Oscillons are spatially stationary, quasi-periodic solutions of nonlinear field theories. (*Informal version: oscillons are what would happen if a stone was thrown into a pond and, rather than creating ripples, the water bounced up and down at the point of impact – this doesn’t (usually!) happen with water, but is possible in systems where the “restoring force” does not increase continuously with displacement.*) Oscillons can be seen in a wide range of settings, including granular systems, low temperature condensates, and in models of the very early universe.

ArXiv:1612.07228 (Liu and Easther) introduces a new class of oscillons which can have an off-centre spatial profile and an envelope whose overall shape evolves with time. These oscillons are exact solutions of the sine-Gordon potential in 1 dimension and have two distinct frequency parameters, ω_{1} and ω_{2}, both of which must be less than one; see equations 4-7 of ArXiv:1612.07228. In the limit where one frequency is exactly unity the solution reduces to the well-known “breather”, shown below:

**Pulsating Oscillons** When both frequencies are close to unity the envelope pulsates slowly with time while the profile is peaked at the centre:

**Double-peak Breathers **For a wide range of parameters these oscillons have a complex “double peak” structure, where the peak field amplitude occurs away from the oscillon centre, overlaid with a complex pulsating envelope:

**Large-amplitude Limit **When one of the frequencies approaches zero we see a fast, small amplitude oscillation, superimposed on a much slower, large amplitude variation.

The above solutions are all plotted from an exact, analytic solution to the sine-Gordon equation, a 1+1 dimensional PDE. (Equations 4-7 of ArXiv:1612xxxx); the full solution space contains a very rich set of possible behaviours.

Solutions with similar properties have previously been seen in variety of settings but their properties had never been understood or analysed in any detail. We use the exact 1-D solution to initialise a numerical solver (with an assumption of rotational symmetry in more than one dimension) for a range of potentials, allowing us to demonstrate that qualitatively similar solutions can be supported by a wide range of potentials.

We are particularly interested in the oscillons seen in simulations of the dynamics following monodromy inflation, an interesting class of string-motivated cosmological models. In these scenarios the resonant growth of perturbations renders the universe very inhomogeneous on microscopically small scales during an epoch roughly 10^{-33} seconds after the Big Bang. A full 3D simulation of this oscillon-dominated phase is shown below; these results are described in more detail in ArXiv:1106.3335. The simulations begin with an initially homogeneous, oscillating inflaton condensate which undergoes parametric resonance, rendering the universe highly inhomogeneous, followed by oscillon formation (the point at which the video below begins.) The oscillons are often initially highly asymmetric but relax to approximately spherical configurations. Once the oscillon dominated phase is well-established the mixture of short- and long period oscillations are clearly visible; the latter are most obvious in the oscillons which intersect the sides of the box. (The orange contours show regions in which local density is six times greater than the average density.)