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).