Understanding oscillons: Standing waves in a ball

Abstract

Oscillons are localized long-lived pulsating states in the three-dimensional ${\ensuremath{\phi}}^{4}$ theory. We gain insight into the spatiotemporal structure and bifurcation of the oscillons by studying time-periodic solutions in a ball of a finite radius. A sequence of weakly localized Bessel waves---nonlinear standing waves with the Bessel-like $r$-dependence---is shown to extend from eigenfunctions of the linearized operator. The lowest-frequency Bessel wave serves as a starting point of a branch of periodic solutions with exponentially localized cores and small-amplitude tails decaying slowly toward the surface of the ball. A numerical continuation of this branch gives rise to the energy-frequency diagram featuring a series of resonant spikes. We show that the standing waves associated with the resonances are born in the period-multiplication bifurcations of the Bessel waves with higher frequencies. The energy-frequency diagram for a sufficiently large ball displays sizeable intervals of stability against spherically symmetric perturbations.