Global dispersive asymptotics for perturbations of the phi^4 kink

Establish global-in-space dispersive asymptotics for perturbations of kink solutions in the classical one-dimensional phi^4 model, namely the cubic wave equation \partial_t^2\phi-\partial_x^2\phi=\phi-\phi^3.

Background

The paper studies a one-dimensional nonlinear KleinGordon equation with a localized quadratic interaction, a nonlocalized quartic interaction, and a linearized operator possessing a single internal mode. Under a Fermi Golden Rule condition, it establishes global dynamics for the internal-mode amplitude and scattering for the radiation component.

The unresolved problem concerns the lower-degree cubic model associated with perturbations of the classical phi^4 kink. In that setting, the radiation self-interacts through a cubic nonlinearity, creating substantially stronger long-range difficulties than in the quartic model treated in the paper. The authors present their quartic analysis as an intermediate step toward understanding this still-unresolved global dispersive behavior.