Understanding oscillons: standing waves in a ball

Oscillons are localised long-lived pulsating states in the three-dimensional $\phi^4$ theory. We gain insight into the spatio-temporal structure and bifurcation of the oscillons by studying time-periodic solutions in a ball of a finite radius. A sequence of weakly localised {\it Bessel waves} -- nonlinear standing waves with the Bessel-like $r$-dependence -- is shown to extend from eigenfunctions of the linearised operator. The lowest-frequency Bessel wave serves as a starting point of a branch of periodic solutions with exponentially localised cores and small-amplitude tails decaying slowly towards 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.