Composite solitary waves in three-component scalar field theory: Three-body low-energy scattering

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Abstract

We study the structure of the manifold of solitary waves in a particular three-component scalar field theoretical model in two-dimensional Minkowski space. These solitary waves involve one, two, three, four, six or seven lumps of energy. The adiabatic motion of the non-linear non-dispersive waves composed of three lumps is interpreted as three-body low-energy scattering of these particle-like kinks.

Introduction

The search for solitary waves is a mandatory topic in a huge number of branches of non-linear science. The reason for this lies in the fact that the method of linearizing non-linear differential equations omits phenomena of essential importance in many different physical problems. The non-linear Klein–Gordon system of coupled PDE equations, generalized to n-real scalar fields, reads: 2ϕat22ϕa+Vϕa=0a=1,2,,N. Here, V(ϕ1,,ϕn) is a differentiable potential function of the scalar fields ϕa, whereas Vϕa are non-linear functions of the fields. It is clear that this system is invariant under Poincarè transformations. For this reason we shall use a system of units where the speed of light is set to one: c=1. Also, due to the potential applications of the PDE system (1) in quantum physics we shall implicitly understand ħ=1, i.e., we shall work in the natural system of units.

We shall focus on physical systems governed by the PDE system (1) in only one spatial dimension. The paradigm with only one real scalar field is the celebrated sine–Gordon equation: 2ϕt22ϕx2+sinϕ=0. Arising in differential geometry in the theory of negative curvature surfaces, the sine–Gordon equation is ubiquitous in the description of one-dimensional physical phenomena. To mention a few, (1) it is the evolution equation for the amplitude of various slowly varying waves, (2) it describes the propagation of a dislocation in a crystal, (3) it provides a model for elementary particles moving on a line, (4) it rules the propagation of magnetic flux in a long Josephson-junction transmission line, and (5) it determines the modulation of a weakly unstable baroclinic wave packet in a two-layer fluid; see, e.g., Ref. [1].

Point (3) deserves special mention and will be the arena within we shall analyze the generalized non-linear Klein–Gordon PDE system proposed in this paper. The similarities between the properties of solitary waves (or solitons) and elementary particles are clear. The field energy density of a non-linear wave (solitary wave or soliton) is localized in a lump. Moreover, being non-dispersive, solitary waves (or solitons) propagate without changes in shape. After collisions, solitary waves almost keep their structure, leaving some radiation in the form of dispersive waves as remnants. Solitons, however, only suffer phase shifts under collisions. In soliton–antisoliton scattering, either both solitary waves may be annihilated or, alternatively, they may form a bound (breather) state. Elementary particles share all of these properties. Thus, if an appropriate system of non-linear field equations admits solitary waves (or solitons) as solutions, these non-dispersive waves behave almost as elementary particles.

Nevertheless, the main feature of these solutions, the confinement of the energy density, has straightforward implementation in many areas. In Condensed Matter, these solutions describe interfaces in magnetic materials [2] and in ferroelectric crystals [3]. They have also been used to understand some bizarre properties of the poly(oxyethylene) [4], widely studied owing to its biotechnical and biomedical applications [5]. Among the impressive features of this polymer, we can mention the ability to entrap metallic ions in aqueous solution and its solubility in water to almost any extent. On the other hand, in Cosmology solitary waves are interpreted as domain walls which seeded the formation of structures in the early universe [6]. They are also of direct interest in high energy physics, for instance, describing gravity in warped spacetimes involving D extra spatial dimensions [7], or the coupling of scalar matter fields with dilaton gravity [8], which seems to be related to the formation and evaporation of black holes [9]. Furthermore, when these defects appear as BPS states in both extended supersymmetric gauge theories [10] and string/M theory [11] they play a crucial rôle in the understanding of dualities between the different regimes of the system. In this framework, they behave as extended states in N=1 SUSY gluodynamics and the Wess–Zumino model [12]. The structure of the solitary waves that we are going to unveil in this paper is richer than the usual kink structure found in simpler models. Thus, our topological defects could describe more subtle effects in each of all these scenarios.

In general, the above-mentioned systems involve a high number of fields. In order to investigate the presence of solitary waves or topological defects, the usual procedure is to obtain an effective scalar field theory, carrying out severe restrictions on the original theory. In most cases one is compelled to pursue an effective theory that corresponds to a single scalar field model, where the existence of topological defects can be checked easily. In general, however, the effective theory depends on several scalar fields, and the truncation can involve a important loss of information about the presence of solitary waves or topological defects. Therefore, it is desirable to investigate the general properties of solitary waves in a multi-scalar field theory. This is an important qualitative step as reported by Rajaraman [13]: This already brings us to the stage where no general methods are available for obtaining all localized static solutions, given the field equations. However, some solutions, but by no means all, can be obtained for a class of such Lagrangians using a little trial and error.

Some work on two-component scalar field theory models equipped with a Minkowskian space–time has been accomplished; see for instance [14], [15], [16], [17], [18], [19], [20], [21]. In some of these models, two-parametric families of solitary waves or kinks are identified. Generally, each member of these families is a composite solitary wave, which involves several lumps, i.e., the energy density is localized at several points. The parameters identifying each solution correspond to a translational parameter, x0, which determines the center of the kink, and an “orbit” parameter b, which specifies the separation between the lumps. Study of Manton’s adiabatic motion of solitary waves [22] is very simple in this case because the geometric metric of the moduli (parameter) space that governs the dynamics does not depend on the translational parameter x0; thus, it is always possible to apply a transformation that leads to an Euclidean metric. On the other hand, there are only a few works that have addressed research into solitary waves in three-component scalar field theory models; see [23], [24], [25], [26]. In this richer case there are three-parametric families of kinks, and the adiabatic evolution of three-body lumps is associated with metrics with curvature. The low-energy dynamics of, in this case, three-body solitary waves is therefore much more intricate.

In a slightly different physical context, where the solitary waves are understood as domain walls (2-branes) in (3+1) dimensions, models with three scalar fields have been discussed in Ref. [27]. Considering topological defects (0-branes) in more spatial dimensions (p-branes) allows for complex structures having defects nested inside defects. It is plausible that the composite solitary waves to be discussed in this paper will give rise to even richer structures of nested defects inside defects if considered in more dimensions. An interesting work is also available on networks of topological defects in a model with two complex scalar fields; see [28]. The potential energy density is a polynomial in the fields of arbitrary order chosen in such a way that the model enjoys a U(1)×Zn symmetry, networks of domain walls arising in the surface of Q-balls. In other direction, defects inside defects in models with two scalar fields have been considered before; see [29] and [30]. Very recently, networks of topological defects arising in a model with 9n scalar fields in nine spatial dimensions have been studied in [31] and the evolution of these (or similar) structures has been analyzed in [32].

In this paper, we shall address a three-component scalar field model that generalizes the two-component model discussed in Ref. [14] to three fields. The organization of the paper is as follows: in Section 2 we introduce the model and describe the spontaneous symmetry breaking pattern as well as the spectrum of dispersive waves. In Section 3 we describe the variety of solitary (non-dispersive) waves that arises in this system. We classify these solutions according to the number of lumps they are made of. There are three basic solitary waves with the energy density confined in one finite region. The rest of the solutions are composite and display several lumps — the energy density is localized at various points. In Section 4 some remarks about the stability of these solutions are given. Finally, in Section 5 we shall describe the adiabatic evolution on the moduli space of solutions of configurations composed of three basic lumps.

Section snippets

Deformation of the λ(ϕϕ)23-model

We shall deal with a three-component scalar field theory model defined in a (1+1) Minkowskian space–time. The dynamics is governed by the action functional: S=d2x[12μϕaνϕaV(ϕ1,ϕ2,ϕ3)] and the potential term is the following sixth-degree polynomial in the fields: V(ϕ)=(ϕ12+ϕ22+ϕ32)(ϕ12+ϕ22+ϕ321)2+2(σ22ϕ22+σ32ϕ32)(ϕ12+ϕ22+ϕ321)+σ24ϕ22+σ34ϕ32.σ2 and σ3 are the non-dimensional coupling constants of the system, chosen, without loss of generality, such that σ2<σ3. ϕa, a=1,2,3, are

The variety of solitary waves

In this section we describe the variety of solitary waves in successive stages. The analogous mechanical system encompasses a vast manifold of finite action trajectories in one-to-one correspondence with the solitary waves of the field theoretical model. One-body solitary waves, made of only one lump, can be found by means of Rajaraman’s trial orbit method, without the need to use elliptic coordinates. On plugging some specific trial curves into the Eq. (11), simple solitary waves that we will

Some remarks about the stability of solitary waves

In order to analyze the stability of the solitary wave solutions, we shall apply different procedures. The direct approach is study of the spectrum of the second-order fluctuation (Hessian) operator (10), now around kink solutions. A non-negative spectrum of the kink Hessian operator ensures that a particular solitary wave solution is stable. When detailed information about the spectrum of the Hessian is lacking, an interesting procedure to study the stability of kink families is the

Three-body low-energy scattering of non-linear waves

The three parameters x0, a and b of the K3OD family of solitary waves fix the center of mass and the relative positions of the basic lumps in the composite kink configuration. We shall now study the evolution of composite solitary waves within the framework of Manton’s adiabatic principle, see [21], [22], by looking at changes in a and b in time. The Manton adiabatic hypothesis can be summarized as follows: the dependence on time of solitary waves develops only in the parameters of the

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