Kink-Kink and Kink-Antikink Interactions with Long-Range Tails

Abstract

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-

Wide class of logarithmic potentials with power-tower kink tails

“…Taking this as a guide, we speculate that corresponding to the power-tower form as given by Eq. (58), the behaviour of φ(x) for large negative x should be of the form the new Manton formalism [16,17], even though developed for integral k is also valid for any real number k. Using this information, one can estimate the force between the (−1, 0) K and the (0, 1) K using the new Manton formalism and show that the KK force would vary like R −9/2 , where R is the distance between the two kinks.…”

“…It is worth pointing out that since for arbitrary m and n, the potential around φ = 0 is of the form φ 2k with k as given by Eq. (64) hence using the new Manton formalism [16,17] one can show that in that case the KK force would go like R −d , where d = 2[1 + m,n + 1/m]. 6.…”

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]].…”