  In this paper we study some classes of second order non-homogeneous nonlinear
differential equations allowing a specific representation for nonlinear Green's
function. In particular, we show that if the nonlinear term possesses a special
multiplicativity property, then its Green's function is represented as the
product of the Heaviside function and the general solution of the corresponding
homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies
of specific non-linearities admitting this representation are derived. The
nonlinear Green's function solution is numerically justified for the
sinh-Gordon and Liouville equations. We also list two open problems leading to
a more thorough characterizations of non-linearities admitting the obtained
representation for the nonlinear Green's function.
