Exact solution to a nonlinear Klein-Gordon equation

Abstract

The nonlinear Klein-Gordon equation ∂μ∂μΦ + M2Φ + λ1Φ1−m + λ2Φ1−2m = 0 has the exact formal solution Φ = [u2m −λ1um/(m − 2)M2+λ12/(m−2)2M4−λ2/4(m − 1)M2]1/mu−1, m ≠ 0, 1, 2, where u and v−1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [aum + b(uv)m/2 + cvm]k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.