Multi-sine-Gordon Models

In this work we study defect structures in relativistic models described by a single real scalar field, which are of current interest in high energy physics b1 ; b2 and in condensed matter b3 ; b4 . We focus on static solutions in $(1,1)$ space-time dimensions, with the motivation to present families of models, and the defect structures they may comprise. The models are described by a single real scalar field, engendering discrete symmetry, so we search for defect structures of global nature, known as kinks and lumps.

In high energy physics, kinks and lumps are of interest in several different contexts b1 ; b2 , including supergravity, branes with a single extra dimension and cosmology of branes sugra ; brane ; cbrane ; etc . In condensed matter, they may contribute to pattern formation b3 , and to the description of interfaces among distinct configurations in magnetic systems b4 . They may also appear in other applications, for instance, in the study of nonlinear excitations in Bose-Einstein condensates bec , and in the construction of magnetic apparatus at the nanometric scale a .

We use the deformation procedure introduced in Ref. dd , and we deform the starting model, which is the $\phi^{4}$ model engendering spontaneous symmetry breaking, to get to new families of sine-Gordon models. We focus on sine-Gordon models because they are of interest in a diversity of contexts; see, e.g., Refs. sg1 ; sg2 . The procedure is implemented via the deformation function and the parameters it comprises, which leads us to the double and triple sine-Gordon models with analytical solutions, etc. Since the potential of the starting model, the $\phi^{4}$ model, can be described by a superpotential which is known explicitly, explicit expressions for the superpotential for deformed models can also be derived, and this leads to important informations on the corresponding BPS states bps , as well as on the energy and stability of the solutions bs of the families of models.

Most of the families of periodic potentials which will be introduced in this work engender periodic substructure, and can be identified by ${\cal M}=1,2,...$, an integer which describes the multiplicity of the substructure of each potential. In this sense, we will describe a diversity of multi-sine-Gordon models, and the defect structures they may comprise. In particular, the double, triple and other multi-frequency sine-Gordon models are of interest to issues concerning integrability msg , and they can also be used to generate braneworld scenarios with a single extra dimension brane ; cbrane ; etc , with substructures that could appear in the form of multi-brane scenarios. The scalar field used to describe the periodic models could also interact with another scalar field, to give rise to defect inside defect did , and to allow for the presence of brane with internal structure bis . In this sense, the present work provides a variety of multi-sine-Gordon models, and study the presence of topological structure at the classical level, offering results which are of current interest to high energy physics, and to other areas of nonlinear science.

We organize the work as follows: In the next Sect. II we describe the main steps of the deformation procedure. In Sect. III we single out the deformation function which will be used throughout the work, and there we study specific sine-Gordon models. In Sect. IV we introduce the superpotentials corresponding to the multi sine-Gordon models studied in the previous Sect. III, and we calculate the energies corresponding to several specific solutions the models comprise. Finally, in Sect. V we present comments and conclusions.

We start the investigation with a quick review of the deformation procedure dd . We consider two real scalar fields $\chi=\chi(x,t)$ and $\phi=\phi(x,t)$ and the corresponding models, described by the Lagrange densities,

Here, $U(\chi)$ and $V(\phi)$ are given potentials, which specify each one of the two models. We introduce a function $f=f(\phi),$ named deformation function, from which we link the model (1a) with the deformed model (1b) by relating the two potentials $U(\chi)$ and $V(\phi)$ in the very specific form

where $U(\chi\to f(\phi))$ means that in the potential $U(\chi)$, we change $\chi$ by $f(\phi)$. In this case, if the starting model has finite energy static solution $\chi(x)$ which obeys the equation of motion

then the deformed model may also have finite energy static solution given by

which obeys

We note that if the deformation function $f(\phi)$ has critical points, the deformation procedure works smoothly if and only if the potential $U(f(\phi))$ has zeros of multiplicity at least two at those critical points. The critical points of $f(\phi)$ are the branching points of $f^{-1}(\chi),$ which is consequently a multivalued function.

The deformation procedure has been used by several authors in a diversity of scenarios, and it appears to work correctly in many distinct situations ddsal ; dd+ .

We go further into the problem using the standard model

This is the $\chi^{4}$ model, with spontaneous symmetry breaking. Here we are using natural units, and we have rescaled the field, and the space and time coordinates to make them dimensionless. This model has as topological defects the BPS states $\chi_{\pm}(x)=\pm\tanh(x-x_{0})$, and we choose the center of the solutions at the origin $x_{0}=0$, for simplicity. The energy density is given by $\varepsilon(x)={\rm sech}^{4}(x)$, which gives the energy $E=4/3$. We also note that the potential has minima at ${\bar{\chi}}_{\pm}=\pm 1$ and obeys: $V({\bar{\chi}}_{\pm})=0,$ $V^{\prime}({\bar{\chi}}_{\pm})=0$, and $V^{\prime\prime}({\bar{\chi}}_{\pm})=4$, where $V^{\prime}(\chi)=dV/d\chi$, etc. Next, if we choose the deformation function in the form $f(\phi)=\cos(\phi)$, we get to the potential

In this case the deformation function depends only on $\phi$, and the corresponding deformed model describes the sine-Gordon model, with no extra parameter involved in the procedure. This model has solutions described by

where $k=0,\pm 1,\pm 2,...$ identifies the particular topological sector of the sine-Gordon model, which has an infinity of sectors. We then see that we go from the $\phi^{4}$ model, which contains a single topological sector, to the sine-Gordon model, which contains an infinity of topological sectors, by the use of the deformation $f(\phi)=\cos(\phi)$, which is a periodic function, which needs the presence of the integer $k$ to identify the particular periodic sector, which is naturally used to identify the particular topological sector of the deformed model, as we have just shown.

Here we follow the methodology of Ref. ddsal , searching for new functions $f=f(\phi)$, from which we can build families of periodic models. We take the starting model described by the potential (6) and the kink-like solution $\chi_{1}(x)=\pm\tanh(x)$, and we choose the deformation function

This function depends on the choice of $g({\phi})$, on $a,$ which is real constant, and on $M$, which can be integer or semi-integer: for $M$ integer, the deformed potential can be written in the form

where $g^{\prime}(\phi)=dg/d\phi$. If we choose $M$ semi-integer, we get

As we show below, using integer values for the parameter $a$ lead the number of vacua of the new model to be fixed at will.

The choice (9) is a generalization of the deformation function used in ddsal . There, the deformation function was taken as $g(\phi)=\phi$, and this allowed to generate families of polynomial potentials. In the present work, below we will take $g(\phi)$ as a periodic function of the scalar field, and this will generate new families of models, with periodic potentials, as generalized sine-Gordon potentials. All the potentials are new periodic potentials, and we solve them finding the defect structures they may comprise directly, using the deformation procedure. In this sense, the present work contribute to offer new models, to study the defect structures they may comprise, and to strengthen the power of the deformation procedure introduced in dd .

In general, the above models identify families of potentials which present static solutions given by

where $g^{-1}$ represents the inverse of $g(\phi)$. Also, $\theta(x)=\arccos(\tanh(x))$, with $\theta\in[0,\pi]$. However, to investigate specific models, let us start with (10) and $g(\phi)=\phi$. Here the potentials are, for $a$ odd,

and for $a$ even,

where $Z_{j}^{a}=\cos((j-1)\pi/a)$.

If we start with (11) and $g(\phi)=\phi$, the potentials are, for $a$ odd,

and, for $a$ even,

where ${Z}_{j}^{a}=\cos((2j-1)\pi/2a)$.

Here we see that the choice $g(\phi)=\phi$ leads us to polynomial potentials, as already investigated in Ref. ddsal . In the present work, however, we want to study periodic potentials, of the sine-Gordon type. For this reason, we consider $g(\phi)$ as a periodic function of the scalar field. If we take $g(\phi)=\cos\phi$, from (10) and (11), we have the potentials

and

which are sine-Gordon potentials with multiplicity ${\cal M}=1$, with the parameter $a$ now controlling the amplitude and periodicity of the potential. In order to obtain new sine-Gordon potentials with higher multiplicity, in the next subsections we take the function $g(\phi)=\alpha\;\cos\phi$, where $\alpha$ will be chosen appropriately.

It is sometimes possible to derive explicit expressions for superpotentials. If $W$ is a function of the scalar field, it is the superpotential if the potential is written in the form

where the prime stands for the derivative with respect to the field. Now, from the deformation procedure we can write that if $W=W(\chi)$ is the superpotential of the starting model, then the superpotential ${\cal W}(\phi)$ of the deformed model has to obey

In the case of the $\phi^{4}$ model which we are using to implement the deformation procedure, we have

and this can be used to get to the superpotentials of the deformed models.

The expressions of the superpotentials are of great use, since they can be used to get the energies of the corresponding defect structures, since we know that the energy of a topological defect $\chi(x)$ with the potential described by the superpotential $W(\chi)$ is given by

where $\chi(\infty)={\rm limit}_{x\to\infty}\chi(x)$, for $\chi(x)$ being the static kinklike solution.

Alternatively, the superpotential can be directly obtained from the integration of Eq. (40),

but the integral does not in general lead to an explicit formula for $W$. However, this is not the case here, and the explicit form of the superpotentials corresponding to the $sine$ family, for $a=3,4,5,$ and $6$ are given by

For the $cosine$ family, the explicit form of the superpotentials for $a=4,$ are given by

The above expressions allow us to get the energy corresponding to each distinct topological sector. To illustrate this issue, let us calculate some energies. We start with the $sine$ family of models. The potential $V^{3,1}_{\sin}$ has multiplicity ${\cal M}=1$, so it has only one kind of kink, with energy $E=1.222$. The potential $V^{4,1}_{\sin}$ has multiplicity ${\cal M}=1$, and the kink has energy $E=0.375$. The potential $V^{5,1}_{\sin}$ which is plotted in Fig. 1 has multiplicity ${\cal M}=2$, so it has two distinct kinks, which we name large and small kinks, the large one representing the kink in the sector where the maxima of the potential has higher height; here the kink energies are given by $E_{l}=0.211$ and $E_{s}=0.123$. The potential $V^{6,1}_{\sin}$ is plotted in Fig. 1, and it also has multiplicity ${\cal M}=2$; here the corresponding energies are $E_{l}=0.139$ and $E_{s}=0.076$. Another potential is $V^{7,1}_{\sin}$, which is plotted in Fig. 3. It has multiplicity ${\cal M}=3$, and the corresponding kink energies are $E_{l}=0.099$, $E_{m}=0.053$, and $E_{s}=0.050$, for the large, medium, and small kink, respectively.