Breather Solutions of the Cubic Klein-Gordon Equation

Abstract

We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein-Gordon equation \begin{align} \partial_t^2 U - ΔU + m^2 U = Γ(x) U^3 \quad \text{on } \mathbb{R} \times \mathbb{R}^3, \end{align} which are weakly localized in space. Various families of such "breather" solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author's knowledge, a new concept and based on the reformulation of the cubic Klein-Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.

Part of author's dissertation thesis (DOI below)

Cited by in corpus (3)

• Variational methods for breather solutions of Nonlinear Wave Equations
• Group classification and exact solutions of a class of nonlinear waves
• On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting