Oscillons and quasibreathers in the ${\varphi }^4$ Klein-Gordon model

Strong numerical evidence is presented for the existence of a continuous family of time-periodic solutions with “weak” spatial localization of the spherically symmetric nonlinear Klein-Gordon equation in $3+1$ dimensions. These solutions are “weakly” localized in space in that they have slowly decaying oscillatory tails and can be interpreted as localized standing waves (quasibreathers). By a detailed analysis of long-lived metastable states (oscillons) formed during the time evolution, it is demonstrated that the oscillon states can be quantitatively described by the weakly localized quasibreathers. It is found that the quasibreathers and their oscillon counterparts exist for a whole continuum of frequencies.

Figure 1

The upper and lower envelope of the oscillations of the field $\varphi (r=0)$ is shown for the evolution of Gaussian initial data of the type (12) with two different sets of initial parameters. It is important to note that the field $\varphi$ oscillates with a nonconstant period $4.6<T<5$. For generic oscillons the period $T$ shows a decreasing tendency towards $T\approx 4.6$.

Figure 2

Upper envelope of the field value for different oscillon states close to the 27th peak of the lifetime curve belonging to ${\varphi }_c=1$. These states are characterized by initial value $r_0$ which is near $r_{27}^{*}\approx 2.39297$. The near-periodic state is approximately at $2100<t<2800$ for the fine-tuned oscillons in the lower plot.

Figure 3

The oscillon lifetime, $\tau$, versus $-\ln |r_0-r_0^{*}|$ is shown for states close to 3 different resonances. The lower lines plotted with a given line type represent subcritical, $r_0<r_0^{*}$, states while the upper lines with the same line type correspond to supercritical, $r_0>r_0^{*}$, solutions.

Figure 4

Upper envelopes of the oscillations of the field value at $r=0$ for three pairs of subcritical and supercritical oscillon states close to the first resonances corresponding to three given ${\varphi }_c$ initial parameters. The fine-tuning was performed using 32 digit arithmetics.

Figure 5

The pulsation frequency, $\omega$, in function of the time for the three pairs of oscillons shown on the previous figure during their corresponding near-periodic states.

Figure 6

The time dependence of the energy contained in spheres of radii $r=19.5$, $r=40.1$, and $r=107.1$ for the ${\varphi }_c=-0.4$ first peak. The thickness of the curve indicates that the high frequency oscillation is still not negligible at these high radii. The different endings of the curves correspond to subcritical and supercritical states.

Figure 7

Radial dependence of Fourier modes obtained by decomposition of a near-periodic state with frequency $\omega =1.41203$.

Figure 8

Convergence of the various results with respect to the precision required when finding the minimum of $|A_2|$.

Figure 9

Value of $A_2$ as a function of the phase ${\phi }_2$, for $\omega =1.37$. The other phases are maintained equal and set to 1.0 (solid line), 1.6 (dashed line), and 2.2 (dotted line).

Figure 10

Same as Fig. 8 but when varying the $N_r$ in each domain.

Figure 11

Same as Fig. 8 but changing the value of $R_{\lim }$.

Figure 12

Same as Fig. 8 but varying the number of modes.

Figure 13

Relative difference between the lhs and rhs members of Eq. (3). The result is averaged over one period and we present the error as a function of $\omega$, for 5, 6, and 7 modes.

Figure 14

Coefficients of the oscillatory homogeneous solutions ${\phi }_n$ ($n=2$, 3, 4, 5) as a function of $\omega$.

Figure 15

Energy density for various values of $\omega$, near the origin (left panel) and in the region where the oscillatory tails dominate (right panel). $R_{\lim }$ is equal to 44.

Figure 16

Same as Fig. 15 but with the volume element included: $r^2\xi$.

Figure 17

The transition radius between the exponential decay and the oscillatory tail (left panel) and the energy inside this radius (right panel), as a function of $\omega$.

Figure 18

The first modes near the origin and in the transition region for $\omega =1.32$, 1.365, and 1.39.

Figure 19

Comparison between the modes coming from solving the system (17) directly and the real part of the modes obtained by performing a Fourier transform of the evolution for $\omega =1.398665$. In order to be able to represent on the same graph the large values in the central oscillon region along with the small tails at high radii, the absolute values are plotted logarithmically. We note that the central value ${\varphi }_n(0)$ is positive for $n=0$ and for odd $n$ while it is negative for other $n$. The peaks pointing downwards on the figure correspond to places where the functions change signature.

Figure 20

Comparison of the higher modes of the two systems presented on the previous figure.

Figure 21

Dependence on the radius $r$ of the relative difference of the modes coming from the mode decomposition and the Fourier transformation for $\omega =1.398665$. The relative difference is defined as the absolute value of the ratio of the difference and the mode decomposition value.

Figure 22

The first two graphs compare the oscillatory region of the mode ${\varphi }_2$ coming from solving the system (17) directly and the complex ${\varphi }_2$ obtained by performing Fourier transform on the result of the evolution code, for $\omega =1.38$. The first graph shows the real and imaginary parts separately, while the second compares the absolute values. The third graph plots the radial dependence of the complex argument of the modes ${\varphi }_1$ and ${\varphi }_2$ obtained by the Fourier transform.

Figure 23

Time evolution of the underlying high frequency oscillation of the oscillon state of Fig. 1 is presented.

Figure 24

The modes of the periodic solution with frequency 1.3 are compared with the Fourier decomposition of two different oscillon states. The first oscillon state with initial data ${\varphi }_c=1$, $r_0=2.70716$ is decomposed between the two subsequent maxima after $t=471.47$, where the calculated frequency is 1.2995. The second state with ${\varphi }_c=1$, $r_0=2.39$ is decomposed just after $t=298.72$, where the frequency is 1.3011.

Figure 25

Comparison of the higher modes of the two systems with frequency $\omega =1.3$ presented on the previous figure.

Figure 26

The modes of the periodic solution with frequency 1.36 is compared with the Fourier decomposition of two different oscillon states. The first oscillon state with initial data ${\varphi }_c=1$, $r_0=2.70716$ is decomposed between the two subsequent maxima after $t=4158.76$, where the calculated frequency is 1.3596. The second state with ${\varphi }_c=1$, $r_0=2.39$ is decomposed just after $t=1526.58$, where the frequency is 1.3613.

Figure 27

Comparison of the higher modes of the two systems with frequency $\omega =1.36$ presented on the previous figure.

Figure 28

On the first graph the upper and lower envelopes of the oscillations of the field at the origin are shown as functions of time for numerical simulations with various number of spatial grid points. The initial data is generated by solving the system (17) for the modes at the pulsation frequency $\omega =1.38$. On the second graph the time dependence of the oscillation frequency is plotted for each simulation.

Figure 29

Initial behavior of the maximum curve of the evolution of the $\omega =1.38$ initial data with numerical resolutions corresponding to various number of spatial grid points. The points on the curves represent the actual maximum points of the oscillating field $\varphi$ at the origin $r=0$.

Figure 30

Initial behavior of the upper envelope of the oscillations at $r=0$ of the evolution of the $\omega =1.30$ and $\omega =1.36$ initial data for numerical simulations with various number of spatial grid points.

Figure 31

Time evolution of the upper envelope curve of the oscillation of $\varphi$ at $r=0$ generated by three different initial data. The first two initial data are provided with the mode decomposition method, solving the system (17) by choosing the frequencies $\omega =1.30$ and $\omega =1.36$. The third evolution corresponds to Gaussian-type initial data (12) with ${\varphi }_c=1$ and $r_0=2.70716$.

Figure 32

Time evolution of the frequency of the oscillation of $\varphi$ at $r=0$ for the three kinds of evolution simulations shown on the previous figure.

Figure 33

Fourier decomposition of the evolution of the $\omega =1.3$ initial data close to the moment of time when its frequency increases to 1.36. The results are compared to the modes of the periodic solution obtained by the mode decomposition method for the $\omega =1.36$ frequency.

Figure 34

Same thing as Fig. 33 but for the higher order modes.