Multi-kinks and composite oscillons in a commensurable and non degenerate double sine-Gordon model

TL;DR

The article introduces a commensurable and non-degenerate double sine-Gordon model in which partial vacuum degeneracy breaking yields static multi-kinks composed of n sub-kinks. It develops a smooth-modulation approximation and a collective-coordinate model that incorporates inter-kink forces and radiation back-reaction to analyze multi-kink collisions, revealing the formation of long-lived composite oscillons and synchronized vibrational modes. The study identifies regimes of elastic-like K_n–K_n collisions and complex K_n–ar{K}_n dynamics including bound states and annihilation, with the oscillon structure mirroring the internal kink composition. The results provide a detailed framework for understanding non-integrable soliton dynamics and emergent coherent structures, with implications for broad nonlinear field theories and potential extensions to higher dimensions.

Abstract

In this paper, we introduce a commensurable and non-degenerate double sine-Gordon model, in which a partial breaking of vacuum degeneracy provides a mechanism for the emergence of static multi-kinks. These multi-kinks $K_n$ are stable field configurations with internal structure, consisting of $n$ localized energy packets with well-defined separations. The properties of the multi-kinks are thoroughly analyzed, including the novel phenomenology that arises during their collisions. In particular, we observe the emergence of long-lived composite oscillons that reflect the original structure of the multi-kinks. The sub-kink's positions and their vibration modes provide collective coordinates that are used to construct a phenomenological model, which offers a good qualitative explanation of the observed oscillon properties. Radiation effects are consistently incorporated, revealing that they play an important role in the observed synchronization of the oscillon's vibrational components.

Figures (17)

• Figure 1: Potential $V_{n,\alpha} (\phi)$, the parameters are selected as: $n= 4$, with: $\alpha=0.1$ (black-continuous line), $\alpha=0.6$ (red-dotted line) and $\alpha=1.5$ (blue-dashed line).
• Figure 2: Kink field configurations $\phi_{k_i } (z)$ obtained from the numerical solution of Eq.(\ref{['BESG']}) (continuous-green lines) and the approximated ansatz Eq.(\ref{['phiAn']}) (dashed-black lines); for the (a) $n=4$ and (b) $n=8$ cases.
• Figure 3: $K_n$ multi-kink: $(a)$ Binding energy $\Delta V_{K_{(n,\alpha)}}$ as a function of $\alpha$, obtained from the numerical integration of the second term of Eq.(\ref{['MKSM1']}) (continuous-green lines) and the analytical expression in Eq.(\ref{['Vint']}) (dashed-black lines). $(b)$ Size $\sigma_{K_{(n,\alpha)}}$ obtained from Eq.(\ref{['sigmaK']}) (continuous-black lines), the dashed lines correspond to the approximated expression in Eq.(\ref{['sigmaK2']}).
• Figure 4: (a) and (c) multi-kinks $\Phi_{K_n}(z)$ obtained from the numerical solution of Eq.(\ref{['Bogo']}) for the potential $V_{n,\alpha} (\phi)$ in Eq.(\ref{['Pot2SG']}) (continuous-green lines) and the analytical ansatz Eq.(\ref{['PhiK']}) (dashed-black lines). (b) and (d) the corresponding energy density distributions. The parameters values are: $n=4$, $\alpha=0.0001$ for the (a) and (b) figures and $n=8$, $\alpha=0.1$ in the (c) and (d) cases.
• Figure 5: Fluctuation spectra around the $k_1-k_2$ sub-kink's pair of $K_4$. $(a)$$V_{n,\alpha} ( \varphi_{(1,2)}(z,a) )$ effective potential (black line) and the $\eta_{S,A} (a,z)$ discrete modes obtained from the solutions of Equation (\ref{['spectra']}) (continuous lines) and Equation (\ref{['eigenfSA']}) (dashed lines) for $a=1.6$. $(b)$ Eigenvalues for the symmetric quasi-zero mode (blue solid line) and anti-symmetric (red dashed line) bound states as a function of the sub-kink's separation $a$. The dots values are obtained from Eqs. (\ref{['w1p1']}) for $\alpha$ values in the range $\alpha \in \{ 0.05-0.5\}$.