We study kink scattering processes in the (1+1)-dimensional ϕ 6 model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical velocities distinguish different regimes of collisions. The exact equation of motion for the ϕ 6 model is also solved numerically with the same initial conditions.We discuss advantages and disadvantages of the collective coordinate approximation, and also outline its applicability limits. Resonance phenomena and the so-called escape windows are also observed in the kink collisions.

In this Letter, we address the long-range interaction between kinks and antikinks, as well as kinks and kinks, in ϕ 2n+4 field theories for n > 1. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the ( 2n n−1 )th power of their separation, and we identify the general prefactor for arbitrary n. Importantly, we test the resulting mathematical prediction with detailed numerical simulations of the dynamic field equation, and obtain good agreement between theory and numerics for the cases of n = 2 (ϕ 8 model), n = 3 (ϕ 10 model) and n = 4 (ϕ 12 model).Introduction. The study of field-theoretic models with polynomial potentials has been a topic of wide appeal across a diverse span of theoretical physics areas, including notably cosmology, condensed matter physics and nonlinear dynamics [1][2][3]. Arguably, the most intensely studied model in this class is the quartic (double well) potential, the so-called ϕ 4 model, connected to the phenomenological Ginzburg-Landau theory [4,5], among numerous other applications [6][7][8][9]. While the ϕ 4 model has a time-honored history in its own right [10], more recently, higher-order field theories have emerged as models of phase transitions [11] relevant to material science [12][13][14] (see also [10, Chap. 11] and [15]), or in quantum mechanical problems (including supersymmetric ones) [16], among others. There, the prototypical example has been the ϕ 6 field-theoretic model, which has led to numerous insights and novel possibilities with respect to the spectral properties [17] and wave interactions [18].Scattering of solitary waves (topological defects or otherwise) is a long-standing topic of active research [19], starting from the early works [7,8]. Our aim here is to go beyond the "classical" models, in a direction that, admittedly, has already seen some significant activity [11,[20][21][22][23][24][25][26]. One of the particularly appealing aspects of this research program (aside from its potential above-mentioned applications in material science or highenergy physics/quantum mechanics) is that higher-order field theories possess topological defect solutions (kinks) with power-law tails, rather than the "standard" exponential tails that we are used to in the ϕ 4 and the (usual variants of) ϕ 6 field theories. The resulting dynamics set by the power-law tails endows topological defects with long-range interactions. Recently, a methodology for quantifying such kink-kink and kink-antikink inter-

Abstract:We study excitation spectra of BPS-saturated topological solutions -the kinks -of the ϕ 8 scalar field model in (1 + 1) dimensions, for three different choices of the model parameters. We demonstrate that some of these kinks have a vibrational mode, apart from the trivial zero (translational) excitation. One of the considered kinks is shown to have three vibrational modes. We perform a numerical calculation of the kink-kink scattering in one of the considered variants of the ϕ 8 model, and find the critical collision velocity v cr that separates the different collision regimes: inelastic bounce of the kinks at v in ≥ v cr , and capture at v in < v cr . We also observe escape windows at some values of v in < v cr where the kinks escape to infinity after bouncing off each other two or more times. We analyse the features of these windows and discuss their relation to the resonant energy exchange between the translational and the vibrational excitations of the colliding kinks.

We studied the kink-antikink collision process for the "double sine-Gordon" (DSG) equation in 1+1 dimensions at different values of the potential parameter R>0. For small values of R we discuss the problem of resonance frequencies. We give qualitative explanation of the frequency shift in comparison with the frequency of the discrete level in the potential well of isolated kink. We show that in this region of the parameter R the effective long-range interaction between kink and antikink takes place.

We study the scattering of the ϕ 8 kinks off each other, namely, we consider those ϕ 8 kinks that have power-law asymptotics. The slow power-law fall-off leads to a long-range interaction between the kink and the antikink. We investigate how the scattering scenarios depend on the initial velocities of the colliding kinks. In particular, we observe the 'escape windows' -the escape of the kinks after two or more collisions, explained by the resonant energy exchange between the translational and vibrational modes. In order to elucidate this phenomenon, we also analyze the excitation spectra of a solitary kink and of a composite kink+antikink configuration.

We present a computational analysis of the long-range interactions of solitary waves in higherorder field theories. Our vehicle of choice is the ϕ 8 field theory, although we explore similar issues in example ϕ 10 and ϕ 12 models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard ϕ 4 model. Upon establishing the power-law asymptotics of the model's solutions' approach towards one of the steady states, we make the case that such asymptotics require particular care in setting up multi-soliton initial conditions. A naive implementation of additive or multiplicative ansätze gives rise to highly pronounced radiation effects and eventually leads to the illusion of a repulsive interaction between a kink and an antikink in such higher-order field theories. We propose and compare several methods for how to "distill" the initial data into suitable ansätze, and we show how these approaches capture the attractive nature of interactions between the topological solitons in the presence of power-law tails (longrange interactions). This development paves the way for a systematic examination of solitary wave interactions in higher-order field theories and raises some intriguing questions regarding potential experimental observations of such interactions. As an Appendix, we present an analysis of kinkantikink interactions in the example models via the method of collective coordinates.the Higgs field [13,14]. Beyond field theoretic models with polynomial potentials, finitegap potentials of Lamé type also lead to scalar field theories with exotic kink solutions, now relevant in the context of sypersymmetric quantum mechanics [15,16] and extended to PT -symmetric situations [17].Although the above-mentioned models with polynomial potentials are non-integrable, studying their properties in (1+1)-dimensional space-time is of common interest because, in this setting, a variety of analytical (and numerical) methods can be straightforwardly deployed to fully understand the dynamics of coherent structures. Moreover, (1+1)dimensional solutions may be relevant to more realistic situations in higher dimensions; for example, the equations for certain five-dimensional brane-world phenomenologies can be reduced to differential equations similar to those of (1+1)-dimensional field theories [18].Such models with polynomial potentials of even degree allow kinks -topological solutions that interpolate between neighboring minima of the potential, i.e. vacua of the model [19].Properties of kinks of the ϕ 4 and ϕ 6 models are well-studied, yielding many important results [4,7,[20][21][22][23][24][25][26][27][28][29][30][31]. At the same time, polynomial potentials of higher degrees have not

The nonbaryonic dark matter of the Universe is assumed to consist of new stable forms of matter. Their stability reflects symmetry of micro world and mechanisms of its symmetry breaking. In the early Universe heavy metastable particles can dominate, leaving primordial black holes (PBHs) after their decay, as well as the structure of particle symmetry breaking gives rise to cosmological phase transitions, from which massive black holes and/or their clusters can originate. PBHs can be formed in such transitions within a narrow interval of masses about 10 17 g and, avoiding severe observational constraints on PBHs, can be a candidate for the dominant form of dark matter. PBHs in this range of mass can give solution of the problem of reionization in the Universe at the redshift z ∼ 5 . . . 10. Clusters of massive PBHs can serve as a nonlinear seeds for galaxy formation, while PBHs evaporating in such clusters can provide an interesting interpretation for the observations of point-like gamma-ray sources. Analysis of possible PBH signatures represents a universal probe for super-high energy physics in the early Universe in studies of indirect effects of the dark matter.

We study the scattering of kink and antikink of the double sine-Gordon model. There is a critical value of the initial velocity v cr of the colliding kinks, which separates different regimes of the collision. At v in > v cr we observe kinks reflection, while at v in < v cr their interaction is complicated with capture and escape windows. We obtain the dependence of v cr on the parameter of the model. This dependence possesses a series of local maxima, which has not been reported by other authors. At some initial velocities below the critical value we observe a new phenomenon -the escape of two oscillons in the final state. Besides that, at v in < v cr we found the initial kinks' velocities at which the oscillons do not escape, and the final configuration looks like a bound state of two oscillons.