Impurity interactions, critical behavior and resonance phenomena in classical φ^4 theory

University of Amsterdam

Report Bachelor Project Physics and

Astronomy, size 15 EC

Conducted between April 3, 2018 and July 5, 2018

Submitted July 4, 2018

Impurity interactions, critical

behaviour and resonance

phenomena in classical φ

4

theory

Author:

Jasper Kager

Student Number:

11037806

Supervisor:

dr. J. van Wezel

Second examiner:

prof. dr. J.S. Caux

Daily Supervisor:

M. Lizunova MSc

Institute for Theoretical Physics Amsterdam

Abstract

The use of φ4 theory as a model for kink dynamics goes back several decades. Even though we know kinks to be present in several condensed matter systems, their dynamics in realistic systems have not been thor-oughly investigated. In particular, little is known about how impurities affect kink dynamics. In this thesis, the influence of inhomogeneities on kink-antikink collisions has been investigated. On one hand, kink-antikink collisions were simulated in φ4 theory with attractive Gaussian impurity at the collision site. Firstly, the dependence of the critical velocity on impurity strength has been investigated. The critical velocity was found to decrease for weakly attractive impurities, and increase for strong ones. A mechanism to explain obtained results has been proposed, and evidence in favor of mentioned mechanism has been obtained from the kink’s vibra-tional spectrum. Secondly, the dependence of 2-bounce window locations on impurity strength has been determined. The center of 2-bounce win-dows was found to shift to lower velocities linearly with impurity strength, suggesting they will disappear for sufficiently high impurity strengths. Then, kink-antikink collisions have been simulated between two strongly repulsive impurities. In this system, kinks were observed to briefly form a bion, after which the bion broke up into a kink-antikink pair.

Contents

1 Introduction and theory 4 1.1 Kinks in φ4 theory . . . 5 1.2 Dynamics of kink collisions . . . 7 1.3 Kink-impurity interactions . . . 10

2 Numerical method 11

3 Dependence of the critical velocity on impurity strength 13 4 Location of bounce windows 16 5 Kink-antikink trap 19

6 Conclusion 20

Dutch summary

Vouw een vel A4-papier een paar keer in de lengte dubbel. Houd het gevouwen papier aan de gevouwen zijden vast. Duw de randen nu aan een zijde naar beneden, en aan de andere zijde naar boven. Als het goed is, is er in het midden van het papier een kink te zien, waar de overgang van naar beneden gevouwen naar naar boven gevouwen zit. Als je een beetje speelt met hoe ver je de randen omvouwt, zal je de kink zien verschuiven.

Dit was een enkel voorbeeld van een kink, maar ze komen op veel plekken in de gecondenseerde materie voor. Als er een kink in een materiaal kan zitten, kunnen er ook meerdere tegelijk zijn. Deze kinks kunnen bewegen, interacties aangaan, en botsen. De vraag is dan ook wat er tijdens zo een botsing gebeurt. Dat blijkt af te hangen van hoe snel de kinks naar elkaar toe bewegen. Bewegen ze sneller dan een zekere “kritische snelheid”, dan botsen ze eenmalig en schieten weg in de richting waar ze vandaan kwamen. Als ze niet zo snel bewegen, blijven ze doorgaans aan elkaar plakken, trillen samen op en neer, en verdwijnen na verloop van tijd als gevolg van energieverlies. Als de ingangssnelheid echter goed wordt gekozen, blijven ze even aan elkaar plakken, om elkaar vervolgens alsnog af te stoten.

Wat we tot nu toe nog niet hebben bekeken, is wat er gebeurt als er een kinkbots-ing plaatsvindt op of nabij een oneffenheid in het materiaal. Blijven ze makke-lijker aan elkaar plakken, of juist niet? Om deze vraag te beantwoorden, zijn in dit stuk simulaties uitgevoerd van kinkbotsingen, met een theoretisch raamwerk genaamd φ4 _{theorie. Daarmee hopen we meer inzicht te krijgen in de rol die}

kinks spelen in de gecondenseerde materie.

1

Introduction and theory

In 1834, while performing experiments to improve the design of canal boats, John Scott Russell observed a solitary wave which moved through the canal for several miles, never changing its velocity and only very slowly changing its shape over time. Since such a wave had never been reported before, Russell’s observa-tion and subsequent experiments on such waves were somewhat controversial. Their existence has since been widely excepted. Solitary waves of permanent shape are nowadays called solitons, and they have been observed and applied in areas such as nonlinear optics [8], Bose-Einstein condensates [17] and biophysics [4]. In addition, they have been theoretically studied in for example mechani-cal systems [20], long graphene nanoribbons [19], long Josephson junctions [11],

seismology [5] and two-dimensional crystal lattices [15].

Two classes of solitons are distinguished; topological solitons, for which lim

x→∞φ(x, t) =x→−∞lim φ(x, t), (1)

(where φ(x, t) represents the soliton) and kinks (or non-topological solitons), for which the above equality does not hold. A plot of both types of waves is given in figure 1a.

Interest in soliton dynamics has been driven in part by the fact that their permanent shape makes them potentially reliable carriers of information. Their applicability is limited by the fact that some kinks undergo dramatic qualitative behavioural changes during interaction with other kinks. The precise meaning of the last statement will become clear in the next few sections, where kink-antikink interactions will be discussed in the case of (1+1)-dimensional φ4_{theory. Due to}

their relative mathematical simplicity, φ4theory and the sine-Gordon model are the most used models for studying solitons. Even though the sine-Gordon model deserves a mention due to its major role in the development of soliton theory, it features only topological soliton solutions. Since these aren’t of interest to us, this model will not be discussed further.

1.1

Kinks in φ

theory

Classical (1+1)-dimensional φ4_{theory may be defined either by its Lagrangian}

density: L(x, t) = 1 2  ∂φ ∂t 2 +1 2  ∂φ ∂x 2 + V (φ), (2) or by the corresponding equation of motion of the field φ:

∂2φ ∂t2 = ∂2φ ∂x2 − ∂V (φ) ∂φ . (3) Here, V (φ) = 1 4(1 − φ 2₎2 ₍₄₎

is the scalar potential, a plot of which is shown in figure 1b. The energy of the system is given by

E = Z ∞ −∞  1 2 ∂φ ∂t 2 +1 2 ∂φ ∂x 2 + V (φ)  dx. (5)

(a) (b)

Figure 1: (a): A topological (orange, dashed) and a non-topological wave (blue, solid). (b): The classical φ4 _{potential, showing global minima at φ = ±1, and a local optimum at φ = 0.}

Equation (3) has three constant solutions, which correspond to the extrema of equation (4). The solutions φ = ±1 are stable, while φ = 0 is unstable. A family of stable, nontrivial solutions of equation (3) is given by

φK(x, t) = tanh h 1 √ 2 x − x0− vt √ 1 − v2 i , (6) φ_K(x, t) = − tanhh√1 2 x − x0− vt √ 1 − v2 i , (7)

where equation (6) represents a kink, moving in the positive x-direction with velocity v while being centered around x = x0 for t = 0. A solution of the

form of equation (7) represents an antikink. It is of interest to analyze the behaviour of small perturbations of equation (6). We set v = 0 and x0= 0 for

simplicity, and define φ(x, t) = φK(x) + η(x, t). Substituting into equation (3)

and linearizing, yields ∂2_{η(x, t)} ∂t2 − ∂2_{η(x, t)} ∂x2 − η(x, t) + 3 tanh 2  _x √ 2  η(x, t) = 0. (8) Assuming η takes the form e−iωtχ(x), we obtain the following:

 − d 2 dx2 − 3 cosh −2_√x 2  χ(x) = (ω2− 2)χ(x). (9) It was proven in Hasenfratz and Klein [16] that all solutions of equation (9) have nonnegative energy. The solution with lowest nonzero energy, given by

χ(x) =  ₃ 2√2 1/2 tanh  _x √ 2  cosh−1  _x √ 2  (10) has ω =p3/2, and may be interpreted as the kinks vibrational frequency.

1.2

Dynamics of kink collisions

Several authors have carried out numerical simulations of kink-antikink colli-sions using equation (3) [1, 7, 14]. The kinks were found to reflect and escape to infinity for v > 0.2598. Several moments during such a collision are illustrated in figure 2. For v < 0.192575, the kinks always form a long-lived bound state (called a bion), in the sense that the energy was found to decay logarithmically with time [13]. For 0.192575 < v < 0.2598, several velocity ranges were found for which the kinks collide and rebound, stop at some finite distance, collide again, and escape to infinity. Velocity ranges where this occurs are called 2-bounce windows. Plots of collisions showing escape, bion formation and a 2-bounce are given in figure 3.

Campbell et al. [7] proposed the following explanation for this behaviour. The first collision causes the excitation of the vibrational mode of the kinks. By conservation of energy, the kinks lose their kinetic energy, making it impossible to escape. If the kinks vibrate in-phase during the second collision, vibrational energy is converted back to kinetic energy, enabling the kinks to escape to infinity. This mechanism has since been confirmed by numerical experiments [6].

Figure 2: Plot of φ(x, t) during a simulation of a kink-antikink collision for several values of t, for v = 0.4. (a)-(c) The kinks moving towards each other. (d) Shows the moment of collision, and figures (e) and (f) show the kinks moving away from each other.

Figure 3: Space-time plots of kink-antikink collisions, showing rebound for v = 0.4 (top left), bion formation for v = 0.23 (top right), and a 2-bounce resonance for v = 0.2 (bottom left). Figures show numerical solutions of equation (3).

Following the aforementioned discoveries, the subcritical regime was investigated further by Anninos et al. [2]. In particular, the existence and relative locations of n-bounce windows was investigated. It was found that several 3-bounce windows may be found near each 2-bounce window. Furthermore, each 3-bounce window is accompanied by several 4-bounce windows, etc. The fractal structure of bounce windows is shown schematically in figure 4. In figure 4a, black bands indicate capture regions, while white bands indicate bounce windows. The continuous black band on the right is the region where the initial velocity of the kinks exceeds the critical velocity. The region in the box is magnified in figure 4b. The black bar on the left indicates the position of a 2-bounce window, while

all the other black bands are now 3-bounce windows. The part of figure 4b inside the box is magnified in figure 4c, which shows several 4-bounce windows. Repeating this procedure would then reveal higher order bounce windows.

Figure 4: Fractal structure of locations and widths of bounce windows. Figure adopted from [2].

1.3

Kink-impurity interactions

Since the discovery of long-lived bion states, bounce windows and their fractal structure in φ4_{theory, similar behaviour has been observed in several other}

mod-els that admit soliton solutions, including φ6 _{theory [12, 18], the sine-Gordon}

model and modifications thereof [3], and the Frenkel-Kontorova model, which can be used to model crystal lattices [15].

Another model, where kink dynamics has been studied, is classical φ4 _model

with an additional, localized inhomogeneity. The equation of motion of such a model is given by equation (3), with modified potential

V (φ) = 1 4(1 − φ

2

)2(1 − µδ(x)), (11) where δ(x) represents the impurity, and µ is a constant. This parameter may be considered a measure of the “strength” of the impurity, in the sense that increasing µ will cause the impurity to have a greater influence on kink dynamics. Kink-impurity interactions are natural to study, since media through which kinks propagate are rarely completely pure. In the case of a water wave, an inhomogeneity may be some obstacle in the water; an optical soliton may prop-agate through a lens that is not completely clean, and a sample that is used to make a Bose-Einstein Condensate may not be perfectly purified.

Kink-impurity collisions in φ4_{theory were first studied numerically by Fei et al.}

[9], where it was shown that while kinks are generally transmitted through attractive impurities, velocity ranges exist where the kink is reflected instead. Later, they also found such behaviour in the sine-Gordon model [10].

Thus, we have two different situations (kink-antikink collisions on one hand and kink-impurity collisions on the other), both of which show resonance phenomena.

The next natural step in studying systems like these, is to consider kink-antikink collisions in the presence of an impurity, to see how the behaviour of the kinks changes due to interactions with the impurity. This will be the goal of the current work. The focus of this work has been to carry out numerical simulations to examine how the critical velocity and the location of bounce windows is influenced by impurity interactions. The next few sections are structured as follows. In sections 2 and 3, a description of the used model and numerical methods will be provided. Section 4 will cover the dependence of the critical velocity on the strength of the impurity, and the way the location of bounce windows changes will be outlined in section 5.

2

Numerical method

Non-topological kink collisions were modelled numerically in Mathematica (ver-sion 10.3.1.0) using equation (3), where the potential V (φ) is given by equation (11), and the initial conditions were

φ(x, t = 0) = tanh  ₁ √ 2 x + x0 √ 1 − v2  − tanh  ₁ √ 2 x − x0 √ 1 − v2  − 1, (12) φ(x, t = τ ) = tanh  ₁ √ 2 x + x0− vt √ 1 − v2  − tanh  ₁ √ 2 x − x0+ vt √ 1 − v2  − 1, (13)

which represents a kink and antikink, placed at positions −x0 and x0

respec-tively, colliding with the same velocity v. The term −1 was introduced to ensure φ(x, t = 0) and φ(x, t = τ ) go to −1 in the limit of large |x|. In equation (11), δ(x) represents the impurity, which is given by a Gaussian of variance σ2_,

centered around x = a:

δ(x) = 1 σ√2πe

−(x−a)2_/2σ2

. (14)

The factor µ in equation (11) now represents the area between the Gaussian curve and the x-axis.

Equation (3) was solved using the method of finite differences: ∂2φ ∂t2 = φ(x, t + τ ) − 2φ(x, t) + φ(x, t − τ ) τ2 , (15) ∂2_φ ∂x2 = φ(x + h, t) − 2φ(x, t) + φ(x − h, t) h2 , (16)

where τ and h represent steps in time and distance respectively, which are assumed to be small. Substituting equations (15) and (16) into equation (3) and solving for φ(x, t + τ ), we obtain

φ(x, t+τ ) = 2φ(x, t)−φ(x, t−τ )+τ 2 h2(φ(x+h, t)−2φ(x, t)+φ(x−h, t))−τ 2∂V ∂φ, (17) where φ(x, t + τ ) satisfies the equation of motion at time t + τ . A numerical solution of equation (3) is then constructed by recursively applying equation (17) to the initial conditions (equations (12) and (13)). Since equation (17) cannot be applied to the outer values of the interval being modelled, the outer points are dropped at each time step.

Conservation of energy was verified by numerically evaluating the expression

Z ∞ −∞ " 1 2  ∂φ ∂t 2 +1 2  ∂φ ∂x 2 + V (φ) # dx − Z tc 0  ∂φ ∂t ∂φ ∂x    xupper xlower dx. (18)

The first term is simply equation (5). The second term requires some clari-fication. Since the outer points of the modelled interval are dropped at each time step, the interval over which the simulation is valid is significantly smaller than the initial interval size. A significant part of the energy that was initially present, may propagate beyond the region of validity, where it will be deleted eventually. The second term in the above equation keeps track of the amount of energy that leaks out of the region of validity (the boundary points of which are indicated by xlower and xupper in equation (18)) in this manner.

During each simulation, equation (18) was evaluated using Mathematica’s built-in NIntegrate function. The built-integration method used was the global adaptive Monte-Carlo method.

The escape velocity of the kinks was estimated by calculating the position of the center of the kink at two different times after the collision. Let us denote these positions by x2 and x1 respectively. Furthermore, we define t2 and t1 as

the times at which x2and x1are calculated. The escape velocity was estimated

using

vout=

x2− x1

∆t , (19)

where ∆t = t2− t1. Note that the difference between t1and the time of collision

should be chosen sufficiently large to be able to neglect any further interactions between the kinks. This raises the question what values of t1 and t2 should be

chosen. During simulations, the velocity of the kinks was observed to change during escape due to interactions. Accordingly, t1and t2should be chosen such

that the kinks are sufficiently far away from each other to be able to neglect further interactions. That way, the velocity at infinity will not be significantly different from the one obtained using equation (19).

Another factor to be taken into account, is available hardware. Simulations of 200 units of time could be carried out in just over a minute. Computation time quickly grows beyond that point, such that simulations longer than 500 units of time were unfeasible.

Taking the above arguments into consideration, the following procedure was used. Each simulation was carried out for either 200 units of time, or until the center of the antikink reached x = 30. If the antikink passed x = 30 after 200 units of time, we would set t1= 150 and t2= 200. If the center of the antikink

did not reach x = 30 after 200 units of time, the simulation was continued until it did so. Then, x2 was set to 30, and t1 was set to 50 units of time before the

end of the simulation.

The initial velocity of the kink was varied, and vout was determined for each

collision where the kinks appeared to escape. The function vout=

p

a + bv + cv2 ₍₂₀₎

was then fitted to the data (where a, b, c are some parameters). The critical velocity is then given by the value of v for which the above function intersects the v-axis.

3

Dependence of the critical velocity on

impu-rity strength

The critical velocity was determined for several values 0 ≤ µ ≤ 0.5. Used parameters were x0= 20, σ = 1/10, a = 0, h = 0.02 and τ = 0.01. The vout,

v-diagram obtained for µ = 1/10 is given in figure 5, and a plot of obtained critical velocities is given in figure 6. The critical velocity vc is observed to decrease

with increasing µ, for 0 ≤ µ . 0.2. For higher µ, the critical velocity begins to rise again. The observation that vc can decrease with increasing µ is surprising.

It is easy to assume that the critical velocity always increases with µ, as the presence of an attractive object should make it harder for the impurities to escape.

Two possible mechanisms that could explain this behaviour have been investi-gated. The first proposed cause for the observed behaviour is that some part of

Figure 5: The escape velocity of a kink after a collision with an antikink while an impurity was located at the collision site (parameters µ = 1/10, a = 0, σ = 1/10), plotted against the initial velocity of the kink. The dots show data from simulations, while the solid line is the corresponding fit. The so-predicted critical velocity is 0.185.

Figure 6: The critical velocity vc as a function of µ. For each simulation, the parameters

the kinetic energy of the kinks is used to excite the impurity vibrational mode rather than the kink mode. Later on in the collision, some of the impurity mode energy is converted back to kinetic energy, giving the kinks enough kinetic en-ergy to escape. As the strength of the impurity is raised, the amount of enen-ergy exchanged between the impurity and the kinks increases, causing the critical velocity to decrease. As the impurity is made more attractive, the amount of energy it absorbs increases. At some point, the impurity absorbs a sufficient amount of energy to dominate the mechanism explained above, causing the critical velocity to increase again.

The second explanation is the following. The natural oscillation frequency of the impurity mode was found to be dependent on impurity strength. Therefore, depending on the strength of the impurity, when the kink-antikink pair has completed one full oscillation, the impurity mode may interfere constructively or destructively with the bion oscillation. Resonance between the two modes may then be responsible for the observed behaviour.

What the above two mechanisms have in common is that the kink and kink-impurity modes play a central role in collisions. The difference between the two is whether the impurity mode is excited near the critical velocity. Con-sider a kink-antikink-impurity collision where the kink velocity is just above the kink-antikink critical velocity. The first mechanism is based on suppression of kink mode excitation, and hence implies the collision will not excite the kink mode. On the other hand, if the second mechanism is correct, the kink mode should be excited. Therefore, to check which one of these two proposed mech-anisms is correct, the post-collision vibrational spectrum of the kink should be investigated. To this end, simulations of a kink-antikink, a kink-impurity and a kink-antikink-impurity collision were carried out with parameters v = 0.26, µ = 1/10, a = 0, σ = 1/10, x0= 20, h = 0.02 and τ = 0.01. In this way, the

collisions occurred 76.9 units of time after the start of the simulation. Then, the value of φ(x = 0, t) was calculated at regular time steps between t = 200 and t = 400, and a discrete Fourier transform was taken. The results are pre-sented in figure 7. The kink mode ω =p3/2 and its first two harmonics are given by the vertical orange lines. It can be seen from the plot that the kink mode is heavily excited in the kink-antikink collision. What is especially striking about figure 7 is that the kink-antikink-impurity spectrum shows a clear peak at the impurity vibrational mode, while the kink mode is not excited. This is consistent with the first mechanism described above, but contradicts the second proposed mechanism. For that reason, it is unlikely that the observed behaviour is caused by resonance between the kink and impurity modes.

Figure 7: Amplitude of the Fourier transform of φ(0, t), for a antikink (blue, solid), kink-impurity (green, dashed) and kink-antikink-kink-impurity (red, dotted) collision, for v = 0.26, x0=

20, a = 0, σ = 1/10, µ = 0.1, h = 0.02 and τ = 0.01. The vertical lines correspond to the kink mode ω =p3/2 and integer multiples thereof. Note that the impurity mode is heavily excited in the kink-antikink-impurity system, while the kink mode is not.

4

Location of bounce windows

In addition to the location of the critical velocity, the location of 2-bounce windows has been determined for several values of µ, where all other parameters have been taken the same as above. Bounce windows were located by simulating kink-antikink-impurity collisions, where the initial kink velocity was varied from 0.08 to the critical velocity for that particular value of µ, with steps of 0.0025. If a 2-bounce resonance was found for some value of v, the step size was reduced to determine the velocity range more precisely.

Bounce windows have been observed for µ ≤ 0.15, and some details of each bounce window are given in table 1. Note that bounce windows of width of around 0.01, 0.0045 and 0.0025 have been observed for every value of µ. Ac-cordingly, these bounce windows are believed to be the same. Consequentially, these results clearly show that the bounce windows occur at lower values of v for increasing µ. The lowest lying bounce windows found for µ = 0 have been observed before (see [1], for example), and has been consistently found to be the lowest lying one by other authors [2, 6, 14]. It is thus believed that, for impurity strength smaller than 0.15, no lower lying bounce windows exist than the ones reported here.

2-bounce windows as a function of µ, to see how their position changes. The location of a bounce window is however not well-defined, since a bounce window has finite width. A natural characteristic of bounce windows that might easily be compared, would be their center. The center of each bounce window found is given in the rightmost column of table 1, and a plot of the centers as a function of µ is given in figure 8. The position of bounce windows appears to decrease linearly with µ. This implies that bounce windows will disappear altogether if the impurity is made strong enough. Estimates of the impurity strength for which each bounce window disappears can be obtained using linear fits. These results are also given in figure 8. Using this procedure, the first, second and third lowest bounce windows are predicted to disappear at µ = 0.179, 0.0265 and 0.305 respectively. Naturally, in order to test these predictions, simulations have to be carried out for greater impurity strengths and smaller initial kink velocities.

Figure 8: The center of the three lowest lying bounce windows for each value of µ. For each simulation, the parameters x0 = 20, a = 0, σ = 1/10, h = 0.02 and τ = 0.01 were used. The

straight lines are given by vmid= 0.199 − 1.16µ (blue, solid), vmid= 0.225 − 0.885µ (orange,

dashed) and vmid= 0.236 − 0.776µ (green, dotted). The fits intersect the µ-axis at µ = 0.179,

µ vlow vhigh ∆v width center 0 0.193 0.203 0.0005 0.01 0.198 0.2245 0.2285 0.0005 0.004 0.227 0.2375 0.2395 0.0005 0.002 0.239 0.244 0.245 0.0005 0.001 0.2445 0.2525 0.2525 0.0005 0.2525 0.255 0.255 0.0005 0.255 0.025 0.1665 0.178 0.0005 0.0115 0.172 0.202 0.2065 0.0005 0.0045 0.204 0.216 0.2185 0.0005 0.0025 0.217 0.2325 0.2325 0.0005 0.2325 0.235 0.235 0.0005 0.235 0.05 0.14 0.15 0.0025 0.01 0.145 0.18 0.1845 0.001 0.0045 0.1823 0.195 0.1977 0.0003 0.0027 0.196 0.08 0.105 0.1175 0.0025 0.0125 0.111 0.153 0.1585 0.0005 0.0055 0.156 0.171 0.1735 0.0005 0.0025 0.172 0.1 0.082 0.09 0.002 0.008 0.086 0.1355 0.1415 0.0005 0.004 0.139 0.155 0.158 0.0005 0.003 0.157 0.1645 0.1662 0.0002 0.0017 0.165 0.17 0.171 0.0005 0.001 0.1705 0.15 0.0975 0.1025 0.0025 0.005 0.1 0.1215 0.124 0.0005 0.0025 0.1223 0.1325 0.134 0.0005 0.0015 0.1333 0.1475 0.1475 0.0005 0.1475

Table 1: All observed 2-bounce windows. For each simulation, the parameters x0 = 20, a =

0, σ = 1/10, h = 0.02 and τ = 0.01 were used. Here, vlowand vhigh indicate the lowest and

highest value of v found to lie inside that bounce window. The error margin is equal to the v step size used to probe that bounce window. The width is simply the difference between vhighand vlow. The center of each bounce window was calculated as (vlow+ vhigh)/2.

5

Kink-antikink trap

Now, we turn our attention to a different system that has been briefly studied, and which has yielded some results that are worth mentioning. Two strongly repulsive impurities were placed in the system, and a kink-antikink pair was placed between the impurities. The potential is now given by

V (φ) = 1 4(1 − φ 2₎2  1 − µ σ√2πe −(x−a)2_/2σ2 − µ σ√2πe −(x+a)2_/2σ2 , (21)

where the Gaussians again represent the impurities. The kinks were made to collide with v = 0.23, such that a bion would be formed for µ = 0. The goal of the simulations was the following. Strongly repulsive impurities can reflect most radiation sent their way. By carrying out these simulations, the influence of reflected radiation on bion behaviour was investigated.

When the system was modelled with v = 0.23, x0 = 10, σ = 1/5, a = 20

and µ = −8, the bion was found to have a natural oscillation time of around t = 7.14. The spacing between the impurities was then adjusted in such a way that the time it took for emitted radiation to make a round-trip between the impurity and the bion was an integer multiple of the time it took for the bion to complete a full oscillation. When a was adjusted to twice this amount, meaning a = 14.28, the bion was observed to split up into a kink-antikink pair after a few oscillations. This is shown in figure 9. These observations suggest that strongly repulsive impurities cause resonance between the oscillation of the bion and the radiation it emits. The value of a was varied in an attempt to observe this effect for a wider range of values. The chosen parameters were v = 0.23, x0 = 10,

µ = −8, σ = 1/5. We then varied a between 13.9 and 17.1, with steps of 0.1. The bion was found to be destroyed when 14.0 ≤ a ≤ 14.9 and for a = 15.4, while the kinks stayed bounded for other values of a.

These observations raise several questions. The first question arises naturally from the simulation shown in figure 9. There, the kink and antikink are ob-served to be reflected by the impurities. If the simulation were extended, the kinks would once again collide. If the resonance mechanism outlined above is correct, the kinks would briefly form a bion, which would then break apart again after several oscillations. To test whether the proposed mechanism is true, the simulation would therefore need to be extended.

The bion splitting up every time the kinks collide would suggest the impurities caused a novel kind of bounce window to have formed. Naturally, the initial velocity of the kinks should be varied to study when these resonances occur.

Figure 9: A kink-antikink collision occuring between two highly repulsive impurities, with v = 0.23. The kinks show resonance behaviour, while they would have formed a bion without the impurities present (see figure 3, top left). Used parameters were x0= 10, µ = −8, σ = 1/5,

a = 14.3, h = 0.02 and τ = 0.01.

6

Conclusion

Over the past few decades, the behaviour of kinks during collisions has turned out to be highly nontrivial. In this work, the influence of an impurity on the dynamics of kink-antikink collisions has been studied through numerical simula-tions. Specifically, the dependence of the critical velocity and 2-bounce window locations on the impurity strength has been determined, in the case where the impurity is centered at the collision point. A mechanism has been proposed to explain the behaviour of the critical velocity, and evidence supporting this mechanism has been found.

Based on the observation that the velocity for which 2-bounce windows occur decreases with increasing impurity strength, bounce windows have been hypoth-esized to disappear for sufficiently large impurity strengths. To prove or disprove this phenomenon, more work needs to be done. Namely, one has to simulate

kink collisions at lower initial velocities, so that the potential disappearance of bounce windows may be observed more directly. Furthermore, simulations of kink collisions with stronger impurities will have to be carried out to see whether the critical velocities will continue to increase. In addition, it would be inter-esting to examine what happens to the critical velocity and bounce windows for repulsive impurities. In addition to studying φ4 _{theory, the above questions}

may also be posed for other field theories with kink solutions, such as φ6_theory.

Then, kink-antikink collisions were considered between two highly repulsive im-purities. In that system, resonances were observed for an initial kink velocity for which a bion is usually formed. This suggests the formation of new bounce windows, where a kink-antikink pair repeatedly collide and separate after a few oscillations, to be reflected by the impurities and collide again. It would be in-teresting to extend described simulations, to see whether described resonances occur.

On the other hand, gaps in our current knowledge on these topics go beyond just the computational aspect. For one, no satisfying theoretical explanation for the existence of bounce windows has been formulated, and most models used to describe kink dynamics are phenomenological in nature. The development of more sophisticated theoretical models might therefore lead to a deeper un-derstanding of kink dynamics, and of classical field theory as a whole. Finally, the applicability of the current model to microscopic systems is limited by the fact that thermal effects have been neglected. By including thermal effects into this model, results obtained will have more direct relevance to the theory of condensed matter physics.

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Appendix A: Higher order bounce windows

Several third, fourth and fifth order bounce windows were found during the simulations described in section 3. While these weren’t used anywhere in the main text, their locations are reproduced here. The table below gives the values of µ and v for which bounce windows were found, along with the number of bounces. The parameters used for all simulations are x0= 20, a = 0, σ = 1/10,

h = 0.02 and τ = 0.01. µ v number of bounces 0 0.204 3 0.224 3 0.237 4 0.2475 3 0.025 0.165 3 0.18 3 0.2 3 0.2015 3 0.207 3 0.208 3 0.2155 3 0.225 3 0.23 4 0.233 5 µ v number of bounces 0.025 0.2355 4 0.2375 3 0.05 0.135 3 0.194 4 0.2025 3 0.2175 3 0.08 0.15 3 0.1525 4 0.16 3 0.1705 3 0.1 0.1 3 0.142 3 0.1642 3 0.1665 3 0.1666 3 0.1825 3