.2c (subscription needed, though). – Hans Lundmark Jun 20 '14 at 16:04