# Solution to a Nonlinear Integral Equation

This particular nightmare of an NIE showed up in my work a week ago and I'm stumped. Not exactly a math specialist, however, so I'm hoping MathSE has some ideas.

$$(\partial^2_x-\Psi(x,t+r)\ast f(x))\Psi(x,t)=\partial_t \Psi(x,t)$$

$\ast$ the bounded convolution operator $f\ast g=\int_0^r f(\xi) g(x-\xi) d\xi$, and $\Psi$ periodic in $x$ with period $r$, i.e. $\Psi(x+r,t)=\Psi(x,t)$. $f(x)=(r^2-x^2)^{-1}$ in context if that makes it easier, but I'll be happy to accept partial answers (or indeed, even a general intuition as to where to look).

• The periodicity and convolution all seem to hint to a Fourier method, but maybe someone who knows more than me can answer Commented Nov 25, 2017 at 0:39
• It's a nonlinear and nonlocal heat equation. Fourier transform may not be so useful due to the nonlinearity. I think it is unlikely you can get a solution formula. What do you hope to do/prove for this equation? (e.g., it should be easy to solve numerically, etc.)
– Jeff
Commented Nov 25, 2017 at 13:28
• @Jeff Unfortunately, what I want most is precisely the solutions - though I'd be happy with some significant subset of them. A numerical approach is fine, but I don't quite know how to go about it and I'm not sure I'd go about knowing that I've found most/all of them. Commented Nov 26, 2017 at 10:40
• Presumably you have an initial condition at $t=0$? Then numerics will give you the unique solution given that initial condition (so this will generate "all" solutions). It would likely not be hard to prove the solution is unique using maximum principle arguments.
– Jeff
Commented Nov 26, 2017 at 15:35
• @Jeff Nope, no initial condition. Certainly I could make one up, but at that point I'm basically trying to classify its behavior and should be looking into linearization options. (Not that I'm not already, but I'm not sure how to linearize something like this either.) Commented Nov 27, 2017 at 1:56