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-

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

Abstract:We explore a variant of the φ 6 model originally proposed in Phys. Rev. D 12 (1975) 1606 as a prototypical, so-called, "bag" model in which domain walls play the role of quarks within hadrons. We examine the steady state of the model, namely an apparent bound state of two kink structures. We explore its linearization, and we find that, as a function of a parameter controlling the curvature of the potential, an effectively arbitrary number of internal modes may arise in the point spectrum of the linearization about the domain wall profile. We explore some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, and how they depend on the principal parameter of the model. We find that the critical velocity exhibits a nonmonotonic dependence on the parameter controlling the curvature of the potential. For the multi-bounce windows, we find that their range and complexity decrease as the relevant parameter decreases (and as the number of internal modes in the model increases). We use a modified collective coordinates method [in the spirit of recent works such as Phys. Rev. D 94 (2016) 085008] in order to capture the relevant phenomenology in a semi-analytical manner.

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We consider the interaction of solitons in a biharmonic, beam model analogue of the well-studied φ 4 Klein-Gordon theory. Specifically, we calculate the force between a well separated kink and antikink. Knowing their accelerations as a function of separation, we can determine their motion using a simple ordinary differential equations. There is good agreement between this asymptotic analysis and numerical computation. Importantly, we find the force has an exponentially-decaying oscillatory behaviour (unlike the monotonically attractive interaction in the Klein-Gordon case). Corresponding to the zeros of the force, we predict the existence of an infinite set of field theory equilibria, i.e., kink-antikink bound states. We confirm the first few of these at the partial differential equations level, and verify their anticipated stability or instability. We also explore the implications of this interaction force in the collision between a kink and an oppositely moving antikink.