#  Collisions between kinks with long-range tails: a simple and efficient
method

João G. F. Campos and Azadeh Mohammadi  Departamento de Física, Universidade
Federal de Pernambuco, Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901,
Brazil

###### Abstract

We construct initial configurations for the scattering between kinks with
long-range tails. For this purpose, we exploit kink solutions in the presence
of BPS-preserving impurities. This approach offers a highly efficient method
and effortless implementation with a negligible computational cost. Our
algorithm has a much smaller complexity than the usual minimization method,
becoming more than a hundred times faster in some scenarios. Consequently,
conducting kink-antikink simulations becomes remarkably straightforward.

††preprint: APS/123-QED

Introduction. In recent years, significant attention has been devoted to
exploring the interactions of kinks, uncovering intriguing phenomena in the
process. Notable contributions include the investigation of kink interactions
with long-range tails [1], the discovery of spectral walls within kink
interactions [2], and the computation of a collective coordinate model for
$\phi^{4}$ kink interactions [3]. These studies have provided valuable
insights into non-perturbative aspects of field theories, spontaneous symmetry
breaking, the dynamics of models with effectively one spatial dimension, and
the behavior of higher-dimensional topological and non-topological structures.
In particular, kinks with long-range tails emerge in field theories when a
certain potential minimum lacks a mass term and, consequently, exhibits no
characteristic length scale. Understanding such systems is a challenging
research field currently under development (see Ref. [4] and references
therein).

The study of interactions between long-range kinks presents a challenge due to
the limitations of usual initial condition approximations. For instance, the
additive ansatz suggests that a kink-antikink configuration can be
approximated by summing the individual kink profiles when they are
sufficiently separated. However, when the kink’s tail decays following a power
law, neighboring kinks do not exhibit a negligible superposition because a
power-law decay lacks a finite range. As a result, the kinks are never truly
well-separated. Therefore, the choice of initial conditions in direct
numerical simulations of long-range kinks’ interactions plays a crucial role
in the observed behavior. Initializing the collision with the standard
additive ansatz may introduce unwanted initial energy, which converts into
radiation. It results in a wrong magnitude and even the sign of the force.
Hence, it has been demonstrated in previous works Refs. [5, 1] that
conventional methods for computing the force between long-range kinks must be
adapted in such cases.

Performing the numerical simulation of kinks’ scattering requires specialized
methods in the long-range regime [6, 7, 8]. The first correct simulation of
the scattering between kinks with long-range tails was performed in Ref. [6].
To that end, the authors developed a new method to construct kink-antikink
configurations. The correct initial configuration $\phi(x,t=0)$ was found by
requiring that it satisfies the static equation of motion as closely as
possible. More explicitly, the following functional was minimized at $t=0$

$I[\phi]=\lVert\phi^{\prime\prime}-V^{\prime}(\phi)\rVert_{2}^{2}+\text{constraints},$
(1)

where $\lVert.\rVert_{2}^{2}$ denotes the two-norm squared of the function.
The constraints are needed to ensure that the kinks’ centers are approximately
fixed in the minimization process. The method was shown to work very well for
the kinks initially at rest. In Ref. [7], the same authors generalized the
method for the case where the kinks have finite initial velocity. It was
achieved by including a second minimization layer for the field
$\chi(x,t)=\dot{\phi}(x,t)$, namely minimizing the following functional at
$t=0$

$J_{1}[\chi]=\lVert\dot{\chi}-\phi^{\prime\prime}+V^{\prime}(\phi)\rVert_{2}^{2}+\text{constraints}.$
(2)

As the functional depends on the time derivative, it was necessary to
integrate the equations of motion in a small time interval at every
minimization step and to take the two-norm. Therefore, it was more
computationally costly than the first minimization layer.

Inspired by the works mentioned earlier, we showed in Ref. [8] that the second
layer of minimization could be performed instead by requiring the field to
obey the zero mode equation as closely as possible. More formally, the
following functional is minimized at $t=0$

$J_{2}[\chi]=\lVert(1-v^{2})\chi^{\prime\prime}-V^{\prime\prime}(\phi)\chi\rVert_{2}^{2}+\text{constraints}.$
(3)

It is a generalization of the single kink case, where $\chi$ is proportional
to the zero mode, related to the model’s translation symmetry. In other words,
the field $\chi$ should be proportional to a generalized translation mode for
multiple kinks. This method is much less computationally costly than the one
presented in Ref. [7] as it does not need integration at every minimization
step.

The issue with minimization methods, however, is that they are quite costly in
general. Although it is possible to perform them on a large scale [9], they
require considerable computational time. In this letter, we propose a new
method to construct kink-antikink initial configurations for long-range kinks
with negligible computational time. To aim for this purpose, we will exploit
impurities that preserve half the system’s Bogomol’nyi-Prasad-Sommerfield
(BPS) property [10, 2].

Half-BPS impurity theories. Consider the following scalar field theory in
(1+1) dimensions

$\mathcal{L}=\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-V(\phi).$ (4)

If the potential is non-negative and contains multiple degenerate vacua, the
model exhibits kink solutions $\phi_{K}(x)$, described by the BPS equation

$\frac{d\phi_{K}}{dx}=\pm W(\phi_{K}),$ (5)

where $W(\phi)=\sqrt{2V(\phi)}$. We fix the center of the kink $\phi_{0}$ at
$-x_{0}$, i.e., $\phi_{K}(-x_{0})=\phi_{0}$. To obtain a kink solution, we
assume that $\phi_{-}<\phi_{0}<\phi_{+}$, where $\phi_{-}$ and $\phi_{+}$ are
two neighboring potential minima.

Now, consider the related field theory

$\mathcal{L}=\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}[\phi_{x}+\sigma(x;\bm{\alpha})W(\phi)]^{2},$
(6)

where $\sigma(x;\bm{\alpha})$ is an impurity containing $p$ free parameters
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{p})$. After its inclusion,
only one BPS equation is preserved. We are interested in the BPS solution
$\phi_{K,\sigma}(x;\bm{\alpha})$, defined by the following equation

$\frac{d\phi_{K,\sigma}}{dx}=\sigma(x;\bm{\alpha})W(\phi_{K,\sigma}),$ (7)

with the condition $\phi_{K,\sigma}(-x_{0};\bm{\alpha})=\phi_{0}$. In terms of
the $\phi_{K}(x)$ and $\sigma(x;\bm{\alpha})$ functions, it reads [11]

$\phi_{K,\sigma}=\phi_{K}(\xi(x;\bm{\alpha})),\quad\xi=-x_{0}+\int_{-x_{0}}^{x}\sigma(x^{\prime};\bm{\alpha})dx^{\prime}.$
(8)

Due to the BPS property, the model possesses a generalized translation
symmetry [10] with the associated zero mode
$\phi_{K}^{\prime}(\xi(x;\alpha))$, where the derivative is with respect to
the function’s argument.

The New Idea. In Ref. [12], the authors proposed that BPS solutions with
specific impurities could describe field configurations containing multiple
kinks. We aim to use such profiles as actual initial data to simulate kink
collisions. We discovered that $\phi_{K,\sigma}(x;\bm{\alpha})$ is an
excellent approximation for the initial condition, even for the long-range
kinks, for some choices of $\sigma$ with an optimal $\bm{\alpha}$.

Our construction offers two significant improvements. Firstly, only a few
parameters require adjustment, which can be accomplished within a negligible
computational time. Secondly, there is no need to add any constraint in the
minimization function if one chooses $\sigma$ appropriately. The BPS equation
already fixes the center of the kink located at $-x_{0}$, and the center of
the opposing kink will automatically be fixed for all $\alpha$ by our choice
of the impurity $\sigma$.

The remaining initial condition, the velocity field $\dot{\phi}$, which we
have already defined as $\chi$, can be similarly obtained at $t=0$. We
consider a family of generalized translation modes of the impurity model.
Namely, we have

$\chi_{K,\sigma}(x;\bm{\beta})=-v\phi_{K}^{\prime}(\xi(x;\bm{\beta})).$ (9)

For some impurities, the BPS solution has a lump character, resembling a kink-
antikink configuration. Changing the integration constant of the BPS solution,
which moves the system in the moduli space, can either increase or decrease
the lump size, moving the kink and the antikink in opposite directions [13].
Hence, the initial velocity field of the original problem can be approximated
remarkably well by $\chi_{K,\sigma}(x;\bm{\beta})$ for some choices of
$\sigma$ with an optimal $\bm{\beta}$.

It is worth mentioning that there is no need to consider the same impurity for
$\phi$ and $\chi$. However, we consider the same impurity $\sigma$ for both
for simplicity. The value of the optimal parameters $\alpha$ and $\beta$ are
not generally equal, which is the case for the models we study here.

Kink-Antikink Collisions. Let us start with a simple impurity profile, denoted
by $\sigma(x;\alpha)=\tanh(\alpha x)$ with one free parameter $\alpha$. We
will refer to the corresponding BPS solution as $\phi_{KA}(x;\alpha)$. The
reason for the notation is clear; it closely resembles a kink-antikink
configuration [12, 14]. Our proposed method involves incorporating a gamma
factor into the BPS equation (7), aiming to include the Lorentz contraction in
the collision process. Then, the following function is minimized

$F(\alpha)=\left\lVert(1-v^{2})\phi_{KA}^{\prime\prime}(\alpha)-V^{\prime}\left(\phi_{KA}(\alpha)\right)\right\rVert_{2}^{2},$
(10)

where the two-norm is taken only in the interval $[-x_{0},x_{0}]$ and we have
omited the $x$ dependence for conciseness. The extra factor $(1-v^{2})$
compared with eq. (1) comes from including the gamma factor in the BPS
equation, essential for finite initial velocities. Then, the optimal $\beta$
is found by minimizing the function

$G(\beta)=\left\lVert(1-v^{2})\chi_{KA}^{\prime\prime}(\beta)-V^{\prime\prime}\left(\phi_{KA}\right)\chi_{KA}(\beta)\right\rVert_{2}^{2},$
(11)

where the $x$ dependence has also been suppressed.

A better fit for the initial field configuration could be obtained by taking
impurities with more free parameters. The impurities should be kink-like with
asymptotic values $\pm 1$ for kink-antikink collisions. Therefore, let us
consider the following impurity with three free parameters

$\sigma(x;\bm{\alpha})=\tanh(\alpha_{1}x+\alpha_{2}\tanh(\alpha_{3}x)).$ (12)

This way, we find remarkable results for the $\phi^{8}$ and $\phi^{12}$
models, as described below.

The $\phi^{8}$ model. In order to test the method described above, we
considered the following $\phi^{8}$ potential

$V(\phi)=\frac{1}{2}\phi^{2n}(1-\phi^{2})^{2},$ (13)

where $n=2$. The model has asymmetric kink solutions with exponential and
$1/x$ asymptotic tails. We construct kink-antikink configurations where the
long-range tails are superposing. We denote the two impurity models
$\tanh(\alpha x)$ and eq. (12) as $BPS1$ and $BPS3$, respectively. The
Euclidean norm of the differential equation at the optimal parameter,
$F(\bm{\alpha}_{o})\equiv e_{\phi}$ and $G(\bm{\beta}_{o})\equiv e_{\chi}$,
are shown in Figs. 1(a) and 1(b) as a function of the half-separation $x_{0}$,
fixing $v=0.1$. As a comparison, the norm of the split-domain (SD) ansatz and
the usual minimized (MIN) solutions are also shown. The SD solution is the
more accurate among the naive initial conditions, such as the additive ansatz
and product ansatz, while the MIN solution gives the reference value of the
field.

Figure 1: (a-b): Values of the $e_{\phi}$ and $e_{\chi}$ as a function of
$x_{0}$ for several methods. (c-d) The initial condition for $\phi$ and
$\chi$. Evolution of the fields in spacetime for the minimized method (e-f),
the BPS1 method (g-h), and the BPS3 (i-j). We fix $v=0.1$.

The BPS1 solution performs much better than the SD one, being much closer to
the MIN. As expected, all solutions converge to the same values as $x_{0}$
increases because the kink-antikink superposition decreases. Remarkably, the
BPS3 solution has an excellent agreement with the MIN, with the two curves
coinciding in the scale shown in the graph. This can also be observed in the
field profiles near the origin in Figs. 1(c) and 1(d).

Let us compare the actual evolution of the field in spacetime for $x_{0}=30$.
Figs. 1(e) and 1(f) show the MIN solution, which evolves smoothly, as
expected. In the BPS1 solution, shown in Figs. 1(g) and 1(h), we find a very
similar evolution, but it is possible to see small oscillations in the
contour. The observed deviation is indeed very small. The contour plot in
black shows the field’s fine scale; without it, it would be impossible to tell
the difference between the graphs. Finally, the BPS3 solution is shown in
Figs. 1(i) and 1(j). It exhibits remarkable similarity with the MIN solution.

We estimate the algorithmic complexity of the BPS method, which is how the
execution time scales with the size of the data set, in the following way. To
mimic the usual simulation procedure in the literature, we fix $x_{0}=25.0$
and perform several minimizations with different $v$. We pick eight equally
spaced velocities in the interval $[0.1,0.8]$. They are picked in an
increasing fashion, using the previous result as an initial guess for the next
minimization. The box is fixed at the interval $[-100.0,100.0]$. The time
execution of the first minimization step as a function of the number of mesh
points N, which is the size of our data set, is shown in Fig. 2(a) for several
methods.

Figure 2: Normalized time cost of the first minimization step for several
minimization schemes considering (a) the $\phi^{8}$ model and (b) the
$\phi^{12}$ model.

The MIN method can be performed using several derivative approximations. We
utilized the pseudospectral method, which can be performed either via matrix
multiplication or the Fast Fourier Transform (FFT) [15]. Finally, we
considered the five-point stencil (5PS) approximation for the derivative. The
5PS and the pseudospectral using the FFT have better performance. The former
scales roughly as $N^{3}$, while the latter does not fit into a power-law
behavior because the algorithmic complexity of the FFT contains a $\log(N)$
factor. On the other hand, the BPS methods scale roughly as $N^{0}$, offering
a significant time gain for sufficiently large $N$. The time cost of the BPS
methods does not increase with $N$ because the most costly step in the process
is, in fact, solving the BPS equation, not taking the derivative.

Remarkably, the BPS3 minimization is at least 160 times faster than the usual
minimization for $N=4096$. This gain is enough to allow simulations on a
moderate scale to be performed even on a personal computer. It is important to
mention that a good initial guess for $\alpha$ and $\beta$ should be provided
to the BPS methods in order to obtain fast and correct results.

We repeat the procedure for several models, including both the $\phi^{10}$
($n=3$) and $\phi^{12}$ ($n=4$) models. We found that the BPS3 is an excellent
method to find minimized initial conditions in general. In order to illustrate
this point, we report our results for the $\phi^{12}$ ($n=4$) with very fat
tail kinks, i.e., with $x^{-1/3}$ asymptotic form.

The $\phi^{12}$ model. To assess the accuracy of the BPS3 method for the
$\phi^{12}$ $(n=4)$ model, we compare it with the MIN solution in Fig. 3
fixing $v=0.1$. The error functions $e_{\phi}$ and $e_{\chi}$ are shown in
Figs. 3(a) and 3(b). The errors are very small and again converge as $x_{0}$
increases. Fixing $x_{0}=50.0$, we obtain an excellent agreement of the
initial conditions near the origin (see Figs. 3(c) and 3(d)), where the
superposition occurs. The evolution of the fields in spacetime is shown in
Figs. 3(e) and 3(f) for the MIN initial condition, and in Figs. 3(g) and 3(h)
for the BPS3. They are almost indistinguishable, except for a small deviation
in the contour closest to the origin.

Figure 3: (a-b): Values of the $e_{\phi}$ and $e_{\chi}$ as a function of
$x_{0}$ for several methods. The SD error does not appear in the graph scale
of (a). (c-d) The initial condition for $\phi$ and $\chi$. Evolution of the
fields in spacetime for the minimized method (e-f) and the BPS3 method (g-h).
We fix $v=0.1$.

We repeat the procedure to estimate the time cost of the first minimization
step with $x_{0}=50.0$ and obtain roughly the same scaling behavior. The
result is shown in Fig. 2(b). Again, remarkably, the BPS3 method is at least
175 times faster than the usual minimization For $N=4096$. The second
minimization step of both $\phi^{8}$ and $\phi^{12}$ models have similar
scaling behavior. However, the time gain is not as significant as in the first
minimization step because minimizing $\chi$ is the less costly step in the
minimization. The reason is simple. Usually, the field $\chi$ is less long-
range than $\phi$ because it is related to the zero mode, that is, the
derivative of the kink configuration in space. Thus, we have, for instance,
that the kink asymptotic behavior in the $\phi^{12}$ model is $x^{-1/3}$,
whereas the zero mode asymptotic behavior is $x^{-4/3}$. Therefore, minimizing
$\chi$ for power-law tails in general is much easier.

To put the accuracy of our construction to a final test, we consider a
relatively small half distance $x_{0}=10.0$ and perform a numerical simulation
using the BPS3 initial condition and the usual minimization method as a
reference. The result is shown in Fig 4 for $v=0.1$. We present the reference
method in panel (a) and BPS3 in panel (b). Even though there is a deviation
from the reference solution in the previous graphs, they are mainly focused on
the evolution before the kinks superpose. That error is very small compared to
the field variation near the bounce. Therefore, it is effectively erased, and
the collision output is virtually identical to the reference one.

Figure 4: Field evolution in spacetime for the initial configuration taking
(a) the usual minimization method and (b) the BPS minimization with the three-
parameter impurity BPS3. Parameters are $x_{0}=10.0$, $v=0.1$, and $n=4$.

Conclusion. We offered a construction to find initial conditions that generate
smooth field evolution of kink-antikink configurations with long-range tails.
This was achieved with the aid of half-BPS preserving impurities. By
considering kink-like impurities, we obtained configurations with a kink-
antikink character. After minimizing an appropriate function, an initial
condition was obtained with remarkable similarity with the reference method.
Our method has a negligible increase in the time cost with the number of mesh
points $N$, becoming more than a hundred times faster for $N=4096$.

Our idea may be generalized to find multi-soliton configurations with long-
range tails in two or higher dimensions. More importantly, it also makes the
simulation of long-range kinks much more accessible, boosting scientific
advances in the field.

###### Acknowledgements.

A. M. acknowledges financial support from the National Council for Scientific
and Technological Development - CNPq, Grant no. 309368/2020-0, the Brazilian
agency CAPES and also Universidade Federal de Pernambuco Edital Qualis A. J.
G. F. C. acknowledges financial support from CNPq, Grant no. 150166/2022-2,
and the Brazilian agency FACEPE, Grant no. BFP-0013-1.05/23.

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