Radial vibrations of BPS skyrmions

We study radial vibrations of spherically symmetric Skyrmions in the Bogomol’nyi-Prasad-Sommerfield Skyrme model. Concretely, we numerically solve the linearized field equations for small fluctuations in a Skyrmion background, both for linearly stable oscillations and for (unstable) resonances. This is complemented by numerical solutions of the full nonlinear system, which confirm all the results of the linear analysis. In all cases, the resulting fundamental excitation provides a rather accurate value for the Roper resonance, supporting the hypothesis that the Bogomol’nyi-Prasad-Sommerfield Skyrme model already gives a reasonable approximate description of this resonance. Furthermore, for many potentials additional higher resonances appear, again in agreement with known experimental results.

Figure 1

Normalized modes (true oscillational and collective) for the ${\varphi }^4$ model.

Figure 2

Relaxation of the field for squeezed soliton initial conditions and their spectra for different values of the parameter $\epsilon$. The solid lines in the right panels represent the Fourier transform for $t\in [100,1800]$; the dashed lines include earlier times $t\in [0,1800]$. The gray areas represent the nonpropagating region below the mass threshold. (a) $\epsilon =0.2$ oscillational mode excited, almost constant amplitude, small radiation via second harmonics; (b) $\epsilon =0$ pure sG case, slow modes on the threshold $\omega =1$ dominate long time evolution; (c) $\epsilon =-0.1$, fast decay of the resonance mode and remaining tail at the mass threshold; (d) similar to (c) but with longer decay of the resonance and a clearly seen difference in frequency; (e) $\epsilon =-0.25$ massless case with decaying resonance and polynomial tail described by Bizon [39].

Figure 3

Relaxation of the field for wobbler initial data for sine-Gordon (2.30) for three different initial separations [(a) $a=2$; (b) $a=3$; (c) $a=5$]. The nonlinear nature of these objects, with basic frequency below the mass threshold and many higher harmonics, is clearly seen.

Figure 4

The Fourier transform of the field at $x=1$ obtained for the time evolution of squeezed soliton initial data. The upper plot is obtained for all times and the bottom plot is obtained for $t>10$ to exclude the decayed resonance mode. The spectra were normalized in such a way that the maximum of the power spectrum for each $\epsilon$ is 1. For $\epsilon >0$, the evolution is clearly dominated by the oscillational mode. For $\epsilon <0$, both the resonance mode and mass threshold evolution dominate the behavior of the field. The resonance mode decays faster and in the bottom plot it continues to be visible only for $\epsilon$ close to $-1/4$. The collective Derrick mode provides a reasonable approximation only for $\epsilon >0$, where it seems to be quite close to the oscillational mode. However, for $\epsilon <0$, it misses both the mass threshold and the resonance mode.

Figure 5

Effective potential $Q$ with the numerically calculated energy levels ${\lambda }_i={\omega }_i^2$ for the step-function Skyrme potential, for the parameter value ${\mathit{l}}^6=2$, corresponding to $R_1=1.494$.

Figure 6

[(1) and (2)] Effective potential $Q$ with the first few energy levels for the potential $\mathcal{U}={\chi }^{2/3}$. [(3) and (4)] Effective potential $Q$ with the first few energy levels for the potential $\mathcal{U}={\chi }^2$. In both cases, the scale constant is ${\mathit{l}}^6=2$.

Figure 7

Effective potentials $Q$ for ${\mathcal{U}}_{\epsilon }=\sqrt{{\epsilon }^2+{\chi }^2}-\epsilon$, for the scale constant ${\mathit{l}}^6=2$.

Figure 8

Effective potential $Q$ for $\mathcal{U}={\chi }^{2a}$, $a>1$, for the scale constant ${\mathit{l}}^6=2$.

Figure 9

Cross section multiplied by ${\omega }^2/4\pi$: ${\sin }^2{\delta }_0$. Maxima indicate approximately real values of the resonant frequency. We can see the drift of the resonance as the barrier is crossed. When the frequency is larger than ${\omega }_{\max }=\sqrt{Q_{\max }}$, the resonance becomes wide. Bright lines show the position of narrow resonances (too narrow to be seen on the scan). We remark that in the left plot the lowest (fundamental) resonance for small values of $a$ is too narrow to be visible.

Figure 10

Resonance frequencies and decay widths for the first (fundamental) resonance for different values of $a$. In the first plot we also compare with the frequency of the collective (Derrick) mode.

Figure 11

Evolution of the field and power spectra for potentials $\mathcal{U}={\chi }^{2a}$ for different $a>1$ [(a) $a=1.80$; (b) $a=1.50$; (c) $$a=1.20$; (d) $a=1.10$]. For $a=1.10$ there is a number of narrow excited resonances, resulting in oscillations with almost constant amplitude, whereas for $a=1.80$ there is a single and much wider resonance that manifests itself with decaying oscillations.

Figure 12

The power spectra scan for different $a$. It is clearly seen that the peaks from the full numerical time evolution precisely coincide with the frequencies of the bound modes for $a<1$ (for compactons) and with the resonance frequencies for $a>1$. It is also visible that for $\omega >{\omega }_{\max }$ the resonances quickly broaden and disappear. For comparison, we also plot the Derrick mode, which is close to the first resonance.

Figure 13

Evolution of the perturbed compacton with $A=0.1$ and $a=0.8$. It can be seen that some radiation escapes the compacton, although, at least for the chosen parameter values, the amplitude of the radiation is very small (observe the logarithmic scale; in the log plot, we actually plot $|\chi |$). Also, the frequency of the radiation outside the compacton seems to increase, and the radiation seems to “freeze” at some finite distance.

Figure 14

Evolution of the field and power spectra for potentials $\mathcal{U}={\chi }^{2a}$ for different $a<1$ (the compacton case) [(a) $a=0.90$; (b) $a=0.80$; (c) $a=0.70$; (d) $a=0.60$]. The amplitude of oscillations does not seem to change showing that hard wall approximation provides an appropriate description. The peaks correspond to bound states of the compacton.

Figure 15

Frequency shift with amplitude due to the nonlinearity for different $a$ [(a) $a=1.50$; (b) $a=1.20$; (c) $a=1.05$; (d) $a=0.95$; (e) $a=2/3$].

Figure 16

Evolution of the perturbed compacton, with the perturbation induced by a time-dependent (energy nonconserving) boundary condition inside the compacton (at $r=2$), $\chi (2,t)={\chi }_0(2)+A\sin (\omega t)$, for $A=0.001$, $\omega =4$, and $a=0.95$. The dashed line is the unperturbed compacton. It can be seen that some radiation escapes the compacton although, at least for the chosen parameter values, the amplitude of the radiation is very small (observe the logarithmic scale; in the log plot, we actually plot $|\chi |$). Also, the frequency of the radiation outside the compacton seems to increase with distance. The radiation no longer freezes at a finite distance, because of the energy nonconserving boundary condition that effectively pumps energy into the system.

Figure 17

Evolution of the perturbed compacton, as in Fig. 16, but for the magnified radiation zone and for a nonlog scale.

Figure 18

A resonance mode found for massless double sine-Gordon $\epsilon =-1/4$ for frequency $1.2665+0.4440i$ in WKB foliation using the rational Chebyshev spectral method for semi-infinite interval $x\in [0,\infty ]$. The frequency is very close to the frequency found using the linear simulation method. Note that the solution tends to 1 as $x$ grows to infinity without any oscillations.

Figure 19

Comparison between the solution in WKB foliation (left) and in the standard $t=\text{const}$ time sheet (right). BPS model, $a=1.7$, $\omega =3.4916+0.0364i$.

Figure 20

Comparison between different stages of a single mode as $a$ changes and $\mathrm{\Gamma }$ grows rapidly for the resonances formed from the first (left) and second (right) bound mode for $a=1$. The solid line is the real part of $\psi (x)$, and the dashed line is the imaginary part. The profiles with smaller $\mathrm{\Gamma }$ are more localized. Profiles in WKB foliation are normalized as $\psi (x\to \infty )\to 1$.

Figure 21

Monochromatic wave ranges for scattering in the effective potential $Q(r)$ for the case $a=1$.

Figure 22

Linear evolution of the perturbed Skyrmion for $a=1$, with oscillating boundary conditions with the frequency $\omega =6$. For this frequency $R_{\text{crit}}\simeq 2$.