Solitary oscillations and multiple antikink-kink pairs in the double sine-Gordon model

We study kink-antikink collisions in a particular case of the double sine-Gordon model depending on only one parameter $r$. The scattering process of large kink-antikink shows the changing of the topological sector. For some parameter intervals we observed two connected effects: the production of up to five antikink-kink pairs and up to three solitary oscillations. The scattering process for small kink-antikink has several possibilities: the changing of the topological sector, one-bounce collision, two-bounce collision, or formation of a bion state. In particular, we observed for small values of $r$ and velocities, the formation of false two-bounce windows and the suppression of true two-bounce windows, despite the presence of an internal shape mode.


INTRODUCTION
Domain walls in (3,1) dimensions and kinks/antikinks in (1,1) dimensions are solutions of nonlinear field theories. Due to their stability and topological structure, these spatially localized configurations can propagate freely without losing their shape [1,2]. Domain walls and kinks have applications in several areas of science including early universe cosmology [3][4][5], pulsar glitches [6], ferroelectrics [7], optical fibers [8] and DNA [9]. In integrable models like the sine-Gordon, the structure of kink-antikink scattering is simple, presenting a totally elastic behavior with at most a phase shift. On the other hand, nonintegrable models show a richer pattern of kink-antikink scattering. The investigation of new patterns of kink scattering is interesting to understand some aspects of nonlinearity connected with physical systems.
Among nonintegrable models, the φ 4 model is the simplest and by far more studied in the literature (see [10] and references therein). It is known that the kink-antikink (KK) collision depends on the initial velocity v. There is a critical velocity v c such that, for v > v c there is an inelastic scattering between the kink and the antikink. For v < v c the kink-antikink remain connected and irradiate continuously. For some values of v, we can observe the annihilation of the pair and the fractal resonance structure. The structure formed, known as escape windows or two-bounce windows, is related to the resonant energy exchange mechanism between the translational and vibrational modes [11]. See Ref. [12] for a review of early works on this complex structure.
More recently, the numerical investigation of other models has been revealed some unexpected aspects of nonlinearity. In particular, in the φ 6 model [13], despite the absence of vibrational mode for one kink, the structure resonant scattering appears. There, the authors explained the results considering the perturbation of the whole kink-antikink pair. In the model of Ref. [14], two-bounce windows are suppressed despite the presence of more than one internal mode. This effect was interpreted as a kind of destructive interference between the several modes. In the Refs. [15,16] we can see once again the importance of vibrational mode in the appearance of two-bounce windows.
The double sine-Gordon (DSG) model was explored long ago in the Ref. [43] to describe the spin dynamics in the superfluid 3 He. There, the longitudinal nuclear magnetic resonance corresponds to the oscillation of an angle φ in a potential well. In the B phase, the longitudinal solitons in the Leggett configuration (where the symmetry breaking axis is parallel to the external field) results in a modified sine-Gordon equation for φ with two types of solutions: small soliton and large soliton. Some results on soliton-antisoliton scattering for both small soliton and large soliton pairs were presented in the Refs. [44,45]. For a more gentle introduction on the subject, see the Ref. [46]. The DSG admits quantization and applications in statistical field theory [47], being closely related to the Ashkin-Teller model that describes two planar Ising models interacting through a local four-spin interaction. Moreover, it is a nice example of a quantum phase transition. This rich phenomenology is addressed considering the kink configurations of the model and their bound states [47]. A recent paper derived the dynamic equation of molecular motion for twisted nematic liquid crystal under applied electric an magnetic fields, showing that it takes the form of a DSG model [48]. The recent observation of a fractional vortex in a superconducting bi-layer [49] indicates that kinks can be formed in such systems. For fractional vortices in superconductors see the Chap. 6 of the Ref. [50]. In the Ref. [51], kinks in a two-band superconductor are described by the DSG Model. Pulsar glitches are sudden changes in the rotation frequency of neutron stars [52]. This is an evidence of the existence of superconductor states in the core of neutron stars.
Collective excitations along a vortex line in neutron 3 P 2 superfluids in neutron stars results, for low energies, in a kink of the DSG model [53]. The generalized sine-Gordon model has been found in the study of the strong/weak coupling sectors of the sl(N, C) affine Toda model coupled to matter fields [54]. The DSG model appears in the reduction to one field of the sl(3, C) generalized sine-Gordon model. In the Ref. [55] the DSG model was proposed as an extended hadron model. That work was extended in the Ref [54] to multiflavor Dirac fields, such that the DSG kink solution describes a multi-baryon. Then, the DSG spectrum and kink-antikink system can be useful for the description of multiflavor spectrum and some resonances in QCD 2 [54]. The DSG model also appears in the effective action describing exotic baryons in QCD 2 [56,57].
The last paragraph illustrated several important applications where the results of kink scattering in the DSG model can be useful. Aspects of kinks in the DSG model such as small oscillations, internal modes, radiation and analytical methods were studied in the Refs. [58][59][60][61][62][63][64][65]. Kink scattering in the DSG model was investigated in the Refs. [66][67][68][69][70][71][72][73][74][75][76][77]. Here we consider a particular case of the DSG model, looking for the structure of the kink-antikink scattering. We found some aspects not reported before, such as the production of up to three solitary oscillations and multiple antikink-kink pairs. We show that these effects are interrelated.
In the Sect. II we discuss the model and two types of kinks: large kinks and small kinks.
For large kinks it is shown the production of multiple extra antikink-kink pairs together with the changing of the topological sector. We report the formation of several oscillations resembling oscillons, but with much shorter lifetimes. For small kinks we investigate the appearance of the structure of two-bounce windows. We present our main conclusion in the Sect. III.

II. THE MODEL
We consider the action with standard dynamics in (1, 1)-dimensions in a Minkowski space- The double sine-Gordon is defined by the potential In this work we consider the potential [65] V (φ) = 2 where 0 < r < 1 is a parameter. This potential is a particular case of the potential (2), with [65] α = (1 − r)/(1 + r), ν = 1/2 and λ = √ 1 − r, and is depicted in Fig. 1 for some values of r. This potential is periodic, with period 2π. For −2π < φ < 2π and for the parameter 0 < r < 1 the potential contains four minima. Note from the figure that the minima are separated by large and small barriers. Corresponding topological solutions are named large and small kinks. For r → 0 the potential goes to the integrable sine-Gordon. The growth of r reduces the height for the small kink until the critical value r → 1, where the potential goes to zero at φ = ±π. In this limit the potential is very close to zero for a finite interval around φ = ±π, as can be seen in the Fig. 1. This agrees with The main motivation of the potential given by the Eq.
(3) was the dependence of only one parameter: if r is small one can explore the vicinity of the integrable sine-Gordon model [65]. Here we will consider this potential in the range 0 < r < 1, exploring the main characteristics of scattering of large kinks and small kinks.

The equation of motion is given by
and static kinks φ S (x) are solutions that connect two sectors of the potential and minimize the energy. Perturbing linearly the scalar field around one static kink solution φ S (x) as where V sch (x) = V φφ (φ S (x)). The study of the Schrödinger-like potential is useful for under- standing some aspects of the scattering structure.
The large kink solution is given by [65] φ l K (x) = 2 arctan and antikink solutions are given by φ lK (x) = φ l K (−x). The Fig. 2a shows some plots of large kink profile for several values of r. The solutions connect the minima ±φ v , with φ v = −π + arccos(r). We note that for large solutions the growth of r contributes to the increasing the asymptotic value of kink. The Fig. 2b depicts plots of Schrödinger-like potential V sch (x) for large kink for several values of r. For r → 0 we have a central minimum around x = 0 and asymptotes to V sch = 4 for x → ±∞, corresponding to the sine-Gordon model. This model has a zero-mode followed by a continuum starting at the mass threshold ω 2 = 4. The increasing of r reduces the asymptotic value of the potential. In particular, for r → 1, we have V sch = 0 for x → ±∞. This agrees with V φφ (φ = ±π) = 0 for r → 1, as noted previously. We can see the process of formation of the volcano-shape potential with the growth of parameter r. For 0 < r < 1 there is only a bound state, the zero mode, followed by a continuum of states at ω 2 corresponding to the asymptotic value of V sch . In addition, the volcano-like potential leads to the possibility of resonances. However, since resonances do not completely store energy during the collision process, we have as a consequence the disappearance of some two-bounces windows [15].
The small kink solution is given by [65] and antikink solutions are given by φ sK (x) = φ s K (−x). The Fig. 3a shows some plots of small kink profile for several values of r. These solutions connect the minima φ v = π ± arccos(r).
We note that the increase in the parameter r reduces the value of φ v . The Fig. 3b depicts the Schrödinger-like potential V sch (x) for small kink for some values of r. We have the same structure of V sch (x) for the antikink φ sK (x). We notice that the increase of r reduces the depth of the minimum and decrease the asymptotic maximum of the potential. The occurrence of bound states for small kink was investigated numerically for several values of r. There is always a zero mode for all values of r. The structure of vibrational mode is summarized in the Fig. 4. For r 0.02 we have the presence of internal mode, whose squared frequency ω 2 decreases with r. For the same values of r, large kinks have larger energies than small kinks. Large kink connecting minima −π + ξ(r) and π − ξ(r), with energy [65] and small kink connecting minima π − ξ(r) and π + ξ(r) with energy [65] The Fig. 5 shows the plots from E L , E S and the difference E L − E S as a function of r. Note from the Eqs. (8) and (9) and from the figure that the difference ∆ E = E L − E S has a monotonic increase with r.
For the kink-antikink scattering process we solved the equation of motion with a 4 th order finite-difference method with a spatial step δx = 0.05. We fixed x = ±x 0 = 12 for the initial symmetric position of the pair. For the time dependence we used a 6 th order symplectic integrator method, with a time step δt = 0.02. We used the following initial conditions where φ K (x, t) = φ S (γ(x − vt)) means a boost solution for the particular static kink solution φ S (x) (which can be φ l K (x) or φ s K (x)), where γ = (1 − v 2 ) −1/2 and φ v is one vacuum of the theory, i.e., a minimum of V (φ). larger values for r favors the occurrence of a larger number of pairs.
In the Fig. 9b we can see the number of solitary oscillatory waves due to the collision process. Note that small values of r are not related to the observation of these oscillations.
Increasing r there is the possibility of occurrence of oscillations, but only in some intervals of r that follows closely the transition regions described in the Fig. 9a for the number of antikink-kink pairs. For example, the Fig. 9b shows that, for 0.4 r 0.5 we can see the formation of one oscillation centered at x = 0, with regions with two oscillations for r ∼ 0.5.
This value of r is roughly in the transition region between the formation of one and two antikink-kink pairs (compare with the Fig. 9a). The same applies for 0.62 r 0.65, with two oscillations for some velocities at r ∼ 0.65. This value of r is roughly in the transition region between the formation of three and four antikink-kink pairs (compare with the Fig.   9a). That is, it seems to have a connection between oscillation formation and the increasing in the number of formed antikink-kink pairs.
Comparing the Figs. 6 and 8, we see that when we have only one produced antikink-kink pair (Fig. 6), it does not radiate, whereas when we have three or four formed antikinkkink pairs (Fig. 8), only the more external pair does not radiate. In the former case, this happens for a thick pair, whereas in the latter, for a thin pair. That is, the thickness of the antikink-kink pair is not a determinant aspect for the absence of emitted radiation. However, since thin antikink-kink pairs that do not radiate have some resemblance with propagating particles, their observation is of more interest. Due to this we investigated some features regarding the width and the field radiation at the midpoint of the thinnest and more external formed antikink-kink pair. The Fig. 10a shows that the average value of the width of the more external antikink-kink pair decreases with the increment of r. Compare for instance the Figs. 8a and 8c, corresponding respectively to r = 0.6 and r = 0.74.
The Figs. 11a-c depicts the Fourier transform of the thinnest antikink-kink pair for a specific initial velocity. The figures show that the frequencies are below the continuum, meaning that the pair is not able to radiate. For low values of r we have two frequencies (Fig. 11a), and we noticed a decline in these frequencies with the growth of r. A third frequency appears for r 0.623 (Fig. 11b). This region coincides with an increase in the number of formed kink-antikink pairs. This behavior is roughly independent of v. We observed that changes in the initial velocity modify the amplitude of the field, not the frequencies. The general behavior of the frequencies with r is depicted in the Fig. 12. Note that the frequencies decrease with r, and are always below the mass threshold. Moreover, the frequencies are usually below that of the vibrational mode (dotted line of the Fig. 12), except for specific regions r ∼ 0.5 and r ∼ 0.64, close respectively, to the transition region of formation of one oscillation and two oscillations.
We also analyzed the solitary oscillations. The Fig. 13a shows the Fourier transform of one of the two oscillations shown in the Fig. 7. We note from the figure that the main frequency is in the discrete spectrum. Also, it appears one small peak in the continuum.
The amplitude of the oscillations decays linearly, as shown for instance in the Fig. 13b.
This means a larger rate than the radiation of an oscillating φ 4 kink, which follows the Manton-Merabet pattern [79].

B. Small kink scattering
The small kink-antikink scattering for small values of r and v > v * can produce large antikink-kink, as depicted in the Fig. 14a. Note from the figure that the velocity of the produced large antikink-kink pair is lower than that of the incident small kink-antikink pair.
This agrees with the large energy of the large kink in comparison with the small kink (see the Fig. 5). Inside the formed large antikink-kink pair one can see emitted radiation. The sector. There are several possibilities: i) for v c < v < v * the collision is almost elastic and produces small radiation inside the produced small kink-antikink pair. This can be seen in the Fig. 16a. In particular, the Fig. 16b shows that the scalar field at x = 0 bounce once around the other topological sector, but returns, oscillating around the initial vacuum state.
This is a one-bounce collision. ii) for v < v c one has the formation of a bion state where, in the long term, the kink-antikink pair annihilates (see the Figs. 17a-b), or a two-bounce collision (see the Figs. 18a-b).
The Fig. 19 shows the velocities v * and v c as a function of r. Note from the figure that v * is a monotonically growing function of r and goes to zero as r → 0. This agrees with the limit of the sine-Gordon model, where small kinks are absent. We also observe that the region v > v * , where a large antikink-kink pair is formed, is restricted to 0 < r 0.38. The critical velocity v c has a maximum around r ∼ 0.3.
The structure of scattering for small kink solutions for v < v c is depicted in the Figs. 20a-c, where we show some plots of the time to first, second and third bounces for small kink-antikink collisions as a function of initial velocity for fixed values of r. In the Fig.   20a, for the case r = 0.05, we can see the formation of bion states and false two-bounce windows (also known as quasiresonances [73]) for v < v c = 0.151. Note that, despite the presence of an internal shape mode, and contrary to the expected by the resonant energy exchange mechanism [11], there are no structures of two-bounce windows. For r = 0.1 (Fig.   20b) we can only see the appearance of two thin two-bounce windows and the occurrence of one-bounce with v > v c = 0.2685. For r = 0.2 (Fig. 20c), we observe the growth of critical velocity (v c = 0.3486) and the increase of quantity of two-bounce windows. In this figure there is the formation of only one false two-bounce windows. That is, the increasing of r contributes to recover the full structure of two-bounce windows expected by the resonant energy exchange mechanism [11]. On the contrary, a decreasing of r shows that the twobounce windows tend to be suppressed, with the formation of false two-bounce windows. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.