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Configurational entropy of generalized sine-Gordon-type models

A family of deformed models of the sine-Gordon-type can be generated by twisting the sine-Gordon model. As a particular case, the 3-sine-Gordon model is here addressed, whose differential configurational entropy and the differential configurational complexity of three topological sectors are discussed, using two complementary approaches. Stability aspects are also discussed.

derived in the context of sine-Gordon potentials [62].
The main aim of this work is to study the DCE and the differential configurational complexity (DCC) underlying deformed sine-Gordon-type models, particularly focusing on the 3-sine-Gordon deformed models, which have several analytical approaches. The DCC concept arises when one realizes that if the power spectrum underlying wave modes is uniform, the inherent complexity is lower, whereas if the power spectrum is distributed non-uniformly among the wave modes, the complexity becomes higher [6]. The underlying DCE will be shown to refine the existing analysis of 3-sine-Gordon deformed models, providing new important physical analyses, focusing on two approaches for the deformation procedure of the original sine-Gordon model. The DCE will be also shown to drive the best choice of the parameters involved in the deformation procedure, associated with the DCE global critical points. This work is organized as follows: Sect. II is devoted to introducing the necessary background for the deformation procedure and constructing the family of deformed sine-Gordon-type models. Sect. III scrutinizes the role played by the DCE in this context, addressing new subtle aspects. Also, it provides and computes the DCE for the 3-sine Gordon model, this time without approximations but in the context of a deformation procedure without a power expansion. Sect. IV presents the conclusions and perspectives about the important results involving the DCE associated with these models.

II. DEFORMED SINE-GORDON-TYPE MODELS
For a scalar field χ = χ(t, x), one can consider models implemented by the Lagrangian where V (χ) denotes the potential governing a scalar field theoretical model. Topological solutions can be particularly introduced when static fields are taken into account. In this case, the equation of motion for χ = χ(x) reads One searches for configurations of the scalar field that sets forth a minimum χ min of V (χ), with boundary conditions lim x→∞ dχ(x)/dx = 0 and lim x→−∞ χ(x) = χ min . Topological solutions usually appear when the potential is non-negative, which can be written, using the superpotential W = W (χ), in the following form, where W χ stands for 1 dW/dχ. It yields the equation of motion [44] which can be solved when solutions of the first-order ODE are employed. As Eq. (3) shows that the scalar potential is invariant under W → −W , Eq. (5) can be also analyzed in the context of this Z 2 -symmetry. Therefore there is a well-defined mapping between the second-order equation of motion into two first-order equations, which in general holds for the topological solution. A topological solution of Eq. (5) on the topological section (jk), has energy minimized to the value E jk BPS = |W (χ j ) − W (χ k )|, which does represent a saturated inequality depending on the homotopy class of the solution at infinity [44,45]. Hereχ j andχ k are two minima. If these minima are adjacent, clearly It represents the Bogomol'nyi bound, and the solution of (5) is named a BPS (Bogomol'nyi-Prasad-Sommerfield) linearly stable state. By insisting that the BPS bound is saturated, one can come up with a simpler set of differential equations to solve, called the Bogomol'nyi equations. Since the superpotential W (χ) is χ-dependent, it can be used to define the topological behavior from the topological current density j µ top = ε µν ∂ ν W (χ). This definition differs from the standard form of the topological current density given by j µ = ε µν ∂ ν χ, which is also trivially conserved, due to the antisymmetry of the Levi-Civita tensor. One can use W (χ) to define the topological current density, yielding the conserved topological charge identical to the energy [63]. This technique of utilizing the superpotential W (χ) can be in some cases more suitable than considering the scalar field χ itself, in the construction of the topological current density [63]. The corresponding topological charge is given by which equals the energy of the static scalar field configurations. The vacuum sector, which is identified by time and space independent field configurations, has zero topological charge.
The concept of conserved topological charge, therefore, introduces topological sectors. Distinct non-vanishing topological charges define different topological sectors, yielding a quite general classification scheme for the topological solutions [63,64]. Besides, the temporal component j 0 top = dW (χ(x))/dx, so that the associated topological charge is directly related to the energy topological current density, see, e. g., Ref. [65].
We will focus our attention on the deformation procedure proposed in Refs. [44,45,55]. A significant advantage of using deformation approaches consists of straightforwardly generating new models and constructing their topological defects analytically, from a given model that hereon will be considered the sine-Gordon one. The scalar field model (1) is supposed to support topological defects. Deformations of the sine-Gordon model, whose potential reads can be therefore implemented. This potential is obtained from the superpotential W (χ) = sin(χ) and has an infinite minima,χ = ±nπ/2, for n ∈ 2N + 1, which can be connected by the topological solutions where k ∈ Z and gd(x) ≡ θ(x) = arcsin[tanh(x)] denotes the Gudermannian function, whose associated energy reads E = 2, with dimensionless units being used. It is worth mentioning that the energy associated to χ(x) can be minimized to when the field configurations obey characterizing BPS states. Hence, the fact that the BPS energy of the minima equals zero comes from the fact that singular points of W (χ) are absolute minima of the potential satisfying W χ (χ) = 0.
The deformation procedure emulates Eq. (1), stating that a family of models can be generated by considering a scalar field Lagrangian density defined by where U (φ) is the potential, governing the new scalar field φ, written in terms of V (χ) as where f (φ) is the deformation function. As χ(x) is a static solution of the original sine-Gordon model, then it can be allocated as a solution of the deformed model whenever the mapping holds. The first-order equations of the deformed model read One can express where is a small parameter that allows a power expansion with respect to it [44,45], and g(φ) is, a priori, an arbitrary function regulating the deformation, being compatible with the -expansion.
Although the deformation procedure holds for a general potential, models controlled by the superpotential W (χ) will be approached, where V (χ) = 1 2 W 2 χ . This expression can be emulated for the new potential as where defining the new superpotential [44] as an -deformation of the superpotential W (χ) = yields Comparing Eqs. (15) and (16) makes it possible to write which can be utilized to derive the minima of the potential by analyzing the deformed ODE The minima can be used to find the corresponding energy in each topological sector, in the form given φ i+1 and φ i representing two adjacent minima of the U (φ). Also, taking into account Eq. (13), up to the first order in , the solution of the new model can be expressed as [55] φ(x) = χ(x) − g(χ(x)).
To introduce a new family of sine-Gordon-type models, Refs. [44,45,55] introduced the original sine-Gordon given by Eq. (7) and implemented the deformation function as where, for deforming the sine-Gordon model, one defines [55] g a (φ) ≡ sin φ a .
The deformation function (20) is parametrized by two real parameters, a = 0 and , with small .
A straightforward advantage of this protocol is the immediate implementation yielding potentials that have interest in relevant applications. Besides, the small parameter identifies topological sector solutions of the first-order equation, with respective energy densities associated with the deformed models. The deformation function defines a family of sine-Gordon-type models described by the potential (14), where the superpotential W has derivative with respect to the scalar field φ given by An additional term arises, being responsible to shift the minima of the potential to new ones, given also modifying the maxima as When a ∈ Z, the number of distinct topological sectors, labeled by p, equals a + 1. Therefore the deformed potential is realized to have periodicity 2πa. Taking into account a = 1 yields the 2-sine-Gordon, whereas, for a = 2, the 3-sine-Gordon model can be obtained, with immediate generalizations for each value of a. For a = 1, the new superpotential reads For the particular case where a = 1, the new superpotential has the straightforward expression Based on the very definition of the deformed superpotential in Eq. (14), the corresponding deformed potential is depicted in Figs  One can then employ W and the minima (23) to compute the energy associated with a topological defect corresponding to each topological sector. The general expression for W in each minimum is which yields the particular case regarding the standard sine-Gordon model for a = 1, Also, topological solutions associated with the first-order equation (5) can be derived using the inverse deformation function. Eq. (20) is hence employed to yield [55] φ where φ 0 (x) = χ(x) denotes the solutions (8) of the original sine-Gordon model. The associated energy density reads with ρ 0 (x) = sech 2 (x) representing the standard energy density associated with sine-Gordon classical solutions.
Now the case of the 3-sine-Gordon model can be considered, which is obtained with a = 2. There are three kinds of topological sectors. The sectors of the first kind are described by the following where is an integer, that connect the minima The first topological sector (31) and the respective minima (32a, 32b) are depicted in Fig. 3. The sectors of the second kind are described by the solutions which connect the minima The third kind of topological sector is described by the solutions

M°2
which connect the minima The third topological sector (35) and the respective minima (36a, 36b) are displayed in Fig. 5. given by [55] whose plots, for various values of , are described in Figs. 6 -8.
To ensure energy positivity, in the first and third topological sectors it must be imposed that E 1 > 0 and E 3 > 0. Hence it implies the bound on for these two sectors It is worth emphasizing that the second topological sector presents energy E 2 = 2. Thus there is no restriction on the parameter and the energy is degenerate.

III. DCE AND THE DEFORMED 3-SINE-GORDON MODEL
Gleiser and Stamatopoulos proposed a precise picture of the DCE for discussing the structure of localized solutions in classical field theories [1,2]. The DCE is here addressed, using the deformed sine-Gordon field configurations, where braneworld scenarios can be studied. Firstly, the DCE setup will be briefly revisited. For it, the Napierian logarithm will be employed, with the DCE quantified in the natural unit of entropy (nat). The DCE regards the number of bits needed to encode localized physical configurations with the best compression. The protocol for computing the DCE consists of evaluating the Fourier transform of the energy density Distinct wavemodes are regulated by the modal fraction, whose expression is given by also circumventing the issue involving complex-valued Fourier transforms. The DCE is therefore given by [2,4,6], where m ρ (k) = m ρ (k)/m sup ρ (k), for m sup ρ (k) denoting the modal fraction supremum. The DCE estimates the dynamical degree of order associated with solutions of the equations of motion regu-lating the scalar field. The configurational stability of the system portrayed by the energy density corresponds to the minima of the DCE. Now one can follow the important contributions in Ref. [6] and additionally introduce the differential configurational complexity (DCC). For it, one must first specify the modal fraction, normalized by the maximum mode contribution, Given a specific power spectrum, the relative contribution of distinct wave modes can be quantified by the modal fraction (42). In fact, this concept arises when one realizes that if the power spectrum underlying wave modes is uniform, the inherent complexity is lower, whereas if the power spectrum is distributed non-uniformly among the wave modes, the complexity becomes higher. The normalization with the maximum mode yields the positivity of the DCC, which is therefore given by [6], As discussed in Ref. [6], the DCC (43) equals zero whenever the wave modes carry the same weight.
For example, uncorrelated noise presents a power spectrum of uniform type, yielding a maximal DCE, with DCC vanishing, whereas for a plane wave, meanwhile, both DCE and DCC vanish. It reinforced the influential interpretation posed in Ref. [6] of the DCC as a useful measure of shape complexity.

A. DCE and DCC of the deformed 3-sine-Gordon model
Here we aim to determine the DCE and the DCC, together with the subsequent information entropic features of the topological sectors provided by Eqs. (31,33,35), with respective energy densities (37a) -(37c) and their corresponding energies (38). It is worth emphasizing that the second sector φ 2 (x) presents degenerate energy with respect to the parameter . We can apply to each topological sector the DCE protocol, with corresponding Fourier transform of the energy density (39), for computing the modal fraction (40), and subsequently, the DCE (41).
The analytical results are explicitly expressed in Appendix A. Now, to completely specify the modal fraction m ρ j (k) (40), for each topological sector i = 1, 2, 3, the value R |ρ j (k)| 2 dk must be computed. Alternatively, one may use the Plancherel theorem, Hence, the denominator entering the modal fraction expression (40) reads, respectively for each topological sector, the following analytical functions of the deformation parameter :   There is a value of minimal complexity with respect to for the second topological sector (33) B. An additional approach to the 3-sine-Gordon model Now the DCE of another deformation of the 3-sine-Gordon model, without a power expansion approach, will be derived. Refs. [44,45] employed the deformation protocol to engender a new family of sine-Gordon-type models, including the 3-sine-Gordon. The general sine-Gordon-type models are regulated by two real parameters. The first one, a, controls both the location and the height of the scalar field maxima, whereas the second parameter, b, indexes the constituents in the family of sine-Gordon-type models, also managing the number of disconnected topological sectors [67]. Turning back to the results in Sec. II, taking into account the χ 4 model with spontaneous symmetry breaking, Eq. (10) yields [48] U (φ) = 1 2 Hence, as f (φ) is the functional deforming the model, 1/f (φ) also deforms the model in the same way.
This function can carry indexes a and b and is given by [44,45] where b ∈ (0, 1) or b ∈ (1, ∞), and a ∈ N, with potential The parameter a plays a prominent role in the generation of further families of models. For a = 1 the double sine-Gordon model is acquired, which contains two topological sectors and for a = 2 the triple sine-Gordon model is obtained, which contains three distinct topological sectors. The parameter b controls the position of the minima and the height of the maxima. The topological solutions are given by, for a given integer a [44,45] φ m k,a = arctan where k, m ∈ N, with giving rise to the defect and anti-defect solutions. For a = 2, namely, for the 3-Gordon model, the superpotential reads The energy densities of the three distinct topological sectors are provided respectively by [44,45] Using the protocol (39,40,41),   18 shows that the topological sector (53a) has the DCE with a peak value S DCE = 0.138 nat, at b = 0.0043. In the interval b ∈ (0, 0.0043), the graphic of the DCE satisfies d 2 S DCE /db 2 < 0 and dS DCE /db > 0, whereas in the range b 0.0043 the DCE is a function of b such that d 2 S DCE /db 2 > 0 and dS DCE /db < 0. Fig. 19 illustrates the topological sector (53b) with qualitatively analog profile of (53a). However, it has the DCE with a peak value S DCE = 0.147 nat, at b = 0.0022. Regarding this plot, the DCE regarding (53b) increases sharply and steeper than the case (53a). Besides, in the range b ∈ (0, 0.0022), the graphic of the DCE satisfies d 2 S DCE /db 2 < 0 and dS DCE /db > 0, whereas when b 0.0022 the DCE satisfies d 2 S DCE /db 2 > 0 and dS DCE /db < 0. In this case, the peak has a higher DCE, attained at a lower value of the parameter b, when compared to the topological sector (53a). Both cases (53a) and (53b) have the DCE asymptotically approaching zero, for b 0.4. We can conclude that for b ∈ (0.4, 1), the associated topological sectors with energy densities (53a) and This is therefore the most stable configuration of the physical system. Now, the DCC is plotted in Fig. 21. Finally, the DCC can be analyzed as a function of the energies of the kinks. Since the energies (38) of the first and the third topological sector are -(linearly)-dependent, the plots are depicted in Fig. 23. The plot of the DCC as a function of the energy E 2 of the second topological sector is omitted, since E 2 (38) is constant.  The protocol of deforming topological defects was here engendered to study two approaches of the 3-sine-Gordon model whose underlying DCE was computed and discussed. Contrary to the approach in Sec. II, where the deformation function could be first-order expanded with respect to the deformation parameter as in Eq. (13), the analytical deformation function (48) in Sec. III does not involve a power expansion. The results involving the underlying DCE and the DCC analysis, for the two deformed models, are complementary. The deformations of the 3-sine-Gordon model can be employed for practical applications in a diversity of scenarios, in particular in the braneworld scenario with one extra dimension, being the brane stabilized by a real scalar field with the best choice of parameters driven by the DCE and the DCC, as detailedly reported in the final discussion of Subsecs. III A and III B, respectively for each different deformation procedure generating the 3-sine-Gordon model. Ref. [55] addressed deformed 2-sine-Gordon models, with applications in gravity localization on thick braneworlds. Here, the results can be used to study gravity localization with deformed 3-sine-Gordon field configurations. The results heretofore derived in this work, using the DCE, can hence drive the best choice of the range of parameters for studying thick branes and field localization on them. As the approach of using deformations has been successfully applied to the study of braneworld scenarios, especially in what concerns the localization of bosonic and fermionic fields onto the brane, one can investigate the relevant results in Refs. [70][71][72][73][74][75], including asymmetric braneworlds in Refs. [76], using the DCE. Finally, higher-dimensional deformation techniques [77,78]