Long-time behaviour of sphalerons in $\phi^4$ models with a false vacuum

Evolution of sphalerons in a class of quartic Klein-Gordon models are studied under a growing perturbation. Sphalerons are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation increases in time and asymptotically approaches the speed of light. To explain this behaviour analytically, a nonlinear collective coordinate method is developed which has three dynamical parameters and leads to an explicit asymptotic solution using a power series expansion. The solution describes the emergence of a spreading tabletop profile whose height approaches the true vacuum while its flanks steepen and accelerate outward. In addition, the energy density is shown to concentrate at the flanks, indicating the onset of a gradient blow-up at large times. These results provide a detailed description of the long-time dynamics of positively perturbed sphalerons, and reveal a universal mechanism for the formation of relativistically expanding structures in nonlinear field theories.