Cyclically deformed defects and topological mass constraints

A systematic procedure for obtaining defect structures through cyclic deformation chains is introduced and explored in detail. The procedure outlines a set of rules for analytically constructing constraint equations that involve the finite localized energy of cyclically generated defects. The idea of obtaining cyclically deformed defects concerns the possibility of regenerating a primitive (departing) defect structure through successive, unidirectional, and eventually irreversible, deformation processes. Our technique is applied on kink-like and lump-like solutions in models described by a single real scalar field, such that extensions to quantum mechanics follow the usual theory of deformed defects. The preliminary results show that the cyclic device supports simultaneously kink-like and lump-like defects into 3- and 4-cyclic deformation chains with topological mass values closed by trigonometric and hyperbolic deformations. In a straightforward generalization, results concerning with the analytical calculation of $N$-cyclic deformations are obtained and lessons regarding extensions from more elaborated primitive defects are depicted.


I. INTRODUCTION
Solutions of a restricted class of nonlinear equations yield an ample variety of topological (kink-like) and non-topological (lump-like) defects of current interest, in particular to mathematical and physical applications as well [1]. More precisely, these defect structures are finite-energy solutions of a nonlinear partial differential equation, which tend to be stabilized by a conserved charge associated with an underlying field theory [2]. To get a more clarifying and effective definition, the equation which leads to the corresponding classical solution needs two main terms: a sharpening nonlinear term and a dispersive term. Essentially, the energy density of the localized solution captures the nonlinear nature of the corresponding Lagrangian, making its dynamics an insightful topic. In this sense, topological and non-topological structures are not only of mathematical interest.
Defect structures are indeed investigated in cosmological scenarios involving seeds for structure formation [3][4][5][6], Q-balls [7,8], and tachyon branes [9][10][11]; in braneworld scenarios with a single extra-dimension of infinity extent [12][13][14][15]; in particle physics as magnetic monopoles [16]; and in some applications on soft condensed matter physics [17] involving, for instance, charge transport in diatomic chains [18][19][20], or even baby skyrmions [21]. Specific studies involving kink-like defects are also used to be related to the symmetry restoration in inflationary scenarios [22,23], or to the fermion effective mass generation induced by some symmetry breaking mechanism [24,25]. Likewise, applications of current interest predominantly include the use of lump-like models to describe the behavior of bright solitons in optical fibers [26,27]. In particular, it represents the most prominent candidate for ultrahigh-speed packet-switched optical networks in the current communication revolution.
Notwithstanding the pop science status of this field -once its proponents seek to put it to use for transporting vast amounts of information -the analysis regarding defect structure solutions may still require the introduction of challenging mathematical devices amalgamating analytical, geometrical, and sometimes complex numerical protocols. Therefore, finding a systematic process for obtaining cyclically deformed defects, constrained by closure relations involving their topological masses, corresponds to the well succeeded purpose of this work. Hereon the nomenclature of topological mass is arbitrarily extended to encompass the finite total energy of non-topological lump-like solutions.
Our work is concerned with a systematic procedure for defect generation defined through trigonometric and hyperbolic bijective deformation functions. They allow the construction of N -finite cyclic deformation chains involving topological and non-topological solutions.
The obtained defects are necessarily generated and regenerated from a primitive defect of an original theory (for instance, the λφ 4 theory).
In this context, several simplificative techniques have been suggested to study and solve non-linear equations through deformation procedures [12,13,28]. The main issues that have stimulated us to investigate cyclically deformed defects have indeed appeared through the BPS first order framework [29][30][31] for obtaining kink-like and lump-like structures.
Such an outstanding process of systematically obtaining cyclically deformed defects allows reedifying a preliminarily introduced defect structure -for instance, the usual kink defect from the λφ 4 theory -through successive, unidirectional, and eventually irreversible, deformation operations. This possibility of regenerating a primitive defect is one of our principal concerns in this work. Our second main contribution consists in systematically defining constraint relations, involving topological masses of the corresponding deformed defects which belong to the cyclically deformed chain. Our technique also supports kink-like and lump-like solutions in models described by a single real scalar field, such that extensions to quantum mechanics follow the usual theory of deformed defects [29].
The manuscript is therefore organized as follows. As a preliminary illustration of the method, in section II we report about some known topological and lump-like solutions to identify the possibilities of finding some 3-cyclic deformation chain, which speculatively engenders some constraints upon the topological masses related to the cyclically deformed defects. Along this section we shall follow the BPS first order framework. Our results for a second set of 3-cyclic deformation allow one to identify a plateau-shaped form for the potential of a deformed kink-solution which, for instance, introduces the possibility of a kind of slow-rolling behavior for the scalar field deformed solution, ϕ, when its dynamics is turned on. A systematic and generalized procedure that simultaneously supports kink-like and lump-like defects into 3-cyclic deformation chains is introduced and scrutinized in section III. The topological conserved quantities are analytically computed and the constraint relations, involving trigonometric and hyperbolic deformation functions, are straightforwardly obtained. The results from section III are extended to a 4-cyclic deformation device which is furthermore driven by hyperbolic and trigonometric deformation chains. Our results shall ratify that the modified topological masses (or the total energy of localized solutions, in the case of lumps), computed for defect structures cyclically deformed from a triggering λχ 4 theory, are mutually constrained. Finally, the hyperbolic and trigonometric deformation procedures are systematically generalized to N -finite cyclically deformed defects in section V. We draw our conclusions in section VI by attempting to some important insights related to the study of defect structures which bring up the novel concept of N -cyclic deformations.

II. CYCLICALLY DEFORMED DEFECTS -NON SYSTEMATIC PROCEDURE
Assuming that the presence of first-order equations evidently simplifies the calculations, along this manuscript we shall report about the BPS first order framework [29][30][31] for investigating topological (kink-like) and non-topological (lump-like) structures.
Let us firstly consider a set of three cyclically deformed defects, namely ψ(s), φ(s), and ϕ(s), with s as the spatial coordinate, in a way that the corresponding 3-cyclic deformation procedure can be prescribed by the following first-order coupled equations, where the subindexes stand for the corresponding derivatives, with α β = 1/β α for α, β = φ, ϕ, ψ, being identified with deformation bijective functions that generate novel families of deformed defects. In this case, x ϕ , y φ , and z ψ are derivatives of the auxiliary superpotentials, and the corresponding BPS form of the cyclically derived potentials shall be given by such that the simplified framework of deformed defects [29] is straightforwardly recovered.
Such a sequence of 3-cyclic deformations naturally obeys the chain rule constraint given by Due to simplificative reasons, the starting point of our discussion will be the dimensionless λφ 4 theory with a scalar potential given by from which, upon solving the equation of motion, one obtains the static solution described by a topological kink (+ sign) (or antikink (− signal)) as for which we shall follow the sign convention adopted in [30], nevertheless reducing our analysis to + sign solutions into the above equation.
As it is well-known, the potential V (φ) engenders one maximal point at φ M ax = 0, and two critical points, φ 0 ± , which are also function zeros, i. e.
that correspond to the asymptotic values of the kink solution, namely its topological indexes. The topological charge is correspondingly given by where any dimensional multiplicative factor was suppressed from the beginning, and the topological mass, which corresponds to the total energy of the localized solution, is given by where the corresponding localized energy density has been identified with At first glance, besides Eq. (5), our description of cyclically deformed defects could follow no additional systematic constraints on the choice of ψ(s), φ(s), and ϕ(s). Just as a preliminary illustration of the method, let us firstly obtain two sets of cyclically deformed defects from the primitive kink solution above.

A. First case
Upon introducing a trigonometric deformation given by and assuming the (+ sign) result from Eq. (6), one obtains with from which one can easily depict the localized energy density as This solution corresponds to a simplified version of the sine-Gordon potential, for which the asymptotic behavior and the conserved topological quantities are easily obtained.
The potential U (ϕ) engenders a set of critical points, ϕ 0 (s) , such that U (ϕ 0 (s) ) = 0 and By identifying ϕ(φ) with one notices that ϕ 0 (s) corresponds to the asymptotic values of the kink solution, i. e.
The maximal points of U (ϕ) are correspondingly provided by and the topological charge is computed through The total energy of the localized solution is given by Taking some advantage of knowing the analytical behavior of λφ 4 solutions, the 3-cyclic deformation chain is accomplished by an ansatz function described by that leads to such that with from which the localized energy density is obtained as This solution corresponds to the first family of models involving non-topological (Q ψ = 0) bell-shaped solutions obtained in [31]. Bell-shaped lump-like defects are non-topological structures generated by nonlinear interactions present at real scalar field models in (1,1) spacetime dimensions [31].
The potential W (ψ) engenders one minimal point, ψ 0 = 0, and the boundary points i. e. it has two zeros for n odd (or n < 1) and three zeros for n even. Since we have a lump-like solution, the minimal point ψ 0 = 0 is analytically connected to the asymptotic values of the λφ 4 kink solution previously obtained. Through the same way, W (ψ) has its maximal points corresponding to Finally, the total energy of the localized solution is given by from which one has, for instance, For the sake of completeness, it is also interesting to notice that the analytical chain described by Eq. (6) leads to from which one obtains One could also notice that, in case of n = 2, a constraint relation involving the values of the topological masses can be established as with arbitrary κ, which introduces a pertinent issue about the possibility of finding some systematic way for constraining topological masses of cyclically deformed defects.
The first column of Fig. 1 summarizes the properties of the above discussed solutions, by introducing an illustrative view of their analytical properties, which will be extended in the following subsection.

B. Second case
Let us now investigate a novel 3-cyclic chain which, as the previous one, is based on a similar ansatz involving some analytical dependence of ψ on y φ of the dimensionless λφ 4 theory. At this point, however, we shall abbreviate some discussions which escape from the scope of this preliminary section.
Upon choosing an ansatz for a deformation function given by one obtains that can be parameterized by where again we are assuming the (+ sign) result from Eq. (6).
The 3-cyclic deformation chain from Eq. (3) can thus be implemented through the following definitions for φ ϕ and ϕ ψ , that leads to and that leads to The BPS states can be obtained through and , so that the respectively associated potentials shall follow the definitions from Eq. (2). Notice that the explicit dependence of x ϕ on ϕ is given in terms The topological charge for the kink-like solution ϕ(x) is given by since it is obtained from a potential U (ϕ) that engenders two critical points, ϕ 0 ± , which also represent function zeros and correspond to the asymptotic values of the primitive λφ 4 kink solution. The maximal point of U (ϕ) is correspondingly given by and is related to the center of an analytical plateau symmetrically shaped around the origin, as one can depict from the second column of Fig. (1).
Otherwise, the non-topological lump-like solution ψ (n) (x) is obtained from a potential W (ψ) that engenders one minimal point, ψ 0 = 0, and one boundary point ψ (n)0 = 1 n ln (2) which can be duplicated to its symmetrical partner, ψ (n)0 = − 1 n ln (2), if 1/n is an even integer for n < 1. Therefore, it can have two or three zeros, depending on the values assumed by n. Again, since we have bell-shaped solutions, the minimal points at ψ 0 = 0 correspond to the asymptotic values of the λφ 4 kink solution previously obtained. The potential W (ψ) also has maximal points corresponding to ψ (n) Finally, the total energy of the localized solution is given by for the kink-like solution, and by for the lump-like solution, from which the equation constraining the values of topological masses can be established as families of non-systematically obtained cyclic deformations that we have discussed above.
For the first family, in the context of 3-cyclic deformations, we have simply recovered the results for sine-Gordon and bell-shaped defects scrutinized by Bazeia et al. [29][30][31]. It satisfactorily works as preliminary analysis. The second family brings up novel results. One can depict, for instance, the plateau-shaped form of the potential for the modified kink solution (blue lines), which denotes the possibility of a kind of slow-rolling behavior for the scalar field ϕ when its dynamical properties are considered. The plots show the results for the primitive defects (thick red lines), φ(s), y φ (s) = dφ/ds, and V (φ); for the kink-like deformed defects (thick blue lines), ϕ(s), x ϕ (s) = dϕ/ds, and U (ϕ); and for the lump-like deformed defects (black lines), ψ (n) (s), z ψ (s) = dψ (n) /ds, and W (ψ). In particular, for solutions related to ψ (n) (s) we have set n = 1/2, 1, 2, 3 and 4 in order to verify all the properties related to their critical points and function zeros discussed above.
In Fig. 2 we have merely illustrated the explicit analytical dependencies of the topological masses, M ψ (n) on n, and compared one each other by including a third case investigated at [31], for which the topological mass is given by It is obtained from the BPS framework for which with Just to summarize this introductory section, an interesting property related to the effective applicability of cyclic deformations can be depicted from Eq. (38). Although one can obtain the explicit analytical dependence of φ on ϕ, the corresponding analytical expression for the inverse function ϕ(φ) = [φ] −1 (ϕ) cannot be obtained. The problem is circumvented by the cyclic chain that relates φ ϕ ψ φ, and therefore allows one to obtain the explicit dependence of x ϕ on s and of U (ϕ) on ϕ.
Finally, for all the above obtained solutions, the quantum mechanics correspondence can be directly obtained. Let us begin by replacing the scalar field ϕ introduced in the previous analysis by a renamed scalar field, χ, which, from this point, will designate the primitive kink defect that triggers the 3-cyclic deformation chain. All the equalities involving ϕ and x ϕ into Eqs. (1) are therefore replaced by such that and the cyclically derived BPS potentials belonging to the 3-cyclic deformation chain are now given by In this context, the novel deformation functions shall be constrained by the chain rule given by ψ φ φ χ χ ψ = 1.
Let us consider the set of auxiliary derivatives described by which upon straightforward integrations lead to with constants chosen to fit ordinary values for the asymptotic limits. After simple mathematical manipulations involving the hyperbolic fundamental relation, with relations from Eq. (53), one easily identifies the following closure relation, which constrains the localized energy distributions for 3-cyclic deformed defect structures.
For the sake of completeness, one should have and, in case of assuming, the BPS deformed functions would concomitantly obtained as with the respective dependencies on φ (n) (s) and ψ (n) (s) implicitly given by Eq. (60).
The non-topological lump-like solution ψ (n) (x) is obtained from a potential W (ψ) that engenders one minimal point, ψ (n)0 = 0, and one boundary point ψ (n)0 = ψ (n) (s → ±∞) = 1 n ln [cosh (n)] which can be duplicated to its symmetrical partner, ψ (n)0 → −ψ (n)0 , if 1/n is an even integer for n < 1. In this case, the total energy of the localized solution is given by Let us now turn to a set of trigonometric auxiliary functions described by which upon straightforward integrations lead to with constants chosen to fit ordinary values for the asymptotic limits. After simple mathematical manipulations involving the ordinary trigonometric fundamental relation, sin (n χ) 2 + cos (n χ) 2 = 1, followed by relations from Eq. (53), one recurrently identifies the closure relation, Again, one should have and, in case of assuming that χ(s) is given by Eq. (61), the BPS deformed functions are given by with the respective dependencies on φ (n) (s) and ψ (n) (s) obtained from Eq. (72).
The second and third columns of Fig. 3 show the analytical dependence on s for the 3cyclically deformed defects obtained through the trigonometric deformation functions from Eq.(69). As in the hyperbolic case, the primitive defect from Eq. (61) leads to kink-like defects for φ (n) and lump-like defects for ψ (n) . We have set n = kπ/8 with k in the range such that the topological charge and the topological mass are respectively given by and The non-topological lump-like solution ψ (n) (x) is obtained from a potential W (ψ) that engenders one minimal point, ψ 0 = 0, and two function zeros given by In this case, the corresponding total energy of the localized solution is given by In correspondence with Fig. 3, the plots of the BPS deformed potentials for hyperbolic Let us hence recover the scalar field ϕ which shall be reintroduced into a novel 4-cyclic deformation chain. The additional scalar field, χ, continues to designate the primitive defect that triggers the 4-cyclic deformations. The coupled system described by Eqs. (1) shall thus be complemented by such that and the cyclically derived BPS potentials belonging to the 4-cyclic deformation chain will be given by Hyperbolic and trigonometric deformation functions shall naturally follow the chain rule given by Let us consider the set of auxiliary functions described by which, for a given w χ substituted into Eq.(81), completes the 4-cyclic chain. Upon straightforward integrations the above deformation functions lead to with constants chosen to fit ordinary asymptotic values. By following the same simple mathematical manipulations involving the fundamental relation from Eq.(58), one easily identifies that These solutions and all the others depicted from the last three columns of Fig. 6 correspond to some kind of non-topological lump-like defect. The plots show the results for the primitive defects, χ(s), w χ (s) = dχ/ds, and ρ(χ(s)); and for the corresponding deformed defects, φ (n) (s), y φ (s) = dφ (n) /ds, and ρ(φ (n) (s)); ϕ (n) (s), x ϕ (s) = dϕ (n) /ds, and ρ(ϕ (n) (s)); and ψ (n) (s), z ψ (s) = dψ (n) /ds, and ρ(ψ (n) (s)). We have set n = kπ/6 with k in the range between 1 and 2, by steps of 0.2 units, in order to depict the analytical dependence on the free parameter n.
In case of assuming that the primitive λχ 4 kink solution from Eq.(87) triggers the 4cyclically deformed chains discussed above, the deformed defect structures have their explicit analytical dependence on s obtained by substituting χ 1 (s) into the results from Eqs. (85).
Through this explicit dependence on s, topological charges (in case of kink-like solutions) and topological masses can be straightforwardly obtained.
The topological charge for the novel kink-like deformed defect obtained from the hyperbolically deformed structure described by Eqs. (91-94) is therefore given by and the topological masses analytically obtained as function of the free parameter n are given by

B. Trigonometric Deformation
Let us now consider the set of auxiliary functions described by which, for a given w χ substituted into Eq.(81), completes the 4-cyclic chain, and upon straightforward integrations leads to with constants chosen to follow the same criteria as for the previous solutions. From Eq.(70), one straightforwardly identifies the closure relation w 2 χ = w 2 χ sin (n χ) 2 + cos (n χ) 2 sin (n χ) 2 + cos (n χ) 2 which constrains the values of the topological masses obtained from trigonometrically deformed defects.  Fig. 6 for hyperbolic deformations. The unique kink-like defect, in this case, is given by with x ϕ (s) = sech (s) 2 sin (n tanh (s)) 2 .
In order to identify and emphasize the oscillatory behavior of such novel deformed defects, we have set n = kπ/6 with k in the range between 12 and 13, by steps of 0.2 units, through which the analytical dependence on the free parameter n can be depicted. One can infer that, for small values of n, namely n < 2, the oscillatory pattern of trigonometric deformation disappears, and hyperbolic and trigonometric deformations are close to each other.
To sum up, once the primitive λχ 4 kink solution from Eq.(87) triggers the 4-cyclically deformed chain discussed above, topological charges (in case of kink-like solutions) topological masses can be straightforwardly obtained through this explicit dependence on s. The topological charge for the novel kink-like deformed defects obtained from the trigonometrically deformed structure described by Eqs. (100-101) is given by and the topological masses are given by The topological charge for the novel kink-like deformed defects related to ϕ (n) and obtained from hyperbolic and trigonometric deformations in dependence on the free parameter n can be depicted from Fig. 9. It also includes the results for the topological charge of the kink-like solution related to φ (n) obtained from the 3-cyclically deformed chain discussed in the previous section.
in both cases which, in the asymptotic limit, n → ∞, leads to for hyperbolic deformations, and to 2 3 for trigonometric deformations.

V. N-CYCLIC DEFORMATIONS
Departing from a triggering defect given by χ (eventually related to the λχ 4 kink solution), the observation of the chain rule constrained by hyperbolic and trigonometric fundamental relations allows one to investigate the possibility of obtaining N -cyclic deformations.
Let us finally introduce a generalized s-deformation operation parameterized by the sderivative given by where φ [s] , with s = 1, 2, . . . , N , are real scalar fields describing defect structures generated from a N -cyclically deformation chain triggered by a primitive defect χ ≡ χ(s), such that The hyperbolic deformation chain shall be given in terms of which upon straightforward integrations results into where 2 F 1 is the Gauss' hypergeometric function, and we have not considered any adjust due to integration constants. From Eqs.(108) one easily identifies that such that from which a very simplified form for the chain rule of the N -cyclic deformation can be obtained as Upon introducing the generalized BPS functions, with w χ = dχ/ds, it is straightforwardly verified that which results into the constraint equation, Consequently the topological masses are constrained by The trigonometric deformation chain can be exactly obtained by substituting tanh (χ) by − sin (χ) and sech (χ) by cos (χ) into Eqs.(109). It shall result into and into the same set of constraints described by Eqs. (116-117).
Essentially, the N -finite set of bijective hyperbolic/trigonometric functions with nonvanishing asymptotically bounded derivatives provides us with the necessary and sufficient tools for defining the N -cyclic deformation chain that systematically results into constraints involving the topological mass of deformed structures.

VI. CONCLUSIONS
We have introduced and scrutinized a systematic procedure for obtaining defect structures through cyclic deformation chains involving topological and non-topological solutions in models described by a single real scalar field. After embedding some previously investigated deformed defect structures, namely the sine-Gordon kink, and the bell-shaped lump-like structure, into a 3-cyclic deformation chain supported by the primitive kink solution of the λφ 4 theory, we have investigated the existence of a systematic technique to obtain novel defect structures. As a preliminary study, we have discussed the kink-like solution engendered by a plateau-shaped potential that is generated from a logarithmic dependent deformation of the λφ 4 theory through an arbitrary (non-systematic) cyclic deformation chain. It subtly mimics the slow-rolling scalar field potentials at 1 + 1 dimensions.
In a subsequent step, we have verified that a set of rules for constructing constraint relations involving topological masses of cyclically generated defects could be systematically The perspective of applying the regeneration mechanism to more complex primitive triggering defects, for instance, the sine-Gordon kink-like solution, connecting it to novel defect structures, may be considered in subsequent analysis.
We have further emphasized that several components of the cyclic chains that we have studied correspond to lump-like solutions. Since they are non-monotonic functions, they induce some subtle features to the non-topological structures. As pointed out by some preliminary issues [29][30][31], for non-topological structures, namely the lump-like ones, the quantum mechanical problem engenders zero-mode functions with one or more nodes. It leads to the possibility of negative energy bound states, providing us unstable solutions, and creating additional difficulties for determining such bound states through the related quantum mechanics.
Whatever the outcome of these investigations, there is no doubt that the study of defect structures through a systematic process of producing N -cyclic deformations leads, at least, to the speculation of novel scenarios of physical and mathematical applicability of defect structures, for instance, at some kind of cyclically connected optical system, or some cyclically regenerative sequence of phase transitions.      FIG. 3: The 3-cyclically deformed defects obtained from hyperbolic (first column) and trigonometric (second and third columns) deformation functions. Results are for the primitive λχ 4 kink solution, χ(s) (thick black line), for the kink-like deformed defects, ψ (n) (s) (black lines), and for the lump-like deformed defects, φ (n) (s) (red lines). The free parameter n was set equal to kπ/8 with k assuming the following integer values in the first/second (third) columns: 1(11) for solid lines, 2(12) for long-dashed lines, 3(13) for short-dashed lines, and 4(14) for dotted lines.   , as function of the parameter n for hyperbolic (thick lines) and trigonometric (dashed lines) deformation chains. Results are for the kink-like solutions obtained from 4-cyclic (blue lines) and 3-cyclic (red lines) deformation chains respectively related to ϕ (n) and φ (n) .