Resonance phenomena in the ϕ8 kinks scattering

E Belendryasova1 and V A Gani1,2

1National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),115409 Moscow, Russia2National Research Center Kurchatov Institute, Institute for Theoretical and ExperimentalPhysics, 117218 Moscow, Russia

E-mail: vagani@mephi.ru

Abstract. We study the scattering of the ϕ8 kinks with power-law asymptotics. We foundtwo critical values of the initial velocity, v

(1)cr and v

(2)cr , which separate different regimes of the

kink-antikink collision. At the initial velocities vin < v(1)cr kinks can not collide due to repulsive

force between them. At vin > v(2)cr the kinks escape to spatial infinities after one collision. In the

range v(1)cr ≤ vin ≤ v

(2)cr we observed kinks capture and formation of their bound state. Besides

that, at these initial velocities we found resonance phenomena — escape windows.

1. IntroductionScattering of one-dimensional topological defects (kinks) is very interesting and fast developingarea. Many field theoretical models in (1+1)-dimensional space-time have kink-like solutions.The most famous are models with polynomial potentials, such as ϕ4, ϕ6 [1, 2], ϕ8 [3]–[6],and ones with non-polynomial potentials — modified sine-Gordon, double sine-Gordon [7, 8],sinh-deformed ϕ4 [9, 10], etc., [11, 12]. Note that models with two or more fields, which havetopologically non-trivial solutions, are also actively studied recently [13]–[21].

The (1+1)-dimensional field-theoretical models are very important for numerous applications.For example, models with polynomial potentials are often used in the description of domainwalls, phase transitions in condensed matter and in the early Universe, etc. Models with non-polynomial potentials, such as sine-Gordon, modified sine-Gordon, and double sine-Gordon,are used to describe disoriented chiral condensate [22, 23], some processes in ferro-magnets,Josephson contacts, in some nematic liquid crystals, and so on. One of the classical examples isthat the dynamics of planar domain walls can be reduced to the kink-(anti)kink interaction [24].Topologically non-trivial field configurations can be responsible for formation of the primordialblack holes in the early Universe [25, 26].

In this short letter we present our first results of numerical simulation of the kink-antikinkcollisions withing the ϕ8 model. Kinks of this model have one power-law and one exponentialasymptotics. In our numerical experiments the kink and antikink are turned to each other bypower-law tails. This leads to long-range interaction between the kinks, which substantiallyaffect their scattering. Here we address only phenomenology of the kink-antikink scattering. Adetailed study of mechanisms, which lead to these phenomena, will be a subject of a separatepublication.

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2. The ϕ8 modelWe consider a field-theoretical model with a real scalar field ϕ(x, t) in two-dimensional space-time with the Lagrangian

L =1

2

(∂ϕ

∂t

)2

− 1

2

(∂ϕ

∂x

)2

− V (ϕ), (1)

where potential has the formV (ϕ) = ϕ4(1− ϕ2)2. (2)

The equation of motion for the field ϕ is partial differential equation of second order:

∂2ϕ

∂t2− ∂2ϕ

∂x2+dV

dϕ= 0. (3)

The static kinks can be obtained from the first order ordinary differential equation

dϕ

dx= ±

√2V (ϕ) (4)

The solution with ϕ(−∞) < ϕ(+∞) is called ’kink’, while ’antikink’ stands for the solution withϕ(−∞) > ϕ(+∞). Nevertheless, in many cases below we call both of them simply ’kinks’.

In a model of ϕ8 the integration of equation (4) leads to implicitly defined kinks:

± 2√

2 x = − 2

ϕ+ ln

1 + ϕ

1− ϕ, (5)

see figure 1 The kinks (5) belong to the topological sectors (−1, 0) and (0, 1), while antikinks —

-8 -6 -4 -2 2 4 6 8 x

-1

-0.5

φ

Figure 1. Kink (−1, 0) and antikink (0,−1) of the ϕ8 model.

to the sectors (1, 0) and (0,−1). Asymptotics of all kinks are the following:

ϕ(−1,0)(x) ≈ −1 +2

e2e2√2 x, at x→ −∞, (6)

ϕ(−1,0)(x) ≈ − 1√2 x

, at x→ +∞, (7)

ϕ(0,1)(x) ≈ − 1√2 x

, at x→ −∞, (8)

ϕ(0,1)(x) ≈ 1− 2

e2e−2√2 x, at x→ +∞. (9)

As one can see from (6)–(9), each kink has one exponential and one power-law asymptotics.

3. Resonance phenomena in the kinks scattering in the sector (−1, 0,−1)We study collisions of kinks with power-law tails, i.e. in our numerical simulation the two collidingkinks are faced to each other by algebraic tails. Let us take the initial condition in the form ofthe kink ϕ(−1,0) and the antikink ϕ(0,−1), which are initially separated by distance 2ξ and aremoving towards each other:

ϕ(x) = ϕ(−1,0)

x+ ξ√1− v2in

+ ϕ(0,−1)

x− ξ√1− v2in

, (10)

see figure 2. The initial velocity of each kink is vin. In our numerical simulation we used theinitial kink-antikink distance 2ξ = 30.

vinvin

-20 -10 10 20 x

-1.0

-0.8

-0.6

-0.4

-0.2

φ

Figure 2. The initial condition of the type of (9).

We solved the second order partial differential equation (3) using the explicit finite-differencescheme,

∂2ϕ

∂t2=ϕk+1j − 2ϕk

j + ϕk−1j

δt2,

∂2ϕ

∂x2=ϕkj+1 − 2ϕk

j + ϕkj−1

δx2, (11)

on a grid with steps δt = 0.008, δx = 0.01.Depending on the initial velocity, the kink-antikink collision looks very different. We have

found two critical values of the initial velocity, v(1)cr ≈ 0.08067 and v

(2)cr ≈ 0.14791, which separate

different regimes of the kinks collisions.

(i) At vin < v(1)cr the kink and the antikink do not collide because they can not overcome mutual

repulsion. The kinks approach, stop, and then move away from each other, see figure 3(a).

(ii) At vin > v(2)cr the kinks collide and escape to infinities, this scenario is shown in figure 3(b).

(iii) At v(1)cr ≤ vin ≤ v(2)cr we observed

(a) kinks capture and formation of their bound state (a bion), see figure 4(a);(b) two-bounce escape windows, see figure 4(b);(c) multi-bounce escape windows — three-and four-bounce escape windows.

The escape windows appear due to resonant energy exchange between translational mode(kinetic energy) and vibrational mode(s) of the colliding kinks. For more detailed discussion ofthis phenomenon see, e.g., [1, 3].

An attentive reader can ask, how can it be that kinks repel, and at the same time they canform a bound state? At the moment we do not have definite answer. This situation is a challengefor us, and we will study the forces between the ϕ8 kinks with power-law tails in the near future.

(a) Scattering at velocity vin = 0.0806 < v(1)cr

(b) Scattering at velocity vin = 0.2000 > v(2)cr

Figure 3. Space-time picture of the kinks scattering at vin < v(1)cr and vin > v

(2)cr .

50 100 150 200 250t

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

φ(0,t)

(a) Bion formation, vin = 0.14510

50 100 150 200 250t

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

φ(0,t)

(b) Two-bounce window, vin = 0.14724

Figure 4. Space-time picture of the kinks scattering at v(1)cr ≤ vin ≤ v(2)cr .

4. ConclusionWe have investigated numerically the scattering of the ϕ8 kinks with power-law asymptotics.Such kinks are highly interactive in the sense that the force between them falls off algebraicwith distance [5]. This, in turn, leads to long-range interaction between kinks — they ’feel’ eachother at much bigger distances than kinks with exponential tails.

The results of our numerical simulation are very interesting. On the one hand, we observed

repulsive forces between kinks. On the other hand, we have found resonant structures — escapewindows, which are typical to models with attractive kink-antikink interaction. Hence, this ourstudy opens wide prospects for future work.

• First, it is interesting to investigate the force between kink and antikink as a function ofkink-antikink distance. This force seems to be repulsive at large distances, and it could beattractive at small distances.

• Second, kinks of the ϕ8 model with the potential (2) have no vibrational mode, which couldaccumulate energy and could be responsible for the appearance of the escape windows. Ifso, where is the kinetic energy stored between the first and the second kinks collisions withinthe two-bounce escape window?

These and other issues will be the subject of a future study.

5. AcknowledgementsThis work was performed using resources of the NRNU MEPhI high-performance computingcenter. The research was supported by the MEPhI Academic Excellence Project under contract№ 02.a03.21.0005, 27.08.2013.

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