Resonance with quasinormal modes in long-range kinks' collisions

TL;DR

This work demonstrates that resonance windows in kink–antikink collisions can be mediated by quasinormal modes even when kinks possess long-range tails on both sides, in both a rational two-parameter model and a $\phi^{10}$ polynomial model. Through spectral analyses, the authors show the absence of normalizable bound states and the presence of QNMs whose energies enable transient storage and later release of kinetic energy during collisions; the resonance windows occur only in limited parameter regions and can disappear as tails become more long-range or internal substructures emerge. In the rational model, windows exist near small $\varepsilon$ and diminish with increasing tail strength, while in the $\phi^{10}$ model a single narrow two-bounce window appears around $a\approx0.55$, with frequency alignment to the lowest QNM. The paper also introduces a Convection–Diffusion initialization method for long-range kink collisions, offering a physically intuitive and computationally efficient alternative to existing minimization-based schemes, enabling more scalable simulations of soliton dynamics.

Abstract

We consider a rational scalar field model in (1+1)-dimensions where the long-range character of the kinks is controllable. We show via numerical simulations that kinks with long-range tails on both sides can exhibit resonance windows. The resonant energy exchange mechanism occurs via the excitation of quasinormal modes, which we obtain via a spectral analysis. Additionally, we locate a resonance window in a family of $φ^{10}$ models with long-range tails on both sides. Moreover, we propose a new algorithm for initializing long-range kink collisions, based on convection-diffusion dynamics.

Figures (19)

• Figure 1: Potential as a function of $\phi$ for several values of $m$ and $\varepsilon$.
• Figure 2: Kink tail on a logarithmic scale as a function of $x$ for several values of $m$ and $\varepsilon$.
• Figure 3: Stability potential (a) for fixed $m=0$ and several values of $\varepsilon$ and (b) for fixed $\varepsilon=0.01$ and several values of $m$.
• Figure 4: (a) Spectral structure (BMs) for $\varepsilon=0.01$ along with the real part of the frequency of the lowest odd QNM. (b) The lowest odd mode, antibound modes and lowest QNMs around the threshold crossing.
• Figure 5: Spectral structure of the lowest QNM with the frequency defined as $\omega=\Omega+i\Gamma$ for $m=0$ in the rational model.