Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it.
By way of some intuition, let's say, I have a very strong feeling that this equation should admit solutions which are periodic in both $x$ and $y.$ These might be in the form of some combination of elliptic functions, theta functions, etc.
Does anyone know any kind of methods that could be of use in going about finding solutions with two periods, given I have some vaguely intuitive sense of being pretty sure they should exist, say by some loose physical argument of plausibility (not being concerned with rigour at this stage)?
I'm not looking for a general method, but any tools at all, references, whatever, anything which might be useful for attacking a PDE if I have good reason to think I can express a solution in terms of e.g. Jacobian elliptic functions, or theta functions, etc.
Thanks to all who give this a go or who can point me in the right direction.