Properties of kinks and kink-impurity interactions in phi^4 theory

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Universiteit van Amsterdam

Report bachelor project Physics & Astronomy, size 15

EC

Conducted between 03-03-2018 and 29-06-2018

Properties of impurities and

kink-impurity interactions in φ

4

theory

Author:

Stan de Lange

Student number: 11015233

University of Amsterdam

Science Park 904

PO box 94216

1090 GE Amsterdam

(Receit date)

Supervisor:

Dhr. dr. Jasper van Wezel

Examinator:

Prof. dr. Jean-S´

_{ebastien Caux}

Daily supervisor:

Mariya Lizunova MSc.

Institute for Theoretical Physics

Faculty of Science University of

Amsterdam

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Abstract

We have investigated behaviour of kinks in a disturbed φ4 potential. The sharp Gaussian impurities affect the trajectory of kinks and oscillating modes can form on positive modes in particular. Using numerical simulations, we have examined the relations between quantities such as kink velocity, mode oscillation, frequency and energy. We also discuss energy-related phenomena: resonant impurity reflection and phase-dependent kink capture and escape.

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Dutch summary

In een systeem met twee stabiele toestanden is een kink is een lokale transitie van de ene naar de andere toestand. Bijvoorbeeld: In een ferromagneet wil een atoom spin-omhoog hebben als zijn buren dat ook hebben. Dit zorgt ervoor dat er in een ferromagneet grote gebieden zijn waarin alle spins of omhoog of omlaag staan. De overgangen tussen deze gebieden heten kinks.

Kinks kunnen ook vrij bewegen door het medium en ze kunnen worden be¨ınvloed door onzuiv-erheden die in het medium zitten. In een kristal, bijvoorbeeld, is dit een atoom wat er niet in thuishoort. Als een kink over een onzuiverheid glijdt, wordt het erdoor aangetrokken of afgestoten. In dit onderzoek zijn verschillende aspecten van deze interacties onderzocht.

Daarnaast kunnen aantrekkende onzuiverheden ook modes hebben. Zo’n mode is een oscillerende beweging rondom de onzuiverheid. Ze trillen met een bepaalde frequentie en amplitude en bevatten energie die ze kunnen uitwisselen met kinks. Energie-uitwisselingen tussen de mode en de kink kunnen bepalen of een kink al dan niet ontsnapt van de onzuiverheid.

English summary

In a system with two stable states, a kink is a local transition from one state to the other. For example: An atom in the lattice of a ferromagnet wants to be spin-up if its neighbours are spin-up as well. This results in areas within the crystal wherein all spins point up, or all spins point down. The transitions between these areas are called kinks.

Kinks can freely move through the medium and they can be affected by impurities in the medium. For example, in a crystal, this may be an atom that doesn’t belong there. A kink may be attracted or repelled by such an impurity. A variety of aspects of these interactions are investigated in this research.

Impurities can also have impurity modes. Such a mode is an oscillatory motion at the site of the impurity. They oscillate with a certain frequency and amplitude, and they contain energy, which can be exchanged with kinks. Energy exchanges between the impurity mode and the kink may determine whether a kink escapes from the attractive power of the impurity.

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1 Introduction

The study of solitons includes a research area known as φ4 _{theory [1]. Kinks, the static,}

non-topological solutions to the corresponding equation of motion, may encounter potential impurities when travelling through the medium. These kinks are found in a diverse set of topics, such as cosmological inflation [2], defects in crystals [3], charge waves [4] and buckled graphene [5]. Due to their ubiquitous presence in physics, it is important that their properties and behaviours and properties are understood. In this paper we perform numerical simulations in order to study kinks in a perturbed φ4_{potential to see what happens when kinks, impurities and impurity modes}

interact, namely considering the next quantities: critical velocities, reflection, transmission and escape velocities, impurity mode frequency and amplitude, and energy distributions.

2 Theory

The (1+1)-dimensional single real scalar field φ(x, t) is given by the following Lagrangian density [1] (using Einstein sum notation where µ ∈ {0, 1}):

L = 1 2(∂µφ)

2_{− V (φ),} ₍₁₎

where V (φ) is the self-interaction of the field φ. From this Lagrangian the following equation of motion is derived:

φtt− φxx+

dV

dφ = 0. (2)

The total energy of a static solution of equation (2) may be calculated with the following functional:

E[φ] = Z ∞ −∞ 1 2 ∂φ ∂t 2 +1 2 ∂φ ∂x 2 + V (φ) ! dx. (3)

To find the energy in a specific region, one integrates not over R but over the desired interval.

In particular, φ4_{theory concerns the following field potential:}

V (φ) = 1 4(1 − φ

2₎2_. ₍₄₎

By substituting equation (4) into equation (2), we obtain the equation of motion:

φtt− φxx+ φ3− φ = 0. (5)

By assuming φ to be a constant we find two topological, stable solutions: φ+ = 1 and φ− = −1;

and one unstable: φ0= 0. Equation (5) has stationary non-topological solutions:

φ(x) = ± tanh x − x√ 0 2

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These solutions are solitary waves with the center x = x0. The plus signed solution is called a kink

(φK) and the minus signed solution an antikink (φ_K¯). Since equation (5) is Lorentz invariant,

kinks may be Lorentz boosted to a velocity v (c ≡ 1) and can be rewritten as:

φ(x, t) = ± tanh x − x0− vt p2(1 − v2₎

!

. (7)

Note that in the limits x → ±∞, we see: φK → φ±. Thus, a kink is a mobile transition between

the two constant solutions. We can add a small perturbation η(x, t) to the kink to investigate its stability:

φ(x, t) = φK(x, t) + η(x), η(x, t) = χ(x)eiωt. (8)

The substitution of equation (8) into equation (2) yields:

−1 2 d2 dx2 + 1 2 d2_V dφ2 _φ K ! χ(x) = 3 2ω 2_χ(x). ₍₉₎

(Due to small η we keep only linear orders.) Equation (9) has two solutions that correspond to bound states: χ0(x) = 9 8 1/4 sech2 x √ 2 , (10)

the translation mode, with ω0 = 0 (the ground state); and:

χ1(x) = 9 8 1/4 tanh _x √ 2 cosh−1 _x √ 2 , (11)

the vibrational mode, with ω1=p3/2 (the excited state). These modes provide a mechanism for

resonant energy transfer that is necessary to explain reflection of a kink by an attractive impurity (section 7). The same mechanism accounts for “bounce windows” in kink-antikink interactions [6].

A single localized impurity in the potential may be introduced by amending the potential in the equation of motion:

φtt− φxx+ (φ3− φ)(1 − δ(x − a)) = 0. (12)

Here, δ is the Dirac delta function, a is the position of the impurity and is its strength that can be either positive or negative. Actually, its effects on the behaviour of are the focus of current research. Naturally, any number of impurities can be considered.

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Figure 1: A typical kink-impurity event. The kink came from the left with v = 0.6 and passed the impurity ( = 0.8, a = 0, indicated by the red dot) leaving behind an oscillating mode at a = 0, as well as scattering off some waves to either side.

3 Method

In order to study kink-impurity interactions, we created simulations by numerically solving the equation of motion. So, these calculations were performed in Mathematica 10.3, using one of two methods. The first is the NDSolve function, which solves the equation of motion for φ(x, t) with determined initial configuration φ(x, 0), its first time derivative φ0(x, 0) and the boundary conditions φ(±100, t) = ±1.

However, Mathematica is highly adaptive and its choice of solution methods, as well as the value of parameters such as integration step size, may depend on the hardware it is being runned on. Since reproducibility is desired, many events are instead simulated with selfmade modules. In these modules the partial differential equation at hand is solved using the explicit Euler method. A precise description of the application of this technique is presented in Appendix A.

4 Parameters and conventions

Here, one may find some clarification for the names of parameters used in the simulations. The impurity, ideally a Dirac delta function, is approximated by a Gaussian distribution:

δ(x − a) = 1 σ√2πexp h −1 2 x − a σ 2 i . (13)

In kink-impurity simulations we set standard deviation (or width) σ = 0.1: although smaller val-ues will resemble a Dirac delta more closely, this value is chosen due to the step size: making σ any smaller would make the impurity unresolvable. (The prefactor, referred to as the impurity strength , is mostly studied between values -2.0 and 1.2.) The center of the Gaussian distribution is called a. The parameter stepsize is the discrete spacing of x-values used for solving the partial

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differential equation. The steps taken in time have size stepsize/2.

Finally, the standard experimental setup (for instance the ones performed to obtain figure 2) assumes x0= −8, a = 0, σ = 0.1 and stepsize = 0.05 or stepsize = 0.02.

5 Impurity strength

The most notable effect in a kink-impurity interaction originates from the sign of . An impurity with positive (called a “positive” impurity for short) is attractive, whereas negative makes the (“negative”) impurity repulsive.

If the kink, travelling at velocity v, encounters a positive impurity, it will be drawn to it and may either pass through it (transmission) or it is unable to escape (capture). The minimal velocity required to escape an impurity is called the critical velocity (vcr). As can be seen in figure 2, a

greater impurity strength results in a higher critical velocity. If the kink is transmitted, it will continue its path, somewhat disturbed with a lower velocity. If the velocity of the kink is sub-critical, it will pass the impurity for some distance but then reverses its direction. It will oscillate around the impurity and continually lose kinetic energy. In rare cases, the kink may be captured but escape on its way back [7]: this “resonant reflection phenomenon” will be elaborated on.

When the kink interacts with a negative impurity, it will either pass through it (transmission) or bounce off and turn back (reflection). In both cases, the kinetic energy loss (measured as loss in velocity) is considerably lower for negative than it is for positive . The small energy loss for negative impurity interactions can be attributed to the fact that impurities can’t form on negative impurities.

Figure 2: The critical velocity v for a kink required to pass an impurity to be transmitted for negative (orange) and positive (blue). The point (0,0) is featured (black) since vcr→ 0 for → 0. stepsize = 0.02

restricts v to be calculated in steps of 0.02, which yields error margins ∆vcr= ±0.01. Due to their small

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We mapped velocity loss as a function of ingoing velocity and in figure 3. Evidently velocity loss (and thus energy loss) due to negative impurity scattering is quite small. In fact the greatest speed loss for positive is vrel = 0.92 for vin= 0.6, = −0.9. For negative , the greatest speed

loss is vrel = −0.95 for vin= 0.56 = −0.1. It appears that strong impurities cause the greatest

relative speed loss. Moreover, faster kinks lose relatively more speed than slower kinks.

Figure 3: The relative outgoing velocity vout/vin as a function of vin, the ingoing velocity, and . vin

is measured between 0.3 and 0.6 with steps of 0.02 in between, and is measured between 0.1 and 1.0 with steps of 0.1 in between. The outgoing velocity, vout is measured by calculating the time difference δt

between the passages of x = ±11 and x = ±10 (the sign depending on whether the kink is transmitted or reflected): vout= 1/δt.

6 Impurity width

Simulations show that for both negative and positive impurities, the critical velocity decreases as the width nearly linearly increases, see figure 4. This negative correlation is to be expected, since making the impurity wider implies lowering the peak, thus having a less powerful maximum. Using a linear fit, we find vcr(σ = 0, = 8) = 0.59 and vcr(σ = 0, = −8) = 0.42. It should be

noted that there is no data for σ > 0.6 due to the great simulation lengths required. It is expected that vcr→ 0 for σ → ∞: this obviously violates the proposed linear trend.

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Figure 4: Critical velocity is dependent on the standard deviation of the impurity: as the impurity becomes wider, the critical velocity becomes lower. Error bars ∆vcr = ±0.01 are displayed.

7 Resonant reflection

There are specific subcritical velocities for which a kink may escape a positive impurity due to resonant energy transfer [7]. Kinetic energy gets transferred from the translational mode to the internal mode and the impurity mode. Due to this, the kink may not have sufficient kinetic energy to escape the impurity, and reverses its path. Then, if there is resonance and energy is transferred back into the translational mode, it escapes, and it is effectively reflected. In fact, the kink can bounce several times before escaping either backwards or forwards. This phenomenon is similar to bounce windows found in kink-antikink interactions [6].

Four cases of resonant reflection have been identified here. Figure 5 is an example of this, with v = 0.15 and = 0.50. This result is in accordance with [7]. Higher-order resonances have not been found.

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Figure 5: The plot of φ(a, t) with v = 0.15 and = 0.5. A resonant energy reflection of the kink by a positive impurity, namely the kink does not have enough speed to be transmitted, but it escapes on its way back.

Since critical velocity depends on σ, we have varied this parameter. Two cases of reflection have been identified for σ = 0.1 and one for σ = 0.2 and σ = 0.5. Making σ any smaller was not possible due to computational limits on step size and simulation runtime. More plots can be found in Appendix B.

8 Impurity modes

When the kink passes over a positive impurity, it leaves behind an oscillating mode at the site of the impurity x = a. It has several physically relevant properties, such as frequency, amplitude and width, as well as its decay over time.

With no known analytic solution to the equation of motion, there are two ways to create a mode: it can be created simply by letting a kink “run through” the impurity, or by finding a function that approximates the mode. A pair of overlapping kink and antikink works well as an approximate initial function [1], see equation (14). (The “±1” term at the end is to ensure that the mode is on a stable level.) Its properties have been studied and compared to those of a “natural” mode. We have checked whether the velocity of the kink affects the oscillation frequency of the mode in some way. For details see figure 6.

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Figure 6: The velocity with which the kink runs through the impurity has no influence on the frequency of the mode impurity formed.

Using equation (14) and varying we found an unexpected relationship between the frequency of the mode and its amplitude, see figure 7b. For low frequency (f = 0.144), the amplitude reaches a value of 1.13, at f = 0.19 it peaks and then quickly drops for higher frequency. Although this suggests a unique relation between frequency and amplitude (and energy), experiments with a kink and a pre-existing impurity mode suggest that far greater amplitudes are possible (see figure 11).

As for impurity modes created on an impurity that had no mode before the event, these strong correlations indicate these are defined equivalently by their energy, frequency and amplitude; and since the frequency of a mode is constant and independent of v, we can conclude that such impu-rity modes are uniquely defined for given .

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(a) Mode frequency as a function of impurity strength

. (b) Amplitude as a function of mode frequency f .

(c) Energy E as a function of mode frequency f .

Figure 7: Relationships between mode frequency, mode amplitude and mode energy. The point at the red end of the spectrum, to the left, corresponds to = 1.2 and the one at the violet end, to the right, corresponds to = 0.1, taking steps of size 0.1 in between. a = 0. f is calculated by finding the 9th and 13th peaks of φ(0, t) (located at t9 and t13 respectively) using the built-in Mathematica function

FindMaxArg. (These specific maxima are chosen as a compromise between accuracy (a longer simulation would allow the mode to assume a more constant shape) and simulation duration, while averaging over multiple peaks yields a more accurate result.) Then, f = 4/(t13− t9). E is calculated using equation (3)

over the interval [−4, 4]. The orange dashed lines indicate the kink eigenfrequency f = (2π)−1p3/2.

Figure 8 shows the decay of a mode over time. This is an extremely slow process, which justifies measuring the frequency, ampitude and energy between the ninth and thirteenth maxima.

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Figure 8: The amplitude of a mode as a function of time created by a kink (v = 0.6) going through an impurity with = 0.7. In this simulation, a = 3. The orange dashed lines are for reference and correspond to the third local maximum and minimum.

Hitherto we have researched kink-modeless impurity interactions, but the kink can also interact with impurity modes (this may happen for example because a different (anti)kink ran through this mode earlier). This effect is notable in the following experiment: the kink is first transmitted by a positive impurity, leaving behind a mode and having its speed decreased. Then, it is reflected by a strong negative impurity. On its way back, it encounters the positive impurity again and will either be transmitted or captured, depending on the phase of the mode. In the experiment, the only varied parameter is the spacing between the impurities, which is done in order to delay the kink and thus to vary the mode phase.

We have found phase-dependent capture phenomena for a variety of parameters. The setup consists of two impurities, located at x = ±a. The impurity to the left has strength l= 0.8 and the one

to the right has strength r= −0.7. The kink (v = 0.6) comes in from the left. We found that the

kink will escape this configuration for 4.00 ≤ a ≤ 4.85 but will be captured for 4.90 ≤ a ≤ 5.00, see figure 10. Capture also occurs for a = 3.75, so that the impurity makes one fewer cycle. Furthermore, the capture event also occurs when the left impurity has its strength increased to l= 0.9, as well as for l= 0.7 if v = 0.5. Graphs of these events can be found in Appendix B.

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(a) a = 4 (b) a = 5

Figure 9: φ(x, t) at x = −a (the location of the attractive impurity). In 9a, the kink leaves behind a mode as it escapes to the left (at t = 45). In 9b, the kink is trapped, thus displaying large oscillations.

Taking a more artificial approach, we simulate an impurity mode by means of equation (14). The kink now encouters the impurity with the mode already in place. We observe the same effect.

(a) x0= −10.8 (b) x0= −10

Figure 10: φ(x, t) at a = 0 (the location of the attractive impurity). In 10a, the kink leaves behind a mode as it escapes to the left (at t = 33). In 10b, the kink is trapped, displaying large oscillations.

As mentioned in section 8, it is possible to create impurity modes with amplitudes as great as 0.3 by running the kink through an impurity with a mode, see figure 11. The stability of these impurity modes needs yet to be investigated, as this mode was not isolated but it was located to a slow-moving kink.

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Figure 11: A kink encounters an impurity mode and subsequently leaves behind an unusually large-amplitude impurity mode.

It is uncertain whether a kink can be “boosted” by an impurity mode. It may be possible that energy is transferred from the impurity mode to the kink’s translational mode, thus increasing it speed somewhat. Using the same setup as above, one case has been found where the outgoing velocity was nearly the same as the incoming velocity, but generally energy ensures a velocity drop, or even capture, rather than a gain.

9 Discussion

To ensure accuracy, we measured the total energy of the system for select simulations, typically those with the most extreme parameter values, since these are expected to be the least accurate. In no case did the total energy vary by more than 0.05%. As mentioned in section 7, the reflection windows we found are in accordance with earlier findings [7]. Although the data is accurate, more reflection windows can be found if a smaller step size is used. The error bars in figures 2 and 4 can be reduced by decreasing the step size as well.

10 Conclusion

We have studied kink-impurity interaction and a variety of properties of impurity modes. Positive impurities are attractive, whereas negative impurities are repulsive, and both have an associated critical velocity that is dependent on , as well as on σ. Impurity modes oscillate dampen ex-tremely slowly and are characterized by their frequency.

When a single kink encounters a positive impurity on its path, it is possible for the kink to be reflected. This has been shown to occur for various impurity widths. The phase of a mode is important in the interaction of a kink with an impurity, and may determine whether the kink escapes or will be trapped.

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kink-impurity event, their amplitude, frequency and energy are related and depend on . Contrary to approximations made in the literature, there seems to be only one impurity mode. However, if the kink interacts with a pre-existing mode, it is possible to create a mode with a much greater amplitude.

There is more work to be done. More computational power would allow us to expand our knowl-edge in multiple ways. Firstly, data on low-strength ( < 0.4, see figure 2) have been neglected due to overly lengthy simulations. Also, simulations broke down for > 1.2: the great spatial change in φ near the impurity causes numerical instability, see figure 12. However, there may yet be valuable information to found in this range. Smaller stepsize would yield simulations with smaller σ as well, thus providing for a more accurate approximation of a Dirac delta function.

Figure 12: The kink (v = 0.6) has interacted with the impurity ( = 1.3) and the peak of the mode it created accelerates to −∞. Shortly after, the simulation crashes.

Follow-up research on kink-impurity mode interactions may prove fruitful. Although the nature of impurity modes seems relatively straightforward if they are created when a kink runs through an impurity without a mode, matters become more complicated if an impurity mode is present. Be-sides performing simulations, constructing a theoretical framework for modes will be an important step towards understanding these phenomena.

Appendix A: Euler method

The function φ(x, 0) would be spatially discretized with step size h over a wide range, typically x ∈ [−100, 100]. The first step in time may be obtained using the time-derivative of the kink:

∂φ ∂t _t=0= −v sech x − x0− vt p2(1 − v2₎ !2 · 1 p2(1 − v)2. (15)

Using the discretized differential operators (τ being the temporal step size, h being the spatial step size, and k and j representing the discrete position in time and space, respectively):

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φtt= φk+1_j − 2φk j + φ k−1 j τ , φxx= φk j+1− 2φkj+ φkj−1 h . (16)

Substituting this in equation (2) and rewriting it gives an expression for the next step at any spatial point j: φk+1_j = 2φk_j − φk−1 k + τ2 h2(φ k j+1− 2φ k j + φ k j−1) − τ 2 dV dφ k j . (17)

In order for the explicit method to be stable [8], we must pick 0 < τ2/h2≤ 1. In our simulations τ = h/2.

Since every point j is dependent on the value of the neighboring values j − 1 and j + 1 of the previous time step, and the outermost points do not have two neighbors, these two points must be discarded at every time step. Starting off with an interval of length l consisting of N = 2l/h steps and making M = tmax/τ steps in time, one would obtain a reliable interval of length len:

len = h

2(N − 2M ). (18) In Mathematica, lists of values of φ at one step in time are iteratively created and used for the construction of the next list. Once a list need not be used anymore to calculate values for the next time step, the unreliable points at both ends are discarded. The calculation is now complete and the ListInterpolate function is used to provide smooth graphical representations of the data. The interpolated function is also applied to equation (3) to obtain the energy.

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Appendix B: Miscellaneous graphs

(a) v = 0.35, = 0.78, σ = 0.1 (b) v = 0.25, = 0.66, σ = 0.2

(c) v = 0.20, = 0.92, σ = 0.5

Figure 13: Three more cases of a kink being reflected by a positive impurity. Figure 13b shows the excitation and deexcitation of the impurity mode most clearly.

Figure 13 shows three more kink reflections. Apparently, reflection windows do not only exist for σ = 0.1 (figures 5 and 13a), but also for σ = 0.2 (figure 13b) and σ = 0.5 (figure 13c). We suspect that reflection windows also exist for σ > 0.5: there is no reason to assume that this phenomena suddenly disappears. No reflections have been found for σ = 1, although more precise measurements may yield such an event.

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(a) v = 0.6, a = 3.75, l= 0.8 (b) v = 0.6, a = 5, l= 0.9

(c) v = 0.5, a = 5, l= 0.7

Figure 14: Three more cases of impurity mode phase-dependent capture besides the ones presented in figure 10. These have slightly different values for a, v and l, as indicated below their respective graphs.

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Appendix C: Table of parameter values

Graph no. Method x0 a stepsize σ d v Boundaries

2 Euler -8 0 Variable [-2.0, 1.0] 0.02 for 0.4 ≤ ≤ 0.7

0.05 otherwise 0.1 - Variable [-60, 60]

3 Euler -8 0 Variable 0.05 0.1 - Variable [-100, 100]

4 NDSolve ( = −8/10),

Euler ( = 8/10) -8 0 -8, 8 0.02 Variable [0.1, 1.0] - Variable [-60, 60]

5 Euler -8 0 0.50 0.05 0.1 - 0.15 [-60, 60] 6 Euler -8 0 0.6 0.05 0.1 - Variable [0.25, 0.9] [-80, 80] 7a Euler 0 0 Variable [0.1, 1.2] 0.05 0.1 d = 0.1 - [-100, 100] 7b Euler 0 0 Variable [0.1, 1.2] 0.05 0.1 d = 0.1 - [-100, 100] 7c Euler 0 0 Variable [0.1, 1.2] 0.05 0.1 d = 0.1 - [-100, 100] 8 Euler -8 3 0.7 0.05 0.1 - 0.6 [-60, 60] 9a Euler -8 4 {0.8, -0.7} 0.05 0.1 - 0.6 [-100, 100] 9b Euler -8 5 {0.8, -0.7} 0.05 0.1 - 0.6 [-100, 100] 10a Euler -10.8 0 0.8 0.05 0.1 d = 0.1 0.3 [-100, 100] 10b Euler -10 0 0.8 0.05 0.1 d = 0.1 0.3 [-100, 100] 11 Euler -7.6 0 0.8 0.05 0.1 d = 0.1 0.5 [-100, 100] 12 Euler -8 0 1.3 0.02 0.1 - 0.6 [-60, 60] 13a Euler -8 0 0.78 0.05 0.1 - 0.35 [-60, 60] 13b Euler -8 0 0.66 0.05 0.2 - 0.25 [-60, 60] 13c Euler -8 0 0.92 0.05 0.5 - 0.20 [-60, 60] 14a Euler -8 {0.8, -0.7} 0.6 0.05 0.1 - 0.6 [-100, 100] 14b Euler -8 {0.9, -0.7} 0.6 0.05 0.1 - 0.6 [-100, 100] 14c Euler -8 {0.7, -0.7} 0.6 0.05 0.1 - 0.5 [-100, 100]

Table 1: The parameters used for the plots. If a parameter is called “Variable” followed by an interval (e.g. “Variable [0.1, 1.0]” for σ, that means that values in that interval are used for the calculations. If no interval follows “Variable”, that means the value is chosen precisely to get the desired effect (e.g. the critical velocity plot requires a velocity input but this velocity depends on the occurence of escape or capture). For plots with two impurities, the one to the left is located at x = −a and the one to the right is at x = a. If two values of are given in curly brackets, that means {l, r}.

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