Boundary scattering in the $\phi^{6}$ model

We study the non-integrable $\phi^{6}$ model on the half-line. The model has two topological sectors. We chose solutions from just one topological sector to fix the initial conditions. The scalar field satisfies a Neumann boundary condition $\phi_{x}\left(0,t\right)=H$. We study the scattering of a kink (antikinks) with all possible regular and stable boundaries. When $H=0$ the results are the same observed for scattering for the same model in the full line. With the increasing of $H$, sensible modifications appear in the dynamics with of the defect with several possibilities for the output depending on the initial velocity and the boundary. Our results are confronted with the topological structure and linear stability analysis of kink, antikink and boundary solutions.

I.

INTRODUCTION
Solitary waves/solitons have a large number of realizations in several areas of physics [1,2]. Despite simpler, models in (1,1) dimensions also have applicability in physics at all scales, from dark and bright solitons in cigar-shaped Bose-Einstein condensate [3] to high energy physics [4]. The kink (antikink) is the simplest solution with solitary character.
Despite the existence of analytical results for some classes of potentials [32], the study of kink scattering is mostly numerical.
As a first example, the nonintegrable φ 4 model presents a rich structure, which depends crucially in their initial velocity v i [10]. For large initial velocity, above a critical value v c , the pair KK recedes from each other, i.e, the kinks always escape to infinity after one collision. On the other hand, for initial velocities bellow v c , the structure of the collision is far more complex. In this region, the kink and antikink capture one another, forming the bion state. For v i v c there is the possibility of scattering after a two-bouncing process, where the pair is again able to escape to infinity after colliding twice [5,6,[8][9][10][11]. Each two-bounce process can be identified as corresponding to a particular two-bounce window.
Such windows in velocity are characterized by widths that are reduced with the grow of v i and accumulate around v i = v c . Near to each two-bounce windows usually there is a sequence of three-bounce windows. This strucure is reproduced in a fractal pattern [10].
According to Campbell, Schonfeld and Wingate (CSW) [8], a collision presenting twobounce is described by a resonance effect between the zero mode and the vibrational mode of the kink. Firstly there is the transferring of energy from the translational mode to the internal (vibrational) one. In the sequel, the energy is transferred back to the zero mode and the kink-antikink pair is separated from their mutual attraction.
A counterexample of the CSW mechanism was found in Ref. [12] for the φ 6 model.
There it was shown that certain antikink-kink collisions exhibit resonant scattering, even in the absence of a vibrational mode for one kink. This surprising result was related to the existence of a bound state produced by the antikink-kink pair [12]. A second counterexample was presented in Ref. [22], where some of us showed the total suppression of two-bounce windows, even with the presence of more than one vibrational mode.
The study of nonlinear field theories with boundary have a long tradition in integrable systems, mostly connected to the sine-Gordon model [33][34][35][36][37][38]. One object of investigation is to find suitable boundary conditions compatible with integrability [39]. Supersymmetric theories on the half-line keeping integrability were also considered [40][41][42][43]. One interesting aspect that deserves to be more investigated is the consequence of nonintegrability in the interaction of a kink or antikink with a boundary. In this line there are few examples of previous investigation for the φ 4 [44] and sine-Gordon [45] models.
The kink can be aplied in scenarions with large symmetry in high energy physics. For instance, the collision of relativistic bubbles can be treated as planar walls and described as kink scattering in (1, 1) dimensions [46][47][48]. Motivated by the undetection of topological defects expected to be produced at a large rate via the Kibble mechanism [49] in the early universe, the authors of the Ref. [50] investigated the φ 6 model under a generic perturbation. They showed that, due to the counterintuitive negative pressure effect, any small perturbation could trigger a chain reaction to influence the stability of a system of kinks.
In this way the negative pressure can be the cause of the vanishing of domain walls in some models.
In condensed matter, the kink in an nonintegrable model was proposed theoretically in buckled graphene nanoribbon [51,52]. The possibility of existence of the negative radiation pressure effect in buckled graphene was considered in the Ref. [53], based in the existence of such effect in a kink the φ 4 model [54]. The connection of solitons with conducting polymers is an active area of research, and some interesting reviews can be found Refs. [55,56] (for a a more gentle introduction to the subject, see the Ref. [57]). Kinks have an important role for describing the intriguing metalic properties of heavily doped trans-polyacetilene. In the Su-Schrieffer-Heeger (SSH) model [58], the kink/antikink excitations separate regions of two different classes of dimerization and satisfy the integrable sine-Gordon equation. These defects are spinless and are formed upon charge transfer from the dopant to the polymer.
The transition to a metallic state was explained by Kivelson and Heeger [59] as a crossover from a soliton lattice to a polaron lattice. Recently the interaction between the kink-antikink pair in photoexcited trans-polyacetilene was studied using ab-initio excited state dynamics [60]. After excited, the atoms evolve using an hybrid time-dependent density functional theory (TD-DFT). For T = 0 a soliton/antisoliton nucleates and pass through each other, as expected by a sine-Gordon model. For T = 0 a surprising result appears, with the solitons scattering after bouncing twice, a characteristic of a nonintegrable model. In an ideal conjugated polymer, a soliton is free to move because the total energy is independent on the position of the soliton. For finite chains, however, end effects push the soliton to the center of the chain. In the SSH theory, and for enlarged dimerization parameters, a linear term is introduced in the potential leading to these repulsive effects of the border [61,62].
An extension of the SSH model to include a third-neighbor interaction [63] shows that all conformal excitations in both cis-and trans-polyacetylene are repelled from the chain ends.
In this paper, we will discuss the process of collision of antikinks and kinks in the φ 6 model with a Neumann boundary condition. Since the antikink (kink) φ 6 has no associated vibrational state, our main motivation was to investigate if the boundary condition is able to produce bounce windows. Our numerical analysis shows that this is the case for antikinkboundary scattering, with a very intricate structure emerging from the scattering process.
The kink-boundary scattering leads to different results depending on the type of boundary to be scattered, with the most notable aspect the possibility of he changing of the topological sector and the nature of the boundary due to the scattering. Anoher motivation was to investigate if the boundary has a repulsive character to the scattering deffect. We found that this is true in most cases, but for some configurations the defect can be "trapped" around or even annihilated by the boundary.
In the next section we review the first-order formalism for obtaining static solutions and stability analysis. The Sect. III reviews some known results of solutions and stability analysis for the φ 6 model in the full line. In the Sect. IV we present solutions and stability analysis in the half line. The numerical analysis of antikink boundary scattering is presented in the Sect. V. Corresponding results for kink boundary scattering is presented in the Sect.
VI. We conclude in the Sect. VII.

II. BPS STATES
We consider the following action with standard dynamics The equation of motion is given by where V φ = dV /dφ. If the potential is the solutions of the first-order equation are also static solutions of the second-order equation of motion. The defects formed with this prescription minimize energy, connect adjacent minima of the potential and are known as BPS defects [64,65]. Stability analysis around a static solution φ S (x) considers φ(x, t) = φ S (x) + η(x)e iωt and the first-order corrections in the equation of motion. This results in a Schrödinger-like eigenvalue equation where the potential is The Schrödinger-like equation can be rewritten as Hη(x) = ω 2 η(x). That is, linear stability is assured if the Hamiltonian H is positive definite.
Now we consider the model on the half-line −∞ < x < 0. For this we consider the action Varying the action with respect to φ leads to an equation of motion for x = 0 identical to Eq.(2) obtained previously for the full-line. In addition, when extended and considered around x = 0, the action gives the integrable Neumann condition Static solutions have energies given by or, by Eqs. (3) and (4),

III. THE φ 6 MODEL ON THE FULL LINE
We consider the model given by The associated superpotential is W (φ) = φ 2 /2 − φ 4 /4. The first-order equations are then Note that the potential has vacua given by {−1, 0, 1}. The kink connecting the minima {0, 1} is given by [12] Φ The antikink connecting the minima {1, 0} is given by [12] Φ The antikink and kink connecting the minima {0, −1} and {−1, 0} are given, respectively, . Each of these solutions has energy E = 1/4, and their profiles can be seen in Fig. 1a. In addition we have four other solutions of the second-order equation of motion, divergent at x = 0: The profile of these solutions are presented in Fig. 1b.
Stability analysis for Φ 1± leads to the following potential of perturbations: Similarly, for Φ 2± we have  In particular, the analysis of these states for Φ 1± was presented in detail in the Ref. [66] (see also Ref. [67]). The potentials have just one bound state, the zero-mode connected with the translational symmetry of the solutions. According to the CSW mechanism [8], the absence of a vibrational state would result in the absence of two-bounce windows in collisions involving kinks (or antikinks). However, an interesting counterexample was presented in Ref. [12]. In their paper the authors showed the presence of bounce windows in antikink-kink collisions. The explanation was that the energy after the initial impact was trapped in the composited antikink-kink configuration. For example, the composite antikink-kink system has a potential of perturbations represented in Fig.   3a. This potential has a tower of bound states. On the contrary, the kink-antikink system Now we consider the φ 6 model on the half-line −∞ < x < 0. Firstly we will consider the solutions given by which correspond respectively to Φ 1+ (x) and Φ 4+ (x) on the full line. The Neumann condition (Eq. (8)), impose that The Eqs. (22) and (23)  Similar reasoning can be done with the solutions In this case the Neumann condition (Eq. (8)) gives also Eqs. (22) and (23).
The profiles of the solutions in the region is not regular for x < 0 and will not be considered here. The energy of static solutions can be found using Eq. (10). Then, the energies for solutions (28 -29) are given by with i = 1 . . . 2 whereas those for solutions (31 -33) are given by with i = 1 . . . 3. The Eqs. (34) and (35) shows that the energies presents an implicit The Fig. 6 shows the Schrödinger-like potential for some static configurations of antikink- The lines are for |H| = 0.05H m (red), |H| = 0.50H m (green) and |H| = 0.95H m (blue). We fix a = −12.50.
For the numerical solutions we used a 4 th -order finite-difference method on a grid N = 2048 nodes with the "infinity" at x min = −100 and a spatial step δx ≈ 0.05. For the time dependence we used a 6 th order sympletic integrator method, with a time step δt = 0.02.
We found no modification of the boundaries φ 1 (x),φ 1 (x) andφ 3 (x) with the increasing of H even for large times, which is compatible with stability (see the Figs. 8a-b for the boundaries φ 1 (x) andφ 3 (x)). On the other hand the instability of the φ 2 (x) shows up in the simulations.
One sees, for H 0.72, the emission of radiation by the boundary, as shown for instance in the Fig. 8c. For H 0.72H m one has the production of a kink by the boundary, as shown for instance in the Fig. 8d. For such high values of H there is no emission of radiation before or after the production of the kink. In this case the effect of instability shows up abruptly, with no signal of instability thereafter. This is compatible with a decaying of the unstable φ 2 boundary to the stable φ 1 .
In the remaining of this work we will study of boundary scattering in the φ 6 model. For this purpose only the regular and stable solutions φ 1 (x),φ 1 (x) andφ 3 (x) will be considered.
We will see that even these stable boundaries can emit radiation, produce a kink/antikink or even change their nature due to the scattering. Clearly this is a consequence of the interaction defect-boundary, not a signal of instability.

V. ANTIKINK-BOUNDARY SCATTERING
Here we will consider scattering with an antikink in the sector (0, 1) with the boundary.
The Fig. 5 shows that an antikink in this sector can be connected with solutions φ 1 (x) and   For the numerical solutions we used a = −12.5 and a 4 th -order finite-difference method. For the time dependence we used a 6 th order sympletic integrator method.
The Fig. 9 shows the richness of possibilities of the scattering products. The realization of each scenario depends on the parameters (v i , H). The Fig. 9a shows a bion state at the boundary. Fig. 9b shows an inelastic scattering (one-bounce) with the boundary. In the blue region the antikink is inelastically scattered by the boundary after one bounce.
The red region characterizes the production of a Φ 1+  The effect of H on the structure of two-bounce windows is described in the Figs. 12a-12d, where we see the times for the first three bounces as a function of the initial velocity.
The two-bounce windows are delimited by divergences in the curves of the time of the third bounce. First of all, we observe that when H = 0 we recover the structure of two-bounce windows presented in antikink-kink scattering for the φ 6 model in the full line, as described in the Ref. [12]. As H grows from zero, also grows the critical velocity v c that separated one-bounce inelastic scattering from bion states. We note also that with the grow of H the two-bounce windows accumulate around v i = v c . For H ≃ 0.6H m they disappear, remaining just the phenomena of kink production on the boundary.

VI. KINK-BOUNDARY
Here we will consider scattering with a kink in the sector (0, 1) with the boundary. The kink is the static solution Φ 1+ (x) given by the Eq. (14), now boosted with a velocity v i with x(t = 0) = a and given by: The Fig. 5 shows that this kink can be connected only with solutionsφ 1 (x) andφ 3 (x). In the following we consider separately the two possibilities.

A. Kink-φ 1 boundary
For this solution we consider −H m < H < 0. The initial conditions are with a = −12.50.
In this case the boundary presents an initially-repulsive force. The Fig. 17 (46)  Fig. 18 we have the effect that during the collision the scalar field at the center of mass oscillates around φ = 0, as in the Fig. 17f.

VII. CONCLUSIONS
In this work we have considered the φ 6 model with a Neumann boundary condition characterized by the parameter H. We found several solutions for kink, antikink and boundary.
Stability analysis for the antikink and kink lead to a translational mode. A similar procedure for the boundary solutions gives the behavior of the squared frequency of discrete modes as a function of H. This resulted in three regular and stable solutions for the boundaries.
The study of the unstable boundary was important since the nature of a boundary can be changed to an unstable one due to the scattering process. The changing of the nature of a regular and stable boundary due to the scattering was verified in some examples in two ways: i) a topological argument -the changing of the topological sector, and ii) energy conservation leading to an accurate description of the critical velocity as a function of H. After a period of time without any sensible alteration, and depending on the parameter H, there are two possibilities for the time evolution of an isolated unstable boundary: i) radiation being emitted continuously, or ii) creation of a kink and decaying to a stable boundary. As initial conditions, we considered kink/antikink scattering with regular and stable boundaries.
Our numerical analysis showed that the antikink-boundary scattering model has a rich pattern. We report the following phenomena, depending on the initial velocity and the parameter H: i) the production of a new kink from the boundary at x = 0 that travels with the reflected antikink. Such reflected antikink-kink pair can have a finite number of mutual collisions separating afterwards. Other possibility is the pair travelling together and colliding an infinite number of times, in a bion-like state. Such effect was observed previously in the φ 4 model (see the Ref. [44]). ii) the presence of a critical escape velocity, above which the antikink always receedes to −∞ without the production of a new travelling kink at the boundary. iii) the formation of two-bounce windows.
For the kink-boundary scattering here are two possibilities for a regular and stable boundary. For one boundary the scalar field always changes to the other topological sector. As a result, the new boundary emits an antikink with or without an oscillon. Another possibility for the initial boundary has a richer pattern. In this case the interaction with the kink is repulsive. This means that there is a critical velocity where the kink become trapped by the boundary. Depending on the parameters (v i , H) one can also see the changing of the boundary with the emission of radiation or the changing of the topological sector.
When compared with the results for the φ 4 model [44], despite its higher nonlinearity, the structure revealed by the antikink-boundary collisions for the φ 6 model is simpler. Indeed, as we saw in Fig. 11,