Breather Solutions of the Nonlinear Wave Equation

TL;DR

This paper develops series solutions for breather solutions in Klein-Gordon equations, revealing that the sine-Gordon equation uniquely admits pole-free series expansions, explaining the special nature of its breathers.

Contribution

It constructs all-order series solutions for breathers in Klein-Gordon equations and identifies the sine-Gordon equation as uniquely allowing pole-free expansions, linking PDE properties to breather stability.

Findings

Series solutions constructed for Klein-Gordon breathers.

Pole-free series occur only for sine-Gordon equation.

Explains the uniqueness of sine-Gordon breathers from a PDE perspective.

Abstract

We construct series solutions to all orders for breathers of Klein-Gordon equations, in powers of an amplitude parameter epsilon, under a sign condition on the coefficients of the expansion of the nonlinearity. All terms may be computed thanks to the properties of a 1D Schr\"odinger equation with two-soliton potential. We prove that this series is free of poles in the unit disc of the complex epsilon plane if and only if the equation is essentially equivalent to the sine-Gordon equation. This appears to be the only result that explains both what makes the sine-Gordon equation special from a PDE point of view, and at the same time, why approximate, long-lived breathers exist. In a related paper (Classical and Quantum Gravity, 25 (2008) 245004 ) we have extended this approach to address the soliton star problem in General Relativity.