Techniques for Solving Static Klein-Gordon Equation with Self-Interaction $\lambda {\phi }^4$ and Arbitrary Spherical Source Terms

The Klein-Gordon equation for a scalar field sourced by a static spherically symmetric background is an interesting second-order differential equation with applications in particle physics, astrophysics, and elsewhere. Here we present static solutions for generic source density profiles in the case where the scalar field has no interactions or a mass term. For a $\lambda {\phi }^4$ self-interaction term, we develop the techniques that are necessary numerical computation. We also provide code to perform the numerical calculations that can be adapted for arbitrary density profiles.