Dimensional deformation of sine-Gordon breathers into oscillons

Abstract

Oscillons are localized field configurations oscillating in time with lifetimes orders of magnitude longer than their oscillation period. In this paper, we simulate non-travelling oscillons produced by deforming the breather solutions of the sine-Gordon model. Such a deformation treats the dimensionality of the model as a real parameter to produce spherically symmetric oscillons. After considering the post-transient oscillation frequency as a control parameter, we probe the initial parameter space to continuously connect breathers and oscillons at various dimensionalities. For sufficiently small dimensional deformations, we find that oscillons can be treated as perturbatively deformed breathers. In $D\gtrsim 2$ spatial dimensions, we observe solutions undergoing intermittent phases of contraction and expansion in their cores. Knowing that stable and unstable configurations can be mapped to disjoint regions of the breather parameter space, we find that amplitude modulated solutions are located in the middle of both stability regimes. These solutions display the dynamics of critical behavior around the stability limit.

Figures (23)

• Figure 1: Radial breather profiles \ref{['eqn:breather-profile']} for several choices of the parameter $\omega_{\rm B}$ describing the breather's frequency. For $0 < \mu-\omega_{\rm B} \ll 1$, the profiles have small amplitude at the origin and damp slowly as $\mu r\to\infty$. As $\omega_{\rm B}$ is increased, the breathers become more peaked at the origin and damp more rapidly at large radii. We explicitly illustrate the breathers that just probe the inflection point of the potential ($\omega_{\rm B} = 0.92\mu$) and the nearest maximum of the potential ($\omega_{\rm B} = 0.71\mu$). For reference, we also plot the minimum ($\omega_{\rm B} = 10^{-1}\mu$) and maximum ($\omega_{\rm B} = 0.95\mu$) frequency breathers used as initial conditions for our simulations.
• Figure 2: Showing the spatial structure, time evolution of oscillon cores and determination of the oscillation frequency ($\omega_{\mathrm{osc}}$) for $\varepsilon=0.75$. Left panel: Typical evolution of the radial profile for a long-lived oscillon, deformed from a SG breather with initial frequency $0.3\mu$ and no phase. Middle panel: Radial profile of a decaying oscillon (observe the time axis in logarithmic scale). Right panel (top): Time evolution of the oscillon evaluated at $r=0$ to determine the dominant frequency and the amplitude $\mathcal{A}$. Right panel (bottom): From the temporal power spectrum $\mathcal{P}_{\omega}$ of the top panel, we determine $\omega_{\mathrm{osc}}/\mu$ to be the angular frequency with the highest peak (marked by the red dot) in power.
• Figure 3: Parameter flow at $\varepsilon=0.5$ for four initial breather frequencies. In the first two panels, we show the evolution of $\mathcal{A}$ (i.e., the red envelope in Fig. \ref{['fig:sols_n_freq']}) and the oscillon's frequency $\omega_{\mathrm{osc}}$ as functions of time. We observe that both parameters evolve very quickly towards an attractor. Once the convergence occurs, the flows slow down but do not stop. In the last panel, we see that the flow in parameter space collapses into a line, aligned along the breather flow line (in orange, dubbed as $\varepsilon=0$) given by Eq. \ref{['eqn:breather-amp']}. The arrows only represent the direction of time, since the speed can be inferred from the first two panels of this figure.
• Figure 4: Surfaces showing the oscillon frequency as a function of the phase and frequency of the initial SG breathers for $\varepsilon = 0.125, 0.25, 0.5$ and $0.75$. Regions within isocontours of oscillation frequency are produced from a grid of $50\times 50$ initial configurations of frequencies $(\omega_{\mathrm{ini}})$ and phases $(\theta_0)$, uniformly distributed in $\log_{10}(\omega_{\mathrm{ini}}/\mu)\in [-1,-0.02]$ and in $\theta_0\in [0,\pi)$. The frequency of stable oscillons (colored in ivory in all of the panels) formed by breathers with $\omega_{\mathrm{ini}}=\omega_{\rm B}\lesssim 0.3\mu$ increases with $\varepsilon$. For the span of initial conditions considered here, the quantity of scenarios where oscillons (with $\omega_{\mathrm{osc}}< \mu$) form tends to decrease as $\varepsilon$ increases. Solutions within the transition regions (in red) may have non-trivial modulation in their core during the transient. Fig. \ref{['fig:diag_n_mod']} focuses in the cases enclosed in the blue rectangle ($\varepsilon=0.75$, in the right bottom panel), which exhibit time-dependent modulation in their amplitude. In the upper right panel (dubbed as $\varepsilon=0.25$), we plotted a green dashed curve as an inset (to the right) to show how the frequency changes for a frequency span at constant phase ($\theta_0=0.64$). Observing a constant frequency plateau which extends over the whole span of $\omega_{\mathrm{ini}}$ in the ivory region, and breaks as $\omega_{\mathrm{ini}}/\mu\rightarrow 1$. Throughout the remaining sections of this paper, the dependence of the oscillation frequency in $\omega_{\mathrm{ini}}$ transforms in various ways to represent the dynamical state of the oscillating field. The upper bound in $\omega_{\mathrm{ini}}$ is set to resolve oscillons within a simulation box of length $\ell=200\mu^{-1}$.
• Figure 5: Left panel: Time evolution of $\phi(0,t)/\phi_{\star}$ for some of the solutions in the blue rectangle (case $\varepsilon=0.75$ of Fig. \ref{['fig:ct_eps_freq']}). We are using the same color code as in Fig. \ref{['fig:ct_eps_freq']}, showing that the solutions with modulated amplitude serve as "transition solutions" between the stable oscillons (in orange) and the fast decaying profiles (in gray). Interestingly, the frequency of the modulating envelope reduces as one approaches the unstable solutions. Central panel: Spatial structure of an amplitude modulated solution for $\varepsilon=0.75$. Amplitude modulation is associated to periodic phases of contraction and expansion of the oscillon core. Right panel: Energy density as a function of radius and time for the solution in the middle panel. Modulation occurs as energy leaves the core in a discrete number of bursts. In the middle and right panels, the black dashed lines correspond to constant-time snapshots of the field (central panel) and energy density (right panel), rescaled to fit in both panels. Rasterization suppresses most of the high-frequency structures in the evolution of field and energy density. Shaded areas below the dashed lines give a qualitative estimate of the field and energy density values. To show the peaks and throughs in the central and right panels, time slices in the middle and right panels do not match.