I am interested in solving coupled differential equations with the following shape:
\begin{equation} \ddot y_p(x) =\sum_{lmn=1}^N C(l,m,n,p)y_l (x)y_m (x)y_n(x) \end{equation} where $p=1\cdots N$. The coefficient $C(l,m,n,p)$ control the general coupling between the functions involved, and can be written as a combination of delta functions in in its arguments.
Are there formal solutions to such class of o.d.e.'s, or at least convenient ansatz to tackle the problem?