Figure 1

Schematic description of the linear radiation peak centered on ${\omega }_{\mathrm{lin}}$ being shifted to the left by the presence of nonlinearities. An oscillon will form if the peak is shifted far enough into the nonlinear region such that it no longer overlaps significantly with the linear peak.Reuse & Permissions

Figure 2

Minimum oscillon energy as a function of core amplitude for a double-well potential in $d=3$. The minimum of this curve is the attractor point with $E_{\infty }\simeq 37.69$.Reuse & Permissions

Figure 3

The thick solid line represents the locus of points satisfying the condition ${\omega }_{\mathrm{gap}}={\Gamma }_{\mathrm{lin}}$ in $d=2$ (“line of existence”) calculated analytically. The thin solid line represents the locus of points that have the attractor energy $E\simeq 4.44$. The dashed line is the numerical minimum radius based on Gaussian initial configurations. Oscillons can only exist if their energy $E\ge E_{\infty }$ and if they have a core amplitude and average radius lying above the line of existence. This is why the numerically measured minimum radius follows the line of existence for $A\gtrsim 1$ but follows the line of minimum energy for $A\lesssim 1$. The arrow indicates the value of the (constant) minimum radius calculated in 8. The circle represents the location of the attractor point; since it lies above the line of existence, oscillons will be perturbatively stable in this system.Reuse & Permissions

Figure 4

The thick solid line represents the locus of points satisfying the condition ${\omega }_{\mathrm{gap}}={\Gamma }_{\mathrm{lin}}$ in $d=3$ (line of existence) calculated analytically. The thin solid line represents the locus of points that have the attractor energy $E\simeq 37.69$. The dashed line is the numerical minimum radius based on Gaussian initial configurations. The arrow indicates the value of the (constant) minimum radius calculated in 8. Oscillons can only exist if their core amplitude and average radius are above the curve (wherein $E\ge E_{\infty }$ is automatically satisfied). The circle represents the location of the attractor point; since it lies below the line of existence, oscillons will not be absolutely stable in this system.Reuse & Permissions

Figure 5

Schematic of an oscillon frequency distribution showing the tail penetrating the radiation region. The graph is plotted in units of ${\omega }_{\mathrm{mass}}$. In the inset we show a close-up view of $\mathcal{F}(\omega )$ and $b(\omega )$, respectively, (dashed lines), and their product $\Omega (\omega )$ (solid line) for the system given by Eq. (1) in $d=3$ for typical values of the various parameters. Note that the curves have been vertically scaled so that all are visible on the same graph.Reuse & Permissions

Figure 6

Analytical calculation of $g={\rho }_{\dot{E}}-1$ (top curve) and ${\rho }_A$ (bottom curve) using Eq. (32), plotted against the reference amplitude $A_r$ along the line of existence. This gives the spectrum of possible values of $g$ and ${\rho }_A$ across the various oscillons in this system. The dots mark the theoretical values of $g\simeq 2.67$ and ${\rho }_A\simeq .66$ assumed by the longest-lived oscillon, which has $A_r\simeq .92$ (thicker line in Fig. 8).Reuse & Permissions

Figure 7

Analytical calculation of ${\gamma }_{\dot{E}}$ (top graph) and ${\gamma }_A$ (bottom graph) using Eq. (33), plotted against the reference amplitude $A_r$ along the line of existence. This gives the spectrum of possible values of ${\gamma }_{\dot{E}}$ and ${\gamma }_A$ across the various oscillons in this system. The dots serve to mark the theoretical values of ${\gamma }_{\dot{E}}\simeq 2.0\times {10}^{-6}$ and ${\gamma }_A\simeq .21$ assumed by the longest-lived oscillon, which has $A_r\simeq .92$ (thicker line in Fig. 8).Reuse & Permissions

Figure 8

Various oscillon trajectories. Note that they all tend to the attractor point but intersect the line of existence (dashed line) before doing so. The thicker trajectory marks the longest-lived oscillon (it intersects the line of existence at $A_r\simeq .92$ and $R_r\simeq 3.25$).Reuse & Permissions

Figure 9

Analytical results for critical values of the amplitude (top) and radius (bottom) along the line of existence (dashed lines) as a function of lifetime are plotted with several examples of short-and long-lived oscillons, all with initial amplitudes $A_0=2$. From left to right, the initial radii for the oscillons are 2.35, 2.41, 2.53, 2.65, and 2.86. It is quite clear that the oscillons decay as they cross the critical values computed analytically.Reuse & Permissions

Figure 10

Oscillon lifetimes vs decay energy $E_D=E_r$. Solid curve is theoretical, dashed line is numerical. The theoretical curve (with an error of $\sim 6\%$ in the horizontal positioning of the peak) correctly predicts the shape of the distribution and that there exists a maximum lifetime in this system on the order of $\sim {10}^4$.Reuse & Permissions

Figure 11

Comparison of theoretical (continuous line) vs numerical (dashed line) radius (top), frequency (middle), and amplitude (bottom) for an oscillon with initial conditions ($A_0=2$, $R_0=2.86$), showing very good agreement [theoretical results are computed with $(A_r,R_r)\simeq (.92,3.25)$].Reuse & Permissions

Figure 12

Comparison of theoretical (continuous line) vs numerical (dashed line) values for the energy of the longest-lived oscillon, obtained with initial conditions ($A_0=2$, $R_0=2.86$) showing excellent agreement [$(A_r,R_r)\simeq (.92,3.25)$].Reuse & Permissions

Figure 13

Comparison of theoretical (continuous line) vs numerical (wavy line) radiation rate for an oscillon with initial conditions ($A_0=2$, $R_0=2.86$), showing excellent agreement. The inset, which is plotted on a log scale, makes it clear that the theory correctly reproduces the rapid initial drop in radiation rate over many orders of magnitude; the linear scale on the larger graph shows that the theory correctly reproduces the extremely small (but finite) radiation rate towards the end of the oscillon’s life [$(A_r,R_r)\simeq (.92,3.25)$].Reuse & Permissions

Figure 14

Comparison of theoretical trajectory (solid curve) vs numerical trajectory (dashed curve) for the longest-lived oscillon ($A_0=2$, $R_0=2.86$), showing very good agreement during the more stable phase of the oscillon’s life ($A\lesssim 1.5$). The great increase in density of data points in the dashed line in the range $.9\lesssim A\lesssim 1.5$ is due to the prolonged period of time spent in this region by the oscillon (i.e., the “plateau” phase). The end point where the dashed line again becomes dashed ($A\simeq .9$, $R\simeq 3.2$) signals the numerical decay point of the oscillon. The thick segment of the solid line highlights the portion of the theoretical trajectory during the low-radiation plateau phase; the end of the thick segment $(A_r,R_r)\simeq (.92,3.24)$ marks the theoretical decay point, showing very good agreement. It is interesting to note that, even after the oscillon decay at $A\simeq .9$, the remaining field configuration continues to tend to the attractor point at $(A_{\infty },R_{\infty })\simeq (.456,4.79)$, as does the theoretical curve.Reuse & Permissions

Figure 15

The top graph shows the theoretical calculation of $\Sigma$ vs time and the bottom shows ${\Sigma }_{\min }$, both for the oscillon with $(A_r,R_r)\simeq (.92,3.25)$. The stability measured by $\Sigma$ is clearly related to the radiation rate of the oscillon: this kind of stability increases in time, since the radiation rate decreases. On the other hand, the stability measured by ${\Sigma }_{\min }$ is related to the resistance of the oscillon to decay: this kind of stability decreases in time as the oscillon moves closer to the line of existence.Reuse & Permissions

Figure 16

The solid curve is the theoretical calculation of ${\mathcal{T}}_{\mathrm{decay}}$ for $(A_r,R_r)\simeq (.92,3.25)$. The dashed curve is the numerically measured period of the superimposed oscillation, showing very good agreement.Reuse & Permissions

Figure 17

The four oscillon time scales plotted over time. From top to bottom we have plotted ${\mathcal{T}}_{\mathrm{relax}}$, ${\mathcal{T}}_{\mathrm{decay}}$, ${\mathcal{T}}_{\mathrm{linear}}$, and ${\mathcal{T}}_{\mathrm{osc}}$. It can clearly be seen here that an oscillon is an object governed by multiple time scales spanning many orders of magnitude.Reuse & Permissions