Long-range interactions of kinks

Abstract

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

Collision of two kinks with inner structure

“…As a result, the original partial differential equation (PDE) (2.2) will be transformed into a system of second-order-in-time ordinary differential equations (ODEs), which can be easily solved by using the ode45 function of Matlab (see also refs. [68,69,99]).…”

Fischer Tropsch Synthesis: The Promoter Effects, Operating Conditions, and Reactor SynthesisThe authors gratefully acknowledge University of Sistan and Baluchestan for helping and supporting this research.Majid Sarkari:Young Researchers Club, Shiraz Branch, Islamic Azad University, Shiraz, Iran. Farhad Fazlollahi: Department of Chemical Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran. Hossein Atashi: Department of Chemical Engineering, Faculty of Engineering, University of Sis

“…Notice that linear combination of the kink solution and the antikink tail solution considered here can be used for kinks with short-range tails, as in our case, but this approximation may not work for kinks with long-range tails, see, e.g., [39,[56][57][58][59]].…”

Non-topological kink scattering in a two-component scalar field theory model

“…Here, if the potentials have global minima of quartic or higher degree, kinks with power-law asymptotics arise in the model, which leads to long-range interactions between the kinks. The exploration of this type of interactions must be carefully addressed and, nowadays constitutes a topic of significant interest [38,39,40,39,41,42,43,44,45,46]. The effect of defects, impurities or inhomogeneities on kink dynamics has been discussed in several models, see references [47,48,49,50,51,52,53,54,55,56,57,58].…”