A General Representation for the Greens Function of Second Order Nonlinear Differential Equations

In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Greens function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Greens 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 Greens 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 Greens function.