Kink-antikink collision in a Lorentz-violating $\phi^4$ model

In this work, kink-antikink collision in a two-dimensional Lorentz-violating $\phi^4$ model is considered. It is shown that the Lorentz-violating term in the proposed model does not affect the structure of the linear perturbation spectrum of the standard $\phi^4$ model, and thus there exists only one vibrational mode. The Lorentz-violating term impacts, however, the frequency and spatial wave function of the vibrational mode. As a consequence, the well-known results on $\phi^4$ kink-antikink collision will also change. Collisions of kink-antikink pairs with different values of initial velocities and Lorentz-violating parameters are simulated using the Fourier spectral method. Our results indicate that models with larger Lorentz-violating parameters would have smaller critical velocities $v_c$ and smaller widths of bounce windows. Interesting fractal structures existing in the curves of maximal energy densities of the scalar field are also found.


Introduction
The domain wall is a simple type of topological soliton that exists in many nonlinear scalar field models. It plays an important role in many branches of physics. For instance, the duality between the sine-Gordon model and massive Thirring model provides the simplest example of bosonization in condensed-matter physics [1]. Cosmic domain walls can exist if the Universe was proceeded by some first-order phase transitions [2,3]. It is even proposed that all of us might live on a four-dimensional domain wall embedded in a five-dimensional space-time [4,5,6,7,8,9]; also see [10,11] for reviews. A domain wall in (1 + 1) dimensions is also called a kink.
The collision between non-integrable kinks is an important topic in the study of kinks. For the simplest kinkantikink collision, it is convenient to take the velocities of the kink and antikink as v 0 and −v 0 , respectively. Such collisions are referred to as velocity-symmetric collisions or symmetric collisions in this paper. All the kink-antikink collisions mentioned below are of this type, only the velocity of the kink v 0 will be specified. tikink). However, some recent works reveal that the CSW mechanism is insufficient to describe bounce phenomena found in some non-integrable models in which the kink either has no vibrational mode at all [19], or only has quasinormal modes [20]. Therefore, by exploring kink-antikink collisions in various kinds of models one may find new phenomena with new physics.
However, as pointed out in Ref. [50], every current candidate for a superunified theory 1 contains some potentials for Lorentz violation, and the same is true for more restricted theories that attempt to treat quantum gravity alone. Theories with the potential for Lorentz violation, including superstring/M/brane theories, canonical and loop quantum gravity, non-commutative spacetime geometry, non-trivial space-time topology, and so on; see [51,52,53,54,55,56] for part of the original works, and [57,58,59,60,61,62] for recent reviews. Thus, it is interesting to search for kink solutions in Lorentz-violating scalar field theories, and study how Lorentz violation impacts the properties of the kinks and their collision.
Analytical static and traveling kink solutions have been found in some Lorentz-violating scalar field models with single or multi-field components [63,64,65,66], and these solutions have been applied in many related issues, e.g., entropic information [67], the Kondo effect [68], and trapping fermions [69].
In this paper, kink-antikink interaction is considered by using the traveling kink solution reported in Ref. [63]. In the next section, the single-field Lorentz-violating model as well as the corresponding kink solution is reviewed. After an analysis of the linear stability, a numerical simulation of the kink-antikink collision is conducted.
Obviously, the Lorentz-violating term does not alter the form of a static solution. Therefore, it is trivial to find a static kink solution. For example, a standard static kink solution is obtained by taking the potential as The real challenge is to find a traveling kink solution. The breaking of Lorentz invariance makes it difficult, in general, to find a boost transformation that helps derive traveling solutions from static ones. Fortunately, as mentioned in Ref. [63], the present model is invariant under a deformed boost transformation x → x = γ(x−vt), where the deformed Lorentz factor is given by γ ≡ 1/ √ 1 − v 2 + 2αv in the natural units 3 . Therefore, the traveling kink and antikink solutions take the following forms: φK where x 0 and v 0 are the initial position and velocity of the kink/antikink, respectively. A positive (negative) v 0 means a kink/antikink moving to the right (left). With the definition of γ, the allowed range of initial velocity varies with the value of the parameter α: which is shown in Fig. 1. The energy density for a traveling solution is [63] ρ(x, t) = 1 2 Obviously, in the Lorentz-violating model, the width and energy density of a moving kink depend on both the magnitude and direction of its velocity because of breaking of time symmetry. To see this phenomenon clearly, consider the linear superposition of a pair of a moving kink and antikink: In Fig. 2, the configuration of φ KK and its energy density at time t = 0 is plotted for α = 0, 1, 2. The asymmetry between the right-moving kink and left-moving antikink appears as α increases. An antikink moving to the left has smaller width and larger energy density than those of a kink moving to the right with the same speed for α > 0. It is the opposite for α < 0.

Linear stability and vibrational mode
Before the discussion of kink-antikink collision, it is important to study the linear perturbation of the static kink solution. First, a good kink solution should be stable against the linear perturbation. In addition, according to the CSW mechanism, the existence of vibrational modes in a linear spectrum is closely connected with the bouncewindow phenomenon.
To derive the equation of motion for a small perturbation δφ(t, x) 1 vibrating around the static kink background φ s (x), the action is expanded up to the second order of δφ: where After taking the variation of δ (2) S with respect to δφ, the linear perturbation equation is obtained: The following mode expansion is then introduced: and substituted into Eq. (12) to obtain a Schrödinger-type equation for f n (x): x φs ∂xφs , and Eq. (14) can be rewritten as where is the Hamiltonian operator. It can be further factorized as [70,71] According to supersymmetric quantum mechanics, the eigenvalues of a system with a factorizable Hamiltonian are always non-negative, i.e.,w 2 n ≥ 0, which also means w 2 n ≥ 0. Therefore, the static kink solution φ s (x) is stable against linear perturbation.
As can be seen from Eq. (14), the Lorentz-violating term does not affect the structure of the linear spectrum of the standard φ 4 model. Thus, for the Lorentz-violating φ 4 model considered here, there are two bound states: the translational mode (zero mode) and vibrational mode, and the corresponding eigenvalues and wave functions are [3] However, one should note that the complete spatial wave function for the nth mode is g n (x) ≡ f n (x)e iwnαx , which varies with α, except for the zero mode; ω 0 = 0 and g 0 (x) = f 0 (x). Therefore, the Lorentz-violating term does affect the shape of the vibrational mode, as shown in Fig. 3.
Since the vibrational mode is closely related to the bounce-window phenomenon, it is expected that the results of kink-antikink collision will also change with α. This issue is discussed next.

Simulation and results
Unlike many integrable models, e.g., the sine-Gorden model, where analytical solutions for multiple kinks can be derived by using methods like Bäcklund transformation, it is extremely difficult to find multi-kink solutions in a nonintegrable model. Therefore, to study the collision of a kink-antikink pair in the model proposed herein, one must resort to numerical simulation. First, the initial condition of the system is taken as where φ KK (x, t) defined in Eq. (9) is a superposition of a kink initially at −x 0 with velocity v 0 and an antikink initially at x 0 with velocity −v 0 . The dynamical equation (2) is solved by using the Fourier spectral method described in Refs. [72,73,40]. A reasonable numerical solution should satisfy the energyconservation law, and this fact is used here to test the viability of the proposed numerical solution.
The total energy of the Lorentz-violating field is given by For a well-separated kink-antikink pair, the total energy predicted by theory is where E(v 0 ) = γ(1 + αv 0 )E s is the energy of a soliton moving with speed v 0 , and E s = [(∂ x φ) 2 /2 + V ]dx = 4 3 is the total energy of the static soliton φ s (x).
The conservation of the total energy is checked by evaluating the relative energy error: Here, E num is the numerical result of the total energy, which is obtained by inserting the numerical solution of φ(x, t) into Eq. (20). In this work, our simulations are implemented with a spatial grid step ∆x = 0.2. The time step ∆t is automatically determined by the ode45 solver in MatLab (MathWorks, USA). The precision of the calculation can be controled by tuning ∆x or/and the tolerance options of the ode45 function, i.e., AbsTol and RelTol (see Ref. [40] for details). In our simulation, the tolerance options have been set to ensure that our numerical solutions of φ(x, t) satisfy |δE| 10 −8 .

Impacts on fractal structure
To get a global idea of the influence of the Lorentzviolating term, first consider how the fractal structure would change under different values of α. In Fig. 4, the fractal structures (up to three-bounce windows) are plotted for α = 0, 0.5, and 1. The two-bounce windows (2BWs), three-bounce windows (3BWs), and the inelastic scattering zones are highlighted in green, pink, and gray, respectively. The un-highlighted zones correspond to higher-order bounce windows as well as bions. The critical velocity (the left-hand boundary of the gray zone) decreases from v c (α = 0) ≈ 0.26, to v c (α = 0.5) ≈ 0.23, and eventually to v c (α = 1) ≈ 0.177. As α increases, all the 2BWs (the green zones) move to the left, and their widths decrease as well. The 2BWs 3BWs and inelastic scattering zones are highlighted in green, pink, and gray, respectively. Un-highlighted zones correspond to higher-order bounce windows and bions. Obviously, as α increases, the critical velocities as well as the widths of each 2BW decrease.
As an explicit example, the evolution of field configuration and the corresponding energy density for v 0 = 0.178 are plotted in Fig. 5. With this initial velocity, one would observe bion, two-bounce, and inelastic scattering by taking α = 0, 0.5, and 1, respectively. From the bottom panels of Fig. 5, it can also be seen that the radiation emitted after collisions is asymmetric with respect to the origin.
The critical velocity v c for α ∈ [0, 4.5] is plotted in Fig. 6, which shows that v c is a monotonically decreasing function of α, and it is always below the maximum velocity v 0,max allowed for a velocity-symmetric collision. Figure 7 shows how the widths of the first two 2BWs vary with respect to α. It also decreases monotonically as α becomes larger.

Information from maxima of energy densities
Given a value of α, the fractal structure clearly shows how the collision results vary with the initial velocity v 0 . One may ask the reverse question; that is, given a value of v 0 , can one tell which interval of parameter α corresponds to a two-bounce, three-bounce, or inelastic collision? In Ref. [49], the authors found that the maxima of various kinds of energy densities can provide important information on the collision phenomena. To see this, the energy density defined in Eq. (8) is first divided as follows: 1st 2BW 2nd 2BW Width of the first two 2BWs  V (φ) are the kinetic energy density, elastic strain energy, and on-site potential energy, respectively (also see [74,47]). By simulating velocity-symmetric kink-antikink collisions with v 0 = 0.178 and α ∈ [0, 1.2], curves of the maxima of the aforementioned four types of energy densities are obtained: ρ max , k max , u max , and p max (see Fig. 8). These curves are qualitatively similar; that is, they show a fractal structure, i.e., in some intervals they behave in an orderly manner, but in several others they seem to be chaotic. The ordered intervals correspond to n-bounce windows. The first three 2BWs, which correspond to The first three 2BWs are highlighted (green zones), as are the first three 3BWs around the first 2BW (pink zones). The gray zone corresponds to inelastic scattering, and unhighlighted chaotic segments correspond to higher-order bounce windows and bions.  Fig. 9, the φ(0, t) curves for the two-and three-bounce solutions are plotted, which correspond to some representative values of α that lie in the highlighted bounce windows. To our knowledge, this is the first report on the fractal structure in the maximal energy density graph, despite the fact that the maximal density graph has been used in many other aspects [75,49].

Summary
In this work, kink-antikink collisions of a Lorentzviolating φ 4 model have been studied. After a short summary of the model and the kink solution of Ref. [63], the linear perturbation spectrum of the static kink solution was analyzed. It was found that the Lorentz-violating term of the proposed model does not change the spectrum structure of the standard φ 4 model, so the static kink solution is linearly stable and there exists only one vibrational mode apart from a zero mode. However the Lorentz-violating term does impact the wave function and frequency of the vibrational mode. As a consequence, the kink-antikink collision phenomena of the present model will deviate from those of the standard φ 4 model.
The deviation was studied numerically via two different approaches. In the first approach, fixing the values of the Lorentz-violating parameter to be α = 0, 0.5, 1, the collision with the initial velocity v 0 ranging from 0.12 to 0.27 was scanned. By comparing the fractal structures of each value of α (Fig. 4), it was found that models with larger Lorentz-violating parameters have smaller critical velocities and narrower widths of 2BWs. The values of critical velocity and widths of 2BWs were also calculated for arbitrary values of α (see Figs. 6 and 7).
In the second approach, setting the initial velocity as v 0 = 0.178, the maximal energy densities corresponding to α ∈ [0, 1.2] were scanned. An interesting fractal structure was observed for the first time (see Fig. 8). In the curves of maximal energy densities, the intervals corresponding to bions are more chaotic than those of the two bounces, three bounces, and inelastic scatterings. This indicates that the plot of maximal energy densities might be very useful in analyzing the results of kink collisions.
Phenomenological applications of the present work are worth consideration, but go beyond the scope of the present work.