Nonlinear second order inhomogeneous differential equations in one dimension

TL;DR

This paper investigates inhomogeneous nonlinear second-order differential equations in one dimension, focusing on equations with point or continuous sources, and develops Green's function methods for solving cubic nonlinear cases.

Contribution

It introduces methods to determine Green's functions for nonlinear second-order equations with point and continuous sources, especially for cubic nonlinearities.

Findings

Green's functions can be explicitly constructed for certain nonlinear equations.

Solutions for continuous sources can be derived from point source Green's functions.

The approach applies to equations with cubic nonlinear potentials.

Abstract

We study inhomogeneous nonlinear second-order differential equations in one dimension. The inhomogeneities can be point sources or continuous source distributions. We consider second order differential equations of type $\phi''(x) + V(\phi(x)) = Q \, \delta(x) $, where $V(\phi)$ is a continuous, differentiable, analytic function and $Q \,\delta (x)$ is a point source. In particular we study cubic functions of the form $V(\phi(x)) = A\,\phi(x) + B\,\phi^3(x)$. We show that Green functions can be determined for modifications of such cubic equations, and that such Green's functions can be used to determine the solutions for cases where the point source is replaced by a continuous source distribution.