In the context of nonlinear optics, the Lugiato-Lefever equation yields a model of four-wave mixing. In its stationary and modal form, I am studying a system of equations which looks as follows:
$$(A +n^2)a_n + i B \sum_{k, l, m \in \mathbb{Z}} \delta_{k+n - l - m} a^{*}_{k}a_{l}a_{m} = 0$$
where $n\in\mathbb{Z}$, $A \in \mathbb{C}$, $B\in \mathbb{R}$ and the sequence $(a_n)_{n\in\mathbb{Z}}$ can take complex values.
It seems likely to me that there is no way of completely solving such an equation. I tried basic approaches like using a generating function which yields a differential equation, but it seems very much unsolvable:
$$ f(x) = \sum_{n\in\mathbb{Z}} a_n x^{|n|}$$
$$ x^2f''(x) + x(1+A) f'(x) + B f^{*}\left(\frac{1}{x}\right) f(x)^2 = 0$$
Does anyone have any insight as to how one could approach studying this type of equation?