Breather solutions of the cubic Klein–Gordon equation

We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation
$$\partial_t^2U-\Delta U+m^2U = \Gamma(x)U^3\quad\text{on } \mathbb{R}\times\mathbb{R}^3,$$
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.