# New Models for Two Real Scalar Fields and Their Kink-Like Solutions

Authors: A. Alonso-Izquierdo; D. Bazeia; L. Losano; J. Mateos Guilarte
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

We study the presence of kinks in models described by two real scalar fields in bidimensional
spacetime. We generate new two-field models, constructed from distinct but important
one-field models, and we solve them with techniques that we introduce in the current work. We
illustrate the results with several examples of current interest to high energy physics.


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## Body

## 1. Introduction

The presence of kinks and solitons in models described by real scalar fields is of direct interest to high energy physics [1, 2] and other areas of nonlinear science [3, 4]. To mention specific studies, in high energy physics kinks appear in very interesting systems introduced, for instance, in [5, 6]. In condensed matter one can investigate domain walls in magnetic systems [7, 8], and nonlinear excitations in Bose-Einstein condensates [9, 10], to quote just a few examples.In this work we focus on one-field and two-field models in(1,1) spacetime dimensions. Two very interesting models described by a single real scalar field are known as the sine-Gordon and the ϕ4 models, engendering spontaneous symmetry breaking. The ϕ4 model is described by a fourth-order polynomial potential and supports kink-like solutions, whereas the sine-Gordon model is characterized by a nonpolynomial potential and supports not only solitons but also multisoliton and breather solutions. Fluctuations around the solitons and λϕ4 kinks, however, are governed by the ℓ=1 and ℓ=2 reflectionless Hamiltonians of a general family known from supersymmetric quantum mechanics [11]. Moreover, a rich family of nonpolynomial models with spontaneous symmetry breaking was proposed in [12]. The main feature of the family of kinks arising in this family is that the Hamiltonians governing the kink small fluctuations cover many of the remaining transparent SUSY Hamiltonians; see also [13].We start with these one-field models, which are described by polynomial and nonpolynomialW=W(ϕ), and we then move on to the two-field models constructed from the previous ones. Our aim is to identify kink solutions in these new models, which in general is a very difficult endeavor, as Rajaraman [14] notices: “This already brings us to the stage where no general methods are available for obtaining all localized static solutions (kinks), 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.” In this work we develop a technique which generates two-component kink solutions for two-field models in a straight-forward while way avoiding the use of the trial and error method mentioned by Rajaraman. We mention, however, that there exist two scalar field theory models [15] and even models of three scalar field ones [16] such that all the kink solutions can be found due to the complete integrability of the analogue mechanical problem.For simplicity, we use natural units and then redefine fields and coordinates such that fields and space and time are all dimensionless. The study starts in Section2, and the one-field and two-field models are then used in Section 3 to generate new models, described by two fields. In this section we deal with polynomial potential, and so, to enlarge the scope of the work, in Section 4 we introduce another family of models, containing a nonpolynomial function of the field χ. We end the work in Section 5, where we introduce some comments and conclusions.


## 2. Generalities

Let us first consider one-field models. We take the Lagrange density in the following form:(1)ℒ=12∂μϕ∂μϕ-V(ϕ).
Here we deal with topological solutions, so we write the potential V(ϕ) in the following form:
(2)V(ϕ)=12Wϕ2,
where W=W(ϕ), and Wϕ stands for the derivative with respect to ϕ; that is, Wϕ=dW/dϕ. The equation of motion for static field configuration is given by
(3)ϕ′′=dUdϕ=WϕWϕϕ.
Here we are using ϕ′=dϕ/dx, and so forth. The energy density for static solutions can be written as
(4)ϵ(x)=12ϕ′2+12Wϕ2=12(ϕ′-Wϕ)2+dWdx
for smooth superpotentials. We note that the energy is minimized to the value
(5)EBPS=|W(ϕ(∞))-W(ϕ(-∞))|
for field configurations that obey the first-order equation
(6)ϕ′=Wϕ.
This is the Bogomol’nyi bound, and we can easily see that solutions to (6) also solve the equation of motion (3). The field configurations that solve the first-order equation are named Bogomol’nyi-Prasad-Sommerfeld (BPS) states [17, 18]. We note that since the potential does not see the sign of W, there are in fact two first-order equations, one for W and the other with W changed to -W. This is related to the spatial reflection symmetry x→-x, which provides us with the kink/antikink solutions.Two important models in the previous class of models are theϕ4 model, where
(7)W(ϕ)=ϕ-13ϕ3,
and the sine-Gordon model, where
(8)W(ϕ)=sin(ϕ).
The potentials are, respectively,
(9)V(ϕ)=12(1-ϕ2)2,V(ϕ)=12cos2(ϕ).
These models have solutions in the following form for the ϕ4 model,
(10)ϕ(x)=tanh(x),
and for the sine-Gordon model,
(11)ϕ(x)=arcsin(tanh(x))+kπ,k=0,±1,±2,…,
where k identifies one among the infinity of topological sectors of the sine-Gordon model.Let us now consider two-field models. We start with the Lagrange density(12)ℒ=12∂μϕ∂μϕ+12∂μχ∂μχ-V(ϕ,χ).
For static configurations, the equations of motion become
(13)ϕ′′=∂V∂ϕ,χ′′=∂V∂χ.
We suppose that the potential V(ϕ,χ) is given in terms of the superpotential W(ϕ,χ) by
(14)V(ϕ,χ)=12Wϕ2+12Wχ2,
where Wϕ=∂W/∂ϕ and Wχ=∂W/∂χ. Notice that the critical points of the superpotential W(ϕ,χ) provide us with the set of vacua ℳ={(ϕ,χ)∈ℝ2:V(ϕ,χ)=0} for the field theory model. The energy density has the form
(15)ϵ(x)=12(ϕ′2+χ′2+Wϕ2+Wχ2)=12[(ϕ′-Wϕ)2+(χ′-Wχ)2]+dW.
The minimum energy solutions comply with
(16)ϕ′=Wϕ,χ′=Wχ
leading us to the BPS energy
(17)EBPS=|W(ϕ(∞),χ(∞))-W(ϕ(-∞),χ(-∞))|
for smooth superpotentials. In terms of the superpotential, the equations of motion for static fields are written as
(18)ϕ′′=WϕWϕϕ+WχWχϕ,χ′′=WϕWϕχ+WχWχχ,
which are solved by the first-order equations (16), for Wϕχ=Wχϕ, as we require in this work. Solutions to these first-order equations are BPS states, which solve the equations of motion. The sectors where the potential has BPS states are named BPS sectors.As an example, let us consider the model characterized by the superpotential(19)W(ϕ,χ)=ϕ-13ϕ3-rϕχ2
which has been studied by Shifman et al. in the context of N=1 supersymmetric Wess-Zumino models with two chiral superfields [19, 20]. In the purely bosonic framework the presence of domain walls and its stability have been analyzed in the references [21–23], while in [24, 25] the complete structure of this type of solutions is given in two critical values of the coupling between the two scalar fields by exploiting the integrability of the analogue mechanical system associated with this model. This well-known model will be used in the following sections to illustrate the applicability of the novel procedure introduced in this paper which allows the identification of kink-like solutions in new field theories.The first-order ODE (16) in this case are written as
(20)ϕ′=1-ϕ2-rχ2,χ′=-2rϕχ.
The potential is given by
(21)V=12(1-ϕ2-rχ2)2+2r2ϕ2χ2,
which can be seen as an extension of the ϕ4 model to the case of two fields. Here we consider r to be real and positive. The vacuum set comprises four elements:
(22)ℳ={v1,2=(±1,0),v3,4=(0,±1r)}.
Associated with the superpotential (19) we can find five BPS sectors (here we do not distinguish between kinks and antikinks) by analyzing the first-order ODE (20). Indeed this model is a very special case because an integrating factor can be calculated for the orbit equation extracted from (20). The kink trajectories are given by
(23)ϕ2+r1-2rχ2-γχ1/r=1,whereγ∈(-∞,γC]
with γC=2r(1/2r)+1/(1-2r) and r≠1/2. For the range γ∈(-∞,γC) the formula (23) describes kinks connecting the vacua v3 and v4 with EBPS=4/3 while the value γ=γC yields kinks linking the points v1,2 with v3,4 with EBPS=2/3. From construction all the kinks living in a specific topological sector are energy degenerate.In particular theγ=-∞ and γ=0 members of (23) correspond, respectively, to the one-component kink
(24)ϕ(x)=±tanh(x),χ(x)=0,
and the two-component kink
(25)ϕ(x)=±tanh(2rx),χ(x)=±1r-2sech(2rx)
lying in the topological sector connecting the minima v3 and v4; see Figure 1.Graphics of the potentialV(ϕ,χ) with r=1/4 and orbits of the kinks (24) and (25) in the internal plane.
(a)
(b)As an illustrative sample we introduce a kink solution(26)ϕ=±12(1±tanh(x)),χ=±12(1∓tanh(x))
lying in the topological sectors joining the points v3,4 with the points v1,2 for the case r=1; see Figure 2.Graphics of the potentialV(ϕ,χ) with r=1 and orbits of the kinks (26) in the internal plane.
(a)
(b)We remark that different superpotentials can generate the same potential. Indeed in this model other superpotentials than (19) have been identified for several particular values of the coupling constant r. This fact provides us with new degenerated BPS and non-BPS solutions in the topological sector joining the points v3 and v4; see [24].


## 3. New Models

To generate new two-field models and the accompanying static solutions, we proceed as follows. We start from the one-field model, with the superpotential written in the form(27)W(ϕ)=∫ϕf(y)dy.
This gives the first-order equation
(28)ϕ′=f(ϕ),
and for the ϕ4 and sine-Gordon models we have f(ϕ)=1-ϕ2 and f(ϕ)=cos(ϕ), respectively.Now, to introduce two-field models, we get inspiration from the previous model, given by (19) and we propose the following superpotential:
(29)W(ϕ,χ)=∫ϕf(y)dy-rϕχ2.
This generates the field potential term
(30)V(ϕ,χ)=12[f(ϕ)-rχ2]2+2r2ϕ2χ2.
The critical points of the superpotential, determined by Wϕ=f(ϕ)-rχ2=0 and Wχ=-2rϕχ=0, provide us with the vacua of the model:
(31)ℳ={(ϕ,χ)∈ℝ2:(ϕ(i),0),(0,±f(0)r)},
where ϕ(i), i=1,…,n, are the roots of f(ϕ). Therefore this kind of models involves n+2 vacua assuming that f(0)≠0. The static solutions are obtained from the first-order equations
(32)ϕ′=f(ϕ)-rχ2,χ′=-2rϕχ.
We can manipulate these equations to get
(33)ϕ′′+(4rϕ-dfdϕ)ϕ′-4rϕf(ϕ)=0.
Choosing ϕ(x) as the static solution of a one-field model, such that
(34)ϕ′=U(ϕ),ϕ′′=dU(ϕ)dϕU(ϕ),
we can rewrite (33) as
(35)dfdϕ+P(ϕ)f(ϕ)-Q(ϕ)=0,
where
(36)P(ϕ)=4rϕU;Q(ϕ)=4rϕ+dUdϕ.
The function U(ϕ) is a particular solution of the linear ODE (35), so we can write the general solution in the form
(37)f(ϕ)=U(ϕ)+Ce-∫‍P(ϕ)dϕ
with C being an integration constant. Plugging (37) into (30) we get a one-parameter family of potentials which has a nontrivial two-component kink solution, whose orbit
(38)rχ2=f(ϕ)-U(ϕ)
emerges from the use of (32), (34), and (37). Notice that strictly speaking the ODE (35) must be verified only on the kink orbit; in the rest of the internal plane a natural extension of the potential is considered.We note that the first-order equations (32) support the orbit χ(x)=0, providing us with a second kink solution for our model. In this case the static solutions ϕ(x) connecting neighbor minima located at the ϕ-axis are obtained from
(39)dϕdx=f(ϕ).
The expression f(ϕ) coincides with U(ϕ) only if the integration constant C=0. The general expression of f(ϕ) gives rise to a family of two-component field theory models, which admits a two-component kink solution whose first component coincides with the kink associated with U(ϕ).The key step of the procedure appears in (34), and it is inspired by [26]. It works nicely for a variety of choices of U(ϕ), and the corresponding models include polynomial and nonpolynomial functions.


### 3.1. A First Example

To illustrate the previous procedure with concrete examples, let us start considering(40)U(ϕ)=κ(a-ϕ)(b+ϕ),
where a, b, and κ are real parameters and we obviously assume that a,b≠0 and a≠-b. We use (37) together with (34) and (36) to obtain
(41)f(ϕ)=κ(a-ϕ)(b+ϕ)+C(a-ϕ)n1(b+ϕ)n2,
where
(42)n1=4raκ(a+b);n2=4rbκ(a+b).
This leads to the field potential term
(43)V=12[(b+ϕ)n2-rχ2κ(a-ϕ)(b+ϕ)+C(a-ϕ)n1(b+ϕ)n2-rχ2]2+2r2ϕ2χ2.
Here we have the static solution extracted from (34) for our choice of U(ϕ) in (40); it reads
(44)ϕ(x)=a-b2±a+b2tanh(κ(a+b)x2),
and from (38) and (41) we obtain
(45)χ(x)=±Cr(a-ϕ(x))(n1/2)(b+ϕ(x))(n2/2).
Dilatations and translations in the internal space allow us to relocate two vacua placed in the ϕ-axis at the points (±1,0), such that without loss of generality we can assume that a=1 and b=1. If we restrict ourselves to potentials (43) with a quartic algebraic expression in the fields ϕ and χ we must impose the conditions 4ar/(a+b)κ=1 and 4br/(a+b)κ=1, or equivalently κ=2r. In this case we get the family of potentials
(46)V=12[(2r+C)(1-ϕ2)-rχ2]2+2r2ϕ2χ2,
where ℳ={v1,2=(±1,0), v3,4=(0,±2+C/r)} comprises four elements provided that C>-2r. The two-component kinks
(47)ϕ(x)=tanh(2rx),χ(x)=±Crsech(2rx)
whose kink orbit is given by rχ2=C(1-ϕ2) connect the points v3,4 and EBPS=(4/3)(C+2r). The expression (46) can be written as
(48)V=(2r+C)2[12(1-ϕ2-r¯χ2)2+2r¯2ϕ2χ2],
where r¯=r/(2r+C). A reparametrization of the spatial variable x¯=(2r+C)x allows us to identify the present example with the potential introduced in the previous section. In this sense if we choose C=0 we get the one-component topological kink solutions (24). For any other choice of the constant C the solution (47) plays the role of the two-component kink (25). The comparison is straightforward when the constant 2r+C in (46) is unity see Figure 1. This works as a test for the procedure introduced in this work in a well-known two-field theory model.If we consider the special caseb=0 in (40), such that U=κϕ(a-ϕ), the use of (37) leads to the function
(49)f(ϕ)=κϕ(a-ϕ)+C(a-ϕ)4r/κ,
which generates the field potential
(50)V=12[C(a-ϕ)4r/κ-rχ2κϕ(a-ϕ)+C(a-ϕ)4r/κ-rχ2]2+2r2ϕ2χ2.
Because we are interested in quartic potentials in this section, we set κ=2r. The previous formula becomes
(51)V=12[(a-ϕ)(aC+(2r-C)ϕ)-rχ2]2+2r2ϕ2χ2,
whose zeroes are located at ℳ={v1=(a,0), v2=(-aC/(2r-C),0), v3,4=(0,±a(C/r))}. Equation (38) leads to the kink orbits χ=±C/r(a-ϕ) which connect the points v3,4 with v1; see Figure 3. The kink solutions are
(52)ϕ=a2(1+tanh(arx)),χ=±a2Cr(1-tanh(arx))
whose energy is EBPS=a3(C+r)/3.Graphics of the potentialV(ϕ,χ) with r=a=1, C=3/2 and orbits of the kinks (52) and (53) in the internal plane.
(a)
(b)As previously mentioned it is easy to identify the one-component kink linking the vacuav1,2 for this model. We have
(53)ϕ=a+2arC-2r-e2arx,χ=0,
whose energy is EBPS=4a3r3/3(C-2r)2.Notice that if we considerr=a=C=κ/2=1 in (52), we recover the solutions (26) of the test model introduced in the previous section.The above illustration shows that the procedure works nicely. Thus, below we introduce new families of models using adequate choices of the parameters.


#### 3.1.1. A Family of Models

In the previous section we restrict ourselves to quartic potentials. Here we will introduce expressions of higher degree. Let us choosea=1, b=1, and κ=2r/n in (41) and (42) with n as a positive integer. Besides we redefine the coupling constant r as r=nr¯/2. Here we have
(54)f(ϕ)=(1-ϕ2)(r¯+C(1-ϕ2)n-1),
which determines the potentials
(55)V=12[-nr¯2χ2(1-ϕ2)(r¯+C(1-ϕ2)n-1)-nr¯2χ2]2+n2r¯22ϕ2χ2.
These potentials involve two distinct behaviors depending on n being odd or even.In the case forn even, there are six degenerate minima at the points ℳe={v1,2=(0,±2(r¯+C)/(nr¯)), v3,4=(±1,0), v5,6=(±1+(r¯/C)1/(n-1),0)} while for n odd there are four degenerate minima; ℳo={v1,2,v3,4}. For all models in the above family, we can find the static solutions
(56)ϕ(x)=±tanh(r¯x),χ(x)=±2Cnr¯sechn(r¯x),
which connect the minima v3 and v4 by means of the orbit (38) given by the algebraical curve
(57)r¯nχ2=2C(1-ϕ2)n;see Figure4. These solutions carry the energy
(58)EBPS=4r¯3+πCΓ[n+1]Γ[n+3/2].Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (56) in the internal plane.
(a)
(b)Now for the orbitχ(x)=0 in the BPS sectors connecting neighbor minima at the ϕ-axis, the solutions are obtained from (39) with f(ϕ) given by (41), which may be solved case by case. For n=1, we have the solution ϕ(x)=±tanh[(r¯+C)x], and for n=2 we get the implicit expression
(59)(1+ϕ)(1-ϕ)(1+(r¯/C)-ϕ1+(r¯/C)+ϕ)C/(C+r¯)=e2r¯x
and so on for other values of n. It is remarkable that we can obtain the explicit expression of the two-component kinks (56) for any value of n but not for the one-component kinks.


#### 3.1.2. Another Family of Models

Here we takeb=0, κ=4r/n, a>0, and integer n>0 and we redefine the coupling constant r as r=(r¯n)/4. From (41) we have
(60)f(ϕ)=(a-ϕ)(r¯ϕ+C(a-ϕ)n-1),
which generate a family of models whose potentials have up to n+2 minima: two minima at v1,2=(0,±2(anC)/(r¯n)), one at v3=(a,0), and up to n-1 minima (for n>1) coming from the condition g(ϕ)=r¯ϕ+C(a-ϕ)n-1=0.For all models of this family, we get from (38), (44), and (40) in the sector connecting the minima v1,2 and v3 the static solutions
(61)ϕ(x)=a2(1±tanh(ar¯x2)),χ(x)=±21-(n/2)cr¯nan/2(1-tanh(ar¯x2))n/2,
whose energy is given by EBPS=(r¯a3/6)+(Can+1/(n+1)); see Figure 5.Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (61) in the internal plane.
(a)
(b)


## 3.1. A First Example

To illustrate the previous procedure with concrete examples, let us start considering(40)U(ϕ)=κ(a-ϕ)(b+ϕ),
where a, b, and κ are real parameters and we obviously assume that a,b≠0 and a≠-b. We use (37) together with (34) and (36) to obtain
(41)f(ϕ)=κ(a-ϕ)(b+ϕ)+C(a-ϕ)n1(b+ϕ)n2,
where
(42)n1=4raκ(a+b);n2=4rbκ(a+b).
This leads to the field potential term
(43)V=12[(b+ϕ)n2-rχ2κ(a-ϕ)(b+ϕ)+C(a-ϕ)n1(b+ϕ)n2-rχ2]2+2r2ϕ2χ2.
Here we have the static solution extracted from (34) for our choice of U(ϕ) in (40); it reads
(44)ϕ(x)=a-b2±a+b2tanh(κ(a+b)x2),
and from (38) and (41) we obtain
(45)χ(x)=±Cr(a-ϕ(x))(n1/2)(b+ϕ(x))(n2/2).
Dilatations and translations in the internal space allow us to relocate two vacua placed in the ϕ-axis at the points (±1,0), such that without loss of generality we can assume that a=1 and b=1. If we restrict ourselves to potentials (43) with a quartic algebraic expression in the fields ϕ and χ we must impose the conditions 4ar/(a+b)κ=1 and 4br/(a+b)κ=1, or equivalently κ=2r. In this case we get the family of potentials
(46)V=12[(2r+C)(1-ϕ2)-rχ2]2+2r2ϕ2χ2,
where ℳ={v1,2=(±1,0), v3,4=(0,±2+C/r)} comprises four elements provided that C>-2r. The two-component kinks
(47)ϕ(x)=tanh(2rx),χ(x)=±Crsech(2rx)
whose kink orbit is given by rχ2=C(1-ϕ2) connect the points v3,4 and EBPS=(4/3)(C+2r). The expression (46) can be written as
(48)V=(2r+C)2[12(1-ϕ2-r¯χ2)2+2r¯2ϕ2χ2],
where r¯=r/(2r+C). A reparametrization of the spatial variable x¯=(2r+C)x allows us to identify the present example with the potential introduced in the previous section. In this sense if we choose C=0 we get the one-component topological kink solutions (24). For any other choice of the constant C the solution (47) plays the role of the two-component kink (25). The comparison is straightforward when the constant 2r+C in (46) is unity see Figure 1. This works as a test for the procedure introduced in this work in a well-known two-field theory model.If we consider the special caseb=0 in (40), such that U=κϕ(a-ϕ), the use of (37) leads to the function
(49)f(ϕ)=κϕ(a-ϕ)+C(a-ϕ)4r/κ,
which generates the field potential
(50)V=12[C(a-ϕ)4r/κ-rχ2κϕ(a-ϕ)+C(a-ϕ)4r/κ-rχ2]2+2r2ϕ2χ2.
Because we are interested in quartic potentials in this section, we set κ=2r. The previous formula becomes
(51)V=12[(a-ϕ)(aC+(2r-C)ϕ)-rχ2]2+2r2ϕ2χ2,
whose zeroes are located at ℳ={v1=(a,0), v2=(-aC/(2r-C),0), v3,4=(0,±a(C/r))}. Equation (38) leads to the kink orbits χ=±C/r(a-ϕ) which connect the points v3,4 with v1; see Figure 3. The kink solutions are
(52)ϕ=a2(1+tanh(arx)),χ=±a2Cr(1-tanh(arx))
whose energy is EBPS=a3(C+r)/3.Graphics of the potentialV(ϕ,χ) with r=a=1, C=3/2 and orbits of the kinks (52) and (53) in the internal plane.
(a)
(b)As previously mentioned it is easy to identify the one-component kink linking the vacuav1,2 for this model. We have
(53)ϕ=a+2arC-2r-e2arx,χ=0,
whose energy is EBPS=4a3r3/3(C-2r)2.Notice that if we considerr=a=C=κ/2=1 in (52), we recover the solutions (26) of the test model introduced in the previous section.The above illustration shows that the procedure works nicely. Thus, below we introduce new families of models using adequate choices of the parameters.


### 3.1.1. A Family of Models

In the previous section we restrict ourselves to quartic potentials. Here we will introduce expressions of higher degree. Let us choosea=1, b=1, and κ=2r/n in (41) and (42) with n as a positive integer. Besides we redefine the coupling constant r as r=nr¯/2. Here we have
(54)f(ϕ)=(1-ϕ2)(r¯+C(1-ϕ2)n-1),
which determines the potentials
(55)V=12[-nr¯2χ2(1-ϕ2)(r¯+C(1-ϕ2)n-1)-nr¯2χ2]2+n2r¯22ϕ2χ2.
These potentials involve two distinct behaviors depending on n being odd or even.In the case forn even, there are six degenerate minima at the points ℳe={v1,2=(0,±2(r¯+C)/(nr¯)), v3,4=(±1,0), v5,6=(±1+(r¯/C)1/(n-1),0)} while for n odd there are four degenerate minima; ℳo={v1,2,v3,4}. For all models in the above family, we can find the static solutions
(56)ϕ(x)=±tanh(r¯x),χ(x)=±2Cnr¯sechn(r¯x),
which connect the minima v3 and v4 by means of the orbit (38) given by the algebraical curve
(57)r¯nχ2=2C(1-ϕ2)n;see Figure4. These solutions carry the energy
(58)EBPS=4r¯3+πCΓ[n+1]Γ[n+3/2].Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (56) in the internal plane.
(a)
(b)Now for the orbitχ(x)=0 in the BPS sectors connecting neighbor minima at the ϕ-axis, the solutions are obtained from (39) with f(ϕ) given by (41), which may be solved case by case. For n=1, we have the solution ϕ(x)=±tanh[(r¯+C)x], and for n=2 we get the implicit expression
(59)(1+ϕ)(1-ϕ)(1+(r¯/C)-ϕ1+(r¯/C)+ϕ)C/(C+r¯)=e2r¯x
and so on for other values of n. It is remarkable that we can obtain the explicit expression of the two-component kinks (56) for any value of n but not for the one-component kinks.


### 3.1.2. Another Family of Models

Here we takeb=0, κ=4r/n, a>0, and integer n>0 and we redefine the coupling constant r as r=(r¯n)/4. From (41) we have
(60)f(ϕ)=(a-ϕ)(r¯ϕ+C(a-ϕ)n-1),
which generate a family of models whose potentials have up to n+2 minima: two minima at v1,2=(0,±2(anC)/(r¯n)), one at v3=(a,0), and up to n-1 minima (for n>1) coming from the condition g(ϕ)=r¯ϕ+C(a-ϕ)n-1=0.For all models of this family, we get from (38), (44), and (40) in the sector connecting the minima v1,2 and v3 the static solutions
(61)ϕ(x)=a2(1±tanh(ar¯x2)),χ(x)=±21-(n/2)cr¯nan/2(1-tanh(ar¯x2))n/2,
whose energy is given by EBPS=(r¯a3/6)+(Can+1/(n+1)); see Figure 5.Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (61) in the internal plane.
(a)
(b)


## 3.1.1. A Family of Models

In the previous section we restrict ourselves to quartic potentials. Here we will introduce expressions of higher degree. Let us choosea=1, b=1, and κ=2r/n in (41) and (42) with n as a positive integer. Besides we redefine the coupling constant r as r=nr¯/2. Here we have
(54)f(ϕ)=(1-ϕ2)(r¯+C(1-ϕ2)n-1),
which determines the potentials
(55)V=12[-nr¯2χ2(1-ϕ2)(r¯+C(1-ϕ2)n-1)-nr¯2χ2]2+n2r¯22ϕ2χ2.
These potentials involve two distinct behaviors depending on n being odd or even.In the case forn even, there are six degenerate minima at the points ℳe={v1,2=(0,±2(r¯+C)/(nr¯)), v3,4=(±1,0), v5,6=(±1+(r¯/C)1/(n-1),0)} while for n odd there are four degenerate minima; ℳo={v1,2,v3,4}. For all models in the above family, we can find the static solutions
(56)ϕ(x)=±tanh(r¯x),χ(x)=±2Cnr¯sechn(r¯x),
which connect the minima v3 and v4 by means of the orbit (38) given by the algebraical curve
(57)r¯nχ2=2C(1-ϕ2)n;see Figure4. These solutions carry the energy
(58)EBPS=4r¯3+πCΓ[n+1]Γ[n+3/2].Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (56) in the internal plane.
(a)
(b)Now for the orbitχ(x)=0 in the BPS sectors connecting neighbor minima at the ϕ-axis, the solutions are obtained from (39) with f(ϕ) given by (41), which may be solved case by case. For n=1, we have the solution ϕ(x)=±tanh[(r¯+C)x], and for n=2 we get the implicit expression
(59)(1+ϕ)(1-ϕ)(1+(r¯/C)-ϕ1+(r¯/C)+ϕ)C/(C+r¯)=e2r¯x
and so on for other values of n. It is remarkable that we can obtain the explicit expression of the two-component kinks (56) for any value of n but not for the one-component kinks.


## 3.1.2. Another Family of Models

Here we takeb=0, κ=4r/n, a>0, and integer n>0 and we redefine the coupling constant r as r=(r¯n)/4. From (41) we have
(60)f(ϕ)=(a-ϕ)(r¯ϕ+C(a-ϕ)n-1),
which generate a family of models whose potentials have up to n+2 minima: two minima at v1,2=(0,±2(anC)/(r¯n)), one at v3=(a,0), and up to n-1 minima (for n>1) coming from the condition g(ϕ)=r¯ϕ+C(a-ϕ)n-1=0.For all models of this family, we get from (38), (44), and (40) in the sector connecting the minima v1,2 and v3 the static solutions
(61)ϕ(x)=a2(1±tanh(ar¯x2)),χ(x)=±21-(n/2)cr¯nan/2(1-tanh(ar¯x2))n/2,
whose energy is given by EBPS=(r¯a3/6)+(Can+1/(n+1)); see Figure 5.Graphics of the potentialV(ϕ,χ) with n=4 and orbits of the kinks (61) in the internal plane.
(a)
(b)


## 4. Nonpolynomial Models

Let us now move on to the case of nonpolynomial potentials. Here we consider(62)W(ϕ,χ)=∫ϕf(y)dy-rϕsinχ,
such that the field potential is
(63)V=12[(f(ϕ)-rsinχ)2+r2ϕ2cos2χ],
which is a periodic function in the variable χ as illustrated in Figure 6. The set of zeroes is given by ℳ={v1,m1,v2,m2,vi+2,m3}, where
(64)v1,m1=(0,arcsinf(0)r+2πm2),v2,m2=(0,-arcsinf(0)r+π(2m3+1)),vi+2,m3=(ϕ(i),π(m1+12)),
where ϕ(i) are the roots of the function f(ϕ)-(-1)nr, and mi∈ℤ. In this case, the static solutions are obtained from the first-order equations
(65)ϕ′=f(ϕ)-rsinχ,χ′=-rϕcosχ.Graphics of the potentialV(ϕ,χ) and orbits of the kinks (76) in the internal plane for n=2.
(a)
(b)We can manipulate these equations to get(66)ϕ′′+ϕϕ′2-(2ϕf(ϕ)+dfdϕ)ϕ′-ϕ(r2-f2(ϕ))=0.Again, choosingϕ(x) satisfying (34) we can rewrite the above equation  as
(67)dfdϕ+P(ϕ)f(ϕ)-R(ϕ)f2(ϕ)-Q(ϕ)=0,
where
(68)P(ϕ)=2ϕ,R(ϕ)=ϕU,Q(ϕ)=dUdϕ+ϕU-r2ϕU.
The general solution is given by
(69)f(ϕ)=U(ϕ)+rtanh[r(C-∫‍ϕdϕU(ϕ))].
Then, choosing U(ϕ) we have the constraint
(70)rsinχ=f(ϕ)-U(ϕ)
for the field χ.Also, the first-order equations (65) support the orbit χ(x)=±(2n-1)π/2, n=0,1,2,…. In this case, the static solutions ϕ(x), connecting neighbor minima located at the ϕ-axis, are obtained from the equation
(71)dϕdx=f(ϕ)+(-1)nr.
Let us now illustrate the above procedure with an example. We start using
(72)U(ϕ)=κ(a-ϕ)(b+ϕ),
where a, b, κ are real parameters. By employing (69) we obtain
(73)(ϕ)=U(ϕ)+rC2(a-ϕ)2ar/κ(a+b)(b+ϕ)2br/κ(a+b)-1C2(a-ϕ)2ar/κ(a+b)(b+ϕ)2br/κ(a+b)+1.
Again we restrict ourselves to cases where the exponents in the previous expression are even integers, such that we impose that ar/κ(a+b)=n with n∈ℕ, for instance, by choosing κ=ar/n(a+b). In this case
(74)f(ϕ)=U(ϕ)+rC2(a-ϕ)2n(b+ϕ)2n-1C2(a-ϕ)2n(b+ϕ)2n+1.
The field potential term is obtained by plugging (74) into (63). In spite of the complexity of this expression the kink solution is written by the expression (44) and from (70) as follows:
(75)χ=arcsinC2(a-ϕ(x))2n(b+ϕ(x))2n-1C2(a-ϕ(x))2n(b+ϕ(x))2n+1.
For example for a=b=1 we get
(76)ϕ=tanhrx2n,χ=arcsin(1-21+C2sech2n(rx/2n)),
whose orbit is displayed in Figure 6 for the case n=2.


## 5. Final Comments

In this work we proposed a new method to construct and solve models described by two real scalar fields. The procedure is simple, inspired by the approach introduced in [26], and it works for the construction of polynomial and nonpolynomial models.To illustrate the procedure, we studied several examples, which show how efficient the method is, in order to construct new two-field models with nontrivial two-component kink solutions. We note that the method starts withW, the superpotential, so all the models we construct lead to first-order differential equations, which solve the equations of motion. In this sense, all the solutions we found are BPS states, and they are classically or linearly stable, as proved before in [27].A relevant feature of the procedure is that it is different from the deformation procedure involving two-field models, and it is very simple to be applied in investigations based on two real scalar fields. An issue which deserves further examination concerns the extension of the method to three or more real scalar fields. This is under investigation, and we hope to report the new results in a separate work.



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*Source: 183295-2013-09-03.xml*