A toy model to explain the missing bounce windows in the kink-antikink collisions

We propose a toy model to reproduce the fractal structure of kink and antikink collisions on one topological sector of $\phi^6$ theory. Using the toy model, we investigate the missing bounce windows observed in the fractal structure, and show that the coupling between two oscillation modes modulate the emergence of the missing bounce windows. By numerical calculation, we present that the two bounce resonance corresponds to the lower eigenfrequency of the toy model.


Introduction
In ϕ 6 model,the Lagrangian density can be defined as [2] Correspondingly, the Euler-Lagrange equation on 1+1 dimension is given as The kink solitons are non-trivial solutions of Equation (2), interpolating between two adjacent vacua.We can obtain six soliton configurations by using the BPS bound [5; 6].Among them, one pair of the kink and antikink (KAK) with configurations ϕ (0,1) = √ (1 + tanh x)/2 and ϕ (1,0) = √ (1 − tanh x)/2 can be obtained.Using the perturbation method, we set ϕ = ϕ s + η(x)e iωt , where ϕ s is the static kink configuration.By linearizing the field equation, Dorey et al. presented that [1] where U(x) = 15ϕ 4 s −12ϕ 2 s +1 is the potential.This Schrödinger like equation has merely continuous spectrum, which means that any localized vibration mode cannot be found as in the ϕ 4 model [7].
If we set the kink and antikink with a finite separation and opposite velocities, they would collide and interact with each other, leading to complex phenomena including the resonance and radiation.Numerical simulations demonstrated how the initial velocity and topological sector affect the phenomena of collisions [1].In the (0,1)+(1,0) sector (ϕ=ϕ (0,1) (x+a)+ϕ (1,0) (x− a), a represents the half separation between the kink and antikink), the kink and antikink have no resonance.But in the (1,0)+(0,1) sector, one observes the fractal structure [1].
We recall that, in field theory with resonant scattering, nbounce windows are sets of incident velocities when kink and antikink bounce n times before their separation.3-bounce windows are nested near a 2-bounce window, and for larger n this pattern repeats to compose the fractal structure.In ϕ 4 theory, the first 2-bounce window occurs with the number of oscillations n = 4. Then all other 2-bounce windows with higher value n are found at least up to the critical velocity [3].However, in ϕ 6 theory, Dorey et al. presented there is one "missing bounce windows" (MBWs), when n = 12 and 14 correspond to two 2-bounce windows, but when n = 13 kink and antikink cannot separate and finally turn into the bion state [1] To better understand the fractal structure and the MBW phenomena in ϕ 6 theory, we propose a toy model including two shape modes and one translation mode to reproduce these features and explain the physical mechanism for the emergence of these phenomena.

The toy model
The essence of CCM is to use dynamics of the translation and shape modes to approximate the partial differential equations (PDEs) in Equation ( 2) [3].The spirit of our toy model is to mimic the dynamic of the translation mode by a "particle" passing through a potential well with exponential pattern, and mimic the dynamic of shape mode by an harmonic oscillator.This toy model can successfully reproduce the fractal structure in ϕ 4 KAK collision, and explain the constraint of the critical velocity [10].In the numerical results of ϕ 6 PDE, it is mentioned above that there are missing windows.With more details, the third two-bounce window(ν in ≈ 0.033) is subtly larger than the neighboring ones, which is absent in the ϕ 4 model because the width of 2-bounce windows decreases monotonically as v in [8].In the origin ϕ 6 model, shape modes exist as the form of bulk modes.From freezing the delocalized modes with the half-separation a = 3, Adam et al. constructed an effective collective coordinate model (CCM), which reproduced some characteristics in ϕ 6 theory like critical velocity and 2-bounce windows [14].However, there is no MBW in their model, so they proposed that the missing windows are caused by higher order shape modes or the radiation.That's why we introduced two harmony oscillators to simulate the behaviors of two shape modes.U(a) is chosen to be the Morse potential (See FIG.1), which is ideal to describe the interaction between particle-like kink and antikink.When the kink and antikink are far away, they have mutual attraction, and when they pass through each other, they have the strong repulsion.The Lagrangian of the toy model is consisted of the parts of kinetic energy and the potential, which can be expressed as Here a(t) denotes the coordinate of the particle with the Morse potential U(a) = e −2a(t) − 2e −a(t) , b(t) and c(t) denote two harmomic oscillators with potentials as where ω 1 and ω 2 denote the oscillation frequencies.The coupled terms in the Lagrangian (4) infer that energy could transfer between shape modes.The Euler-Lagrangian equations for the three modes are derived as Here k i j are explicit functions of coefficients including m 1 , m 12 , w 1 , etc.
We study the dynamic of the toy model with numerical calculation.To observe the fractal structure, we show the numerical output of the field We don't consider shape modes in Equation ( 7) because parameters b(t) and c(t) have ignorable contribution to the fractal structure.We observe that when the kink and antikink are separated far enough, the evolution of b(t) and c(t) is independent of a(t).It is obvious that the second and third equations in ( 6) constitute a set of coupled oscillators equations.It means that the system have two independent eigenfrequencies [9], which can be derived by the decoupling technique as

Numerical result
We use the "Ndsolve" in Mathematica to perform the numerical calculation.From the Lagrangian in Equation ( 4), the parameter m 23 can effect the intensity of energy transfer between two oscillation modes.Initially, We set m 1 = m 2 = m 3 = 1 and m 12 = m 13 = 0.1.This ensures that the two oscillators are equivalent when we ignore the frequency.The initial half separation is set as a(0) = 5 across this work.In FIG. 3, we plot the the center field value ϕ(0, t) as a function of the time and incident velocities for m 23 = 0.05, 0.10, 0.15, 0.20, 0.25 in panels (a), (b), (c), (d), and (e), respectively.When m 23 = 0 (see FIG. 2), there are two MBWs near v in = 0.20 and v in = 0.27.As m 23 increases to 0.10, the first MBW gets deeper (If the third collision time of a MBW gets longer, we call that the MBW becomes deeper, inversely we call it shallower) while the second one becomes shallower.Meanwhile, as seen in FIG.3(b), the escaping window at V in ∼ 0.34 turns to a MBW.The current first MBW has 'penetrated the ceiling' and converts into a resonance window when m 12 equals to 0.15.It can also be observed that some MBWs get deeper and some turn back into resonance windows.When m 23 reaches 0.20 (see FIG. 3(d)), the window near v in = 0.18 turns back into a MBW.In the meantime, these appeared MBWs for 0.32 < V in < 0.4 vanish again.The fractal structure tends to be stable as m 23 gets enough larger but less than 1 (FIG.3(e)).We observed that the coefficient m 23 has ignorable effect on the critical velocity, but it can effect the fractal structure significantly observed from FIG. 3. It also plays an important role in the formation of two-bounce windows.
Another difference between the ϕ 6 model and ϕ 4 model is that more than one oscillation modes are contained in the former theory.Although there is no local self-excitation, recent literature showed that delocalized oscillation modes do exist with different frequencies [? ].It is widely known that when resonance scattering exists, there are favorable timings where the shape mode would be excited by the second impact through some characteristic phase angle.Thus, the resonance condition can be expressed as [8]: Where T is the time interval between two collisions, and n represents the number of lower frequency oscillation in the period.δ is the phase position [8].
We propose that the resonance frequency may relate to the eigenfrequencies.To test our proposal, we plot the v in of the centres of each 2-bounce windows versus the oscillation number of times n from the numerical calculation.We don't count for the start number of bounce in the first window.So, when depicting scattering points, n always starts at 1.
In the top panel of FIG. 4, the slope of fitted straight is 2π/ ω, where ω = 2.9522 agrees exactly with ω=2.9522 from Equation (8).While in the bottom panel, ω=1.9813 is also very close to the ω=1.9819.In Table 1, we list pairs of ω and ω for different m 23 to exhibit their correlation.Thus, both two values are close to the minimal frequencies for each case.We conclude that the whole resonance system always responds to the minimum eigenfrequency, since the lower frequency needs lower energy to be excited.Now we study where missing phenomenon could exist.we fix ω 1 = 2, ω 2 = 3 again, and plot the time between the two collisions versus n by using the centres of 2-bounce windows for m 23 equals to 0.04, 0.06, 0.07, 0.08, 0.11 and 0.13.In FIG. 5, we show these plots in panels (a), (b), (c), (d), (e) and (f), respectively.There is no missing phenomenon   The appearance of missing windows observed in FIG. 5 is most probably due to that there is energy transfer between two oscillation modes.We set ω 1 = 2, ω 2 = 3, m 23 = 0.07 (FIG.5(c)) for illustration.When n is larger than 34, although the energy transfer may still exist somewhere, the total energy is large enough that the third collision wouldn't happen.On the other hand, when n is less than 22, there is not enough time to support an effective energy transfer between oscillation modes.That can explain that the missing windows shifts towards n=0 as m 23 rises.In FIG. 6, we list the plots of a(t), b(t) and c(t) for 2-bounce windows and a MBW to show the dynamics.In FIG. 6, panel (a) and (c), the energy is almost completely transferred from the oscillation modes to the translation mode.In panel (b), the energy is transferred between two oscillation modes, so that the kink and antikink have not enough kinetic energy to reach the infinity.The equivalence of the frequency (fitted) and the lowest eigenfrequency is not a coincidence.In our toy model, we take ω 1 = 2, ω 2 = 3, m 23 = 0.15 for illustration.We show the dynamics for the first 2-bounce window in panel (a) of FIG. 6.It can be seen that the maximum half-distance between the two collisions is a max ≈ 3.2 and n = 5.In panel (b) and (c), we can see that a max increases as n been larger.When a max is larger, the Morse potential disappears, and the eigenfrequency plays a prominent role.Thus, the eigenfrequency becomes the fitted frequency for large n limit.
In ϕ 6 theory, there are four delocalized modes at the half separation a ≈ 6 [1].The fitted frequency ω = 1.0452 from the plot of T versus n equals to the frequency of the first delocalized mode.These delocalized modes depends on the value of a.In cases of 2-bounce resonance scattering, a ≈ 6 is the maximum half-distance that kink and antikink could reach in the period of the first and the second impact.Thus, the "eigenfrequency" in the ϕ 6 model may be the lowest one of the four frequencies of delocalized modes at a = 6.
In our toy model, the two frequencies of oscillators are different, and the lower one dominates the 2-bounce windows.The higher frequency is evident in the MBW (see panel (b) in Fig R4 , where the green line denotes the dynamics of the coordinate 'c').Thus, our toy model illustrates that the MBW are the result of multiple eigenfrequencies, and sheds light on the MBW mechanism in ϕ 6 model.
In FIG. 7, we manifest a set of coefficients that reproduce the fractal structure close to those from the PDE in the ϕ 6 model.

Conclusion
In this work, we constructed a toy model with one translation mode and two oscillation modes, which could reproduce the fractal structure and the MBW phenomena observed in the PDE simulation of the ϕ 6 model.From numerical results, we obtained that our toy model resonance always responds to the lower eigenfrequency.Furthermore, we found that m 23 , the coupling coefficient between two oscillators, modulates the MBWs significantly.
In our toy model, we demonstrated a possible mechanism of missing bounce windows in soliton dynamics and showed some similarities compared to the full ϕ 6 model.However, in the full model there are more complex phenomena like spectral walls our toy model cannot reproduce [12; 13].

Figure 1 :
Figure 1: The plot of Morse potential.

Figure 2 :
Figure 2: The plot of ϕ(0, t) for the toy model.ν in denotes the incident velocity.The initial parameters are set as ω 1 = 2, ω 2 = 3, m 23 = 0.First, we fix ω 1 = 2, ω 2 = 3, and try different settings of m 23 to show its role in modulating the MBW.In FIG.2, we first plot the m 23 = 0 case.For 0.32 < ν in < 0.34, the regime between two adjacent two bounce windows, namely the 'three bounce regime', show two different patterns appearing alternatively.The non-monotonicity of the widths of two-bounce windows may be correlated with the dual patterns.

(a) ω 1 4 Figure 4 :
Figure 4: T versus n for two sets of (ω 1 , ω 2 , m 23 ) are plotted in the top and bottom panels, respectively.The straight lines are obtained by using the least square method.
the first 2-bounce window when v in = 0.1805.
the first MBW when v in = 0.266.
the second 2-bounce window when v in = 0.311.

Table 1 :
A tabulation of