A General Representation for the Green's Function of Second Order Nonlinear Differential Equations

TL;DR

This paper introduces a unified way to represent the Green's function for certain second order nonlinear differential equations, enabling easier analysis and numerical solutions for equations like sinh-Gordon and Liouville.

Contribution

It provides a novel representation of nonlinear Green's functions for specific classes of second order nonlinear differential equations, expanding analytical and numerical tools.

Findings

Representation holds for equations with multiplicative nonlinearities.

Numerical solutions demonstrated for sinh-Gordon and Liouville equations.

Open problems identified for broader characterization of applicable nonlinearities.

Abstract

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.