  We consider some non-linear non-homogeneous partial differential equations
(PDEs) and derive their exact solution as a functional Taylor expansion in
powers of the source term. The kind of PDEs we consider are dispersive ones
where the exact solution of the corresponding homogeneous equations can have
some known shape. The technique has a formal similarity with the
Dyson--Schwinger set of equations to solve quantum field theories. However,
there are no physical constraints. Indeed, we show that a complete coincidence
with the statistical field model of a quartic scalar theory can be achieved in
the Gaussian expansion of the cumulants of the partition function.
