Fractal structure in the scalarλ(φ2−1)2theory

Abstract

Head-on collisions of kink and antikink solitons are investigated numerically in the classical onedimensional h(4'-1 )2 model. It is shown that whether a kink-antikink interaction settles to a bound state or a two-soliton solution depends "fractally" on the impact velocity. We discuss the results using the framework of perturbation theory which helps to clarify the nature of the fractal structure in terms of resonances with the internal shape mode oscillations. We also review the technique of collective coordinates used to reduce the infinite-dimensional system to one with just two degrees of freedom. Although we do not expect exact agreement by using such a simplification, we show that the reduced system bears a striking qualitative resemblance to the full infinite-dimensional system, reproducing the fractal structure. The maximum Lyapunov exponents are computed for the bound-state oscillations and found to be -0.3 for both the full and reduced systems, demonstrating the chaotic nature of the bound state.

Kink-antikink collisions for twin models

“…We fixed x 0 = 15 as the initial kink position and we set the grid boundaries at x max = 120. For the φ 4 model we reproduced some results from the literature [6] concerning to the appearance of two-bounce windows. For a particular initial velocity v in , the structure of bounces can be easily verified with a plot of the scalar field at the center of mass φ(0, t) ized by the integer m labeling the number of oscillations in φ(0, t) between the bounces.…”

“…The effect of a phase shift of the collision process for an integrable model can be confronted to the richness of the collision process for nonintegrable models, where most of analysis must be done numerically. As already shown by Anninos et al [6] for the λφ 4 theory, for sufficiently small initial velocities, the kink and antikink capture one another and a trapped bion state is formed. On the other hand for larger velocities a simple collision occurs, and after the contact, the pair of defects retreats from each other.…”

“…In addition to the already cited λφ 4 model [6], one can cite the modified sine Gordon [8] and the φ 6 model [13]. Most of nonintegrable models have internal oscillatory modes, responsible for resonant scattering.…”

Semiclassical description of soliton-antisoliton pair production in particle collisions

“…Let us explain the choice (3.1) of the scalar potential. First, we do not consider the standard φ 4 theory because evolution of the kink-antikink pairs there is known to exhibit chaotic behavior [77,78]. That chaos is related to the fact that the spectrum of linear perturbations around the φ 4 kink contains two localized modes representing its spatial translations and periodic vibrations of its form.…”

Multi-kink collisions in the ϕ 6 model

“…In particular, the kink-(anti)kink scattering and the interactions of kinks with impurities are of growing interest. A wide variety of phenomena emerges in these systems, e.g., escape windows and quasi-resonances in kink-(antikink) collisions [20][21][22][23][24][25][26][27][28][29][30], resonant interactions of kinks with wells, barriers and impurities [31][32][33][34], non-radiative energy exchange in multi-soliton collisions [35][36][37]. It is interesting that the presence of a kink's internal modes does not guarantee the appearance of resonance windows, as it has been recently shown for the deformed φ 4 model [38].…”