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Kink dynamics in the MSTB modelTo cite this article: A Alonso Izquierdo 2019 Phys. Scr. 94 085302

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https://doi.org/10.1088/1402-4896/ab1184

Kink dynamics in the MSTB model

A Alonso Izquierdo

Department of Applied Mathematics and IUFFyM, University of Salamanca, Av. Filiberto Villalobos 119,Salamanca, Spain

E-mail: alonsoiz@usal.es

Received 14 January 2019, revised 6 February 2019Accepted for publication 20 March 2019Published 6 June 2019

AbstractIn this paper kink scattering processes are investigated in the Montonen–Sarker–Trullinger–Bishop (MSTB) model. The MSTB model is in fact a one-parametric family of relativistic scalarfield theories living in a one-time one-space Minkowski space-time which encompasses twocoupled scalar fields. Among the static solutions of the model two kinds of topological kinks aredistinguished in a precise range of the family parameter. In that regime there exists one unstablekink exhibiting only one non-null component of the scalar field. Another type of topological kinksolutions, stable in this case, includes two different kinks for which the two components of thescalar field are non-null. Both one-component and two-component topological kinks areaccompanied by their antikink partners. The decay of the unstable kink to one of the stablesolutions plus radiation is numerically computed. The pair of stable two-component kinks livingrespectively on upper and lower semi-ellipses in the field space belongs to the same topologicalsector in the configuration space and provides an ideal playground to address several scatteringevents involving one kink and either its own antikink or the antikink of the other stable kink. Bymeans of numerical analysis we shall find and describe interesting physical phenomena. Bion(kink–antikink oscillations) formation, kink reflection, kink–antikink annihilation, kinktransmutation and resonances are examples of these types of events. The appearance of thesephenomena emerging in the kink–antikink scattering depends critically on the initial collisionvelocity and the chosen value of the coupling constant parametrizing the family of MSTBmodels.

Keywords: non-linear Klein–Gordon equation, kink dynamics, two-coupled scalar field theorymodels, bounce resonant windows

(Some figures may appear in colour only in the online journal)

1. Introduction

Over the last fifty years, topological defects behaving assolitary waves in nonlinear scalar field theories, but neveroccurring in linear systems, have been understood as thecornerstone in explaining the existence and the role of walland/or brane structures in condensed matter [1, 2], cosmol-ogy [3], optics [4], molecular systems [5], etc. One dimen-sional solitons or kinks, becoming domain walls in 3D space,are accompanied in different nonlinear gauge theories orsigma models by the existence of vortices and cosmic stringsas line topological defects, monopoles and skyrmions, pointdefects, and instantons or textures, (−1)-brane defects, all ofthem sharing the essential feature of living in nonlinear sce-narios. We shall focus in this paper on solitons and kinks,

whose paradigms are the solitary waves arising in the sine-Gordon and f4 models. The impact of its study has beenenormous in both the physical and mathematical literature indiverse contexts, despite the fact that these models involveonly one real scalar field. The search for static kinks in N-scalar field theories proved to also be an active research area,see, for instance [6–9]. The discovery of kink solutions forwhich N-components of the scalar field arranged in an iso-vector were non-null opened a window to new possibilitiesranging from its use to get a better knowledge of knownphenomena to its application to understand other physicalproperties. In this paper we shall deal with the particularlyinteresting one-parametric family of relativistic (1+1)-dimensional N=2 scalar field theories known as theMontonen–Sarker–Trullinger–Bishop (MSTB) model. This

Physica Scripta

Phys. Scr. 94 (2019) 085302 (20pp) https://doi.org/10.1088/1402-4896/ab1184

0031-8949/19/085302+20$33.00 © 2019 IOP Publishing Ltd Printed in the UK1

https://orcid.org/0000-0002-3882-3702https://orcid.org/0000-0002-3882-3702mailto:alonsoiz@usal.eshttps://doi.org/10.1088/1402-4896/ab1184https://crossmark.crossref.org/dialog/?doi=10.1088/1402-4896/ab1184&domain=pdf&date_stamp=2019-06-06https://crossmark.crossref.org/dialog/?doi=10.1088/1402-4896/ab1184&domain=pdf&date_stamp=2019-06-06

system arises as a deformation of the O(2) linear sigma modeland has been the focus of study by many researchers fordecades. It constitutes a natural generalization of the f4

model. The potential as a function of the two scalar fieldspresents two absolute minima. Therefore, the existence of twodegenerate vacua and the associated spontaneous symmetrybreakdown in the quantum version of the model is envisaged.The part of the potential energy density independent of thefield derivatives in this system is the fourth-degree poly-nomial isotropic in quartic field powers but anisotropic inquadratic powers: U , 11 2

1

2 12

22 2 1

22

22f f f f s f= + - +( ) ( ) .

The anisotropy parameter σ2 is the family parameter. A briefchronology of the main works concerning the family ofMSTB models is introduced in the following points:

(i) Birth of the model: In 1976 Montonen discovered thisfamily of models in his search for charged solitons incomplex scalar field theories with a global U(1) phasesymmetry [10]. In his paper two different classes ofstatic topological kinks were identified in the parameterrange σ ä (0, 1). There are one-null component kinks,for which the second scalar field component f2vanishes, whereas the f1 kink profile is precisely thesame as the kink profile in the standard N=1f4-model. There also exist two non-null componentkinks, such that the f1 and f2 kink profiles are bothnon-null and constrained to live in one of the two semi-ellipses: 11

2 1

1 22

2f f+ =s- with f2>0 or f2

instantaneous pictures in a movie. The central theme in thispaper is the description of the kink dynamics in the MSTBmodel. For example, the scattering between a pair of two-component topological kinks will be one of the main pro-blems to be studied and will be thoroughly discussed. Pur-suing this endeavor we shall encounter a great difficulty.Contrarily to the analogue mechanical system governingstatic solutions in the MSTB model, the search for MSTBsolutions evolving in time is not an integrable problem. TheMSTB field theory is a nonintegrable system rather differentto the integrable sine-Gordon field theory which admits aninfinite number of conserved charges. The consequence is thatwe cannot apply analytical tools to study the dynamics of anyobject, extended or not, in the MSTB field theory. Therefore,we shall rely on a mixture of numerical and symbolic com-putations to accomplish this task.

Rather than meson scattering (collisions between linearplane waves emerging in a vacuum background) we areinterested in the study of the kink–kink scattering in theMSTB model, which gives rise to very intriguing and com-plex dynamical processes. Collisions between infinitelyextended objects may bring us to contemplate highly non-trivial and exotic evolution patterns. Kink–kink and kink–antikink collisions have been thoroughly studied in one-component scalar field theory models. Indeed, this subjectdrew great attention towards the seminal paper by Campbelland collaborators [28]. In this work, Campbell, Schonfeld andWingate investigated the dynamical interactions betweenkinks and antikinks in the archetypical f4 model by varyinginitial collision velocities. For initial collision velocity greaterthan a critical velocity vc≈0.259 8 kink reflection takesplace. If the initial collision velocity is large enough the kinkand the antikink collide, bounce back and escape respectivelytowards x = -¥ and x = +¥, losing a certain amount ofenergy through plane wave (meson) radiation emission. Ifv0

operator of the stable kinks is numerically analyzed in thispaper. As previously mentioned, the one-component f4-kinksembedded in this model are unstable. The study of the dis-integration of these unstable kinks and the description of theemerging objects in the final state for this event is also a newcontribution.

The organization of this paper is as follows: in section 2the MSTB model is introduced and its variety of kink solitarywaves is described; in section 3 the kink dynamics in theMSTB model is thoroughly discussed, in particular the dis-integration of the one-null component kinks and the two typesof kink/antikink scattering processes are explained. Finally,some conclusions are drawn and future prospects are outlinedin section 4.

2. The MSTB model and its static kink variety

The dynamics of the one-parameter family of MSTB modelsis governed by the action

S x Ud1

2, , 1a a

21 2ò f f f f= ¶ ¶ -m m

⎡⎣⎢

⎤⎦⎥( ) ( )

where the functional U[f1, f2] as a function of the fields is thefourth-degree polynomial

U ,1

21

2. 21 2 1

222 2

2

22f f f f

sf= + - +( ) ( ) ( )

Here :a 1,1 f , a 1, 2= , are two dimensionless realscalar fields and the Minkowski metric gμν is chosen asg00=−g11=1 and g12=g21=0. The notation x

0≡ t andx1≡ x will be used from now on. The coupling constant σarising in the second summand of (2) is a real parameter,

s Î . The MSTB model is thus a deformation of the O(2)-linear sigma model, where explicit symmetry breaking of O(2) to the discrete subgroup 2 2 ´ generated respectivelyby (f1→−f1, f2→f2) and (f1→f1, f2→−f2) takes placedue to the last summand in (2).

The system of coupled PDE equations

t x2 1 , 3

21

2

21

2 1 12

22f f f f f

¶¶

-¶¶

= - -( ) ( )

t x2 1 , 4

22

2

22

2 2 12

22

2

2f ff f f

¶¶

-¶¶

= - - - s( ) ( )

encompasses the two Euler–Lagrange equations of the actionfunctional (1). The configuration space is defined as theset of finite energy maps from the Minkowski space-time tothe field space, i.e., x t x t x t, , , ,1 2 f f= F º Î{ ( ) ( ( ) ( ))

E x tMaps , : ,1,1 2 F < +¥( ) [ ( )] }. The energy densitycarried by a particular configuration x t,F =( )

x t x t, , ,1 2f f( ( ) ( )) is :

x tt t

x xU

,1

2

1

2

1

2

1

2, ,

12

22

12

22

1 2

f f

f ff f

F =¶¶

+¶¶

+¶¶

+¶¶

+

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

[ ( )]

( )

whereas its spatial integration along the whole real linedefines the total energy:

E x x t, d , . 51 2 òf f = F-¥¥

[ ] [ ( )] ( )

Evolution of the different elements in the configuration spacetaken as initial values of the system (3)–(4) is determined bysolving the corresponding Cauchy problems for this PDEsystem. The energy finiteness condition forces configurationsto satisfy the following asymptotic conditions:

x t

t

x t

xx t

lim,

lim,

0 and

lim , ,x x

x

¶F¶

=¶F

¶=

F Î¥ ¥

¥

( ) ( )

( )

where A A1, 0 , 1, 0 = = + = -+ -{ ( ) ( )} is the set ofzeros or absolute minima of the MSTB potential U(f1, f2).Since is a discrete set, the values t,F ¥( ) are constantsof the motion because any variation of the asymptotic valuesof the fields would cost infinite energy. The configurationspace is therefore the union of four disconnected topologicalsectors È È È= ++ +- -+ -- distinguished by thefour admissible values of the fields at the two ends of the realspatial line. It is standard to define the ‘topological’ charge

q t t1

2, , ,1 1f f= +¥ - -¥( ( ) ( ))

as the invariant distinguishing between the different sectors ofthe configuration space1. Configurations carrying nonzerotopological charges (living in +- for q=+1 or -+ forq=−1), stay at their sector and are unable to evolve in timeto configurations belonging to ++ and --, all the forbiddenevolutions would require infinite energy.

The simplest solutions of the partial differential equationsystem (3)–(4) are static and homogeneous, precisely theelements of the set, which are the absolute minima of U:Φ±(x, t)=A±=(±1, 0). Therefore, these zero energysolutions are classically stable and provide bona fide groundstates to quantize the MSTB model: the choice of one of thetwo degenerate absolute minima of U as the vacuum of thequantum version of the model spontaneously breaks furtherthe remaining symmetry 2 2 ´ to the 2 subgroup gen-erated by (f1→f1, f2→−f2). The potential U(f1, f2) as afunction of the fields is nonnegative and admits critical pointsthat are thus static homogeneous solutions of the fieldequations. The character of the critical points depends on theranges of σ. If σ2ä (0, 2) the potential term U(f1, f2) has twodegenerate absolute minima A±=(±1, 0), a local maximumlocated at the internal plane origin (0, 0) and two saddlepoints placed at 0, 1 22s-( ), see figure 1 (left). If

2,2s Î ¥[ ) the potential term U(f1, f2) has two absoluteminima A±=(±1, 0) again, but now the origin (0, 0)becomes a saddle point of U(f1, f2) and the saddle points ofthe previous range become imaginary, losing their physicalsense, see figure 1 (right). Only the absolute minima will playa role in the quantum realm because attempts to use the other

1 To distinguish between ++ and --, both having q=0, one needs to fix,e.g., t,1f -¥( ) as well.

4

Phys. Scr. 94 (2019) 085302 A A Izquierdo

types of classical solutions as quantum ground states will beplagued with tachyons in at least one of the two phononbranches.

The next step is the search for static but space-dependentsolutions to the field equations (3) and (4), which become:

x

x

d

d2 1 ,

d

d2 1 . 6

21

2 1 12

22

22

2 2 12

22

2

2

ff f f

ff f f

=- - -

=- - - - s( )( )

( )

Reinterpreting x as mechanical time and thinking of (f1, f2)as the coordinates of a particle moving in a plane, the ODEsystem (6) is no more than the Newton equations for a particlemoving in the force field created by the potential V=−U.Kinks, which are static solutions of the field equations offinite energy, or, localized energy density, correspond in thisway to finite mechanical action trajectories of this Newtoniansystem. It happens that mechanical systems with two degreesof freedom isotropic in quartic powers of the coordinates butanisotropic in the quadratic powers are Hamilton–Jacobiseparable problems by using elliptic coordinates. In the Eulerversion the variable u ,sÎ +¥[ ) measures half the sum ofthe distances of the particle position to two fixed points in theplane F±=(±σ, 0) and vä[−σ, σ] is half the differencebetween these distances. The change of coordinates is tanta-mount to a map from the infinite strip to the upper half plane:

: , , 2 2 r s s s¥ ´ - [ ) [ ] . By allowing negative signsalso in the f2 coordinates, the 1: 2 map reads:

uv

u v

1,

1. 7

1 1

2 22 2 2 2

*

*

r f fs

r f fs

s s

º =

º = - -

( )

( ) ( )( ) ( )

The energy functional (5) expressed in these elliptic coordi-nates

E x xu v

u

u

x

u v

v

v

x u v

u u v v

d1

2

d

d

1

2

d

d

1

2

1 1

2 2

2 2

2

2 2

2 2

2

2 2

2 2 2 2 2 2 2 2

ò s

ss s

F =--

+--

+-

´ - - + - -

-¥

¥⎜ ⎟

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

[ ( )]

( )[( ) ( ) ( ) ( )]]

may be rewritten á la Bogomolny in the form:

E x xu v

u

u

x

u

u vu

u v

v

v

x

v

u vv T

d1

2

d

d

1 11

2

d

d1 1

8

a

b

2 2

2 2

2 2

2 22

2 2 2

2 2

2 2

2 22

2

ò ss

s

s

F =--

- ---

- +--

´ - ---

- +

-¥

¥

⎪

⎪

⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

[ ( )]

( ) ( )

( ) ( ) ∣ ∣( )

where a, b=0,1 and

T xu

xu

xv

xv

dd

d1

dd

d1 . 9

2

2

ò

ò

= -

+ -

-¥

¥

-¥

¥

∣ ∣ ( )

( ) ( )

Given the structure of the functional (8), one sees that theBogomolny–Prasad–Sommerfeld bound T∣ ∣ is saturated bystatic configurations complying with any of the following foursystems of first-order ODE’s:

u

x

u

u vu

v

x

v

u vv

d

d1 1 ,

d

d1 1 . 10

a

b

2 2

2 22

2 2

2 22

s

s

= ---

-

= ---

-

( ) ( )

( ) ( ) ( )

After finding the finite ‘action’ solutions of the ODE system(10) one immediately obtains all the kink solitary waves ofthe MSTB model by going back to Cartesian coordinates bymeans of the change of variables (7). The coordinate linesback in the (f1, f2)-plane are confocal ellipses and hyper-bolae whose foci are located at F±=(±σ, σ). We remark thatthe two copies ρ± are needed in (7) to cover the entire plane.Thus, smoothness conditions must be imposed on the solu-tions when crossing the axis f1. The static kink variety of theMSTB model will be analytically identified on these grounds.For this purpose, it is convenient to distinguish two regimesdetermined from the σ parameter where different kink pat-terns arise: (1) regime A: σ ä [0, 1) and (2) regime B:

1,s Î +¥[ ). We start now describing the topological kinks:

Figure 1.Graphical representation of the potential term (2) for σ=0.5 (left) and σ=1.5 (right). Notice that in the first case (f1, f2)=(0, 0)is a local maximum whereas in the second case it is a saddle point.

5

Phys. Scr. 94 (2019) 085302 A A Izquierdo

– One non-null component topological kinks xqstatic ( )( ) . For

any positive value of σ kink solutions whose second fieldcomponent vanishes exist. In this case the kink solutionsare given by

x q xtanh , 0 , 11qstatic =( ) ( ) ( )( )

where x x x0= - with x0 Î being the kink center.Here q=±1 is the topological charge which distin-guishes respectively between kinks and antikinks. Noticethat the mirror symmetry x x:xp - relates these twosolutions. Obviously, the minima A+ and A− areconnected by these kinks by means of the straight linef2=0, see figure 2. The energy density of thesesolutions is depicted also in figure 2. These topologicaldefects consist of only one energy density lump, thus,they may be interpreted as basic extended particles of thephysical system.

The description of these kinks in the elliptic planecomprises two possibilities:(i) In the regime B (σ>1) the vacuum points are

characterized by u±=σ and v±=±1. The condi-tion u=σ solves the u-equation in (10). Theensuing v-equation v1 1v

xad

d2= - -( ) ( ) is easily

integrated in the range −1

fluctuation operator

x

xx

xx

dd

4 6 sech 0

0d

d2 sech

q

2

22

2

22 2

s=

- + -

- + -

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

[ ( )]( )

is diagonal and the spectral problem in this case correspondsto two exactly solvable spectral problems (independentfrom each other) for Schrödinger operators with transparentPöschl–Teller potentials. The longitudinal eigenmodes F1 =

w

f x , 0 t1w( ( ) ) comprise a zero mode F x xsech , 0 t10 2=( ) ( ) , an

excited discrete eigenmode F x x xsech tanh , 013 =( ) ( ) with

eigenvalue 31 2w =( ) and a continuous spectrum whichemerges on the threshold value 4, k4k

212

1w = + with

k1 Î and eigenfunctions which are plane waves times thesecond Jacobi polynomial in xtanh . The discrete spectrumof the transverse fluctuations F f x0, t2 2=

w w( ( )) contains onlythe eigenvalue 12 2 2w s= -( ) with eigenfunction F2 2 =

w

x0, sech t( ) while the continuous spectrum kk2 2 222w s= +with k2 Î is in this case built from similar eigenfunctionsreplacing the second by the first Jacobi polynomial. Infigure 3 the complete spectrum of the operator xq [ ( )]( ) isdepicted as a function of the coupling constant σ in the

interval [0, 3]. The spectra of both the longitudinal and thetransverse kink fluctuation operators have been overlapped infigure 3. Note that discrete longitudinal/tranverse eigenvaluescan be immersed in the continuous spectrum of the tranverse/longitudinal fluctuations. The most relevant result from thisanalysis is that the eigenvalue 12

2 2w s= - (emerging in the

transverse fluctuation operator) is negative if 0

three main points: (1) The translational zero mode

F x x x xsech , sech tanh t0 2 s s s s= ( ) ( ( ) ( ) ( ))

belongs to the kernel of in the range σ ä (0, 1). (2) A pairof doubly degenerate continuous spectra emerging respec-tively on the threshold values 4 and σ2 exist. (3) In addition,numerical investigations reveal the presence of a discreteeigenvalue 1

2n for large enough values of σ in this regime.In figure 5 the spectrum of the operator K xqstatic

, l[ ( )]( ) isplotted for the range σ ä (0, 1]. Observe that the discreteeigenvalue 1

2n is non-negative. Thus, there are no negativeeigenvalues in the spectrum of K xqstatic

, l[ ( )]( ) and these kinksare stable, as previously pointed out.– Two one-parametric families of two non-null componentnontopological kinks: In regime A, there also exists a pairof one-parametric families of nontopological kinksN x x x; , , ,static 1 2g f g f g=

( ) ( ( ) ( )), whose field compo-nents are both non-null:

xx x

x x;

cosh cosh

cosh cosh161f g

s s s ss s s s

= -+

- + + + - -

- + + + - -( ) ( ) ( )

( ) ( )( )

xx

x x;

2 sinh

cosh cosh. 172f g

s ss s s s

=+

+ -

- + + + - -( )

( ) ( )( )

The notations σ±=1±σ and x x 1gs s= - ( )have been used in (16) and (17) to emphasize theregularity of these expressions. To derive the kinkprofiles (16) and (17) the separability of the ODE system

(10) has been used. A particular member belonging tothese families is singled out by the value of the realparameter g Î and the asymptotic value reached bythese kink trajectories at both ends of the spatial line. Infigure 6, the field components, the energy density and theorbit have been displayed for the particularN x;static g

+ ( )-kink with γ=6. In general, all theN x;static g

( )-kinks connect one of the two vacua A± withitself by means of closed orbits, all of them crossing oneof the two foci Fm.

In the elliptic strip the projection of these kink trajec-tories runs twice over the uä[−σ, σ] and v ä [σ, 1] intervals,see figure 6. Therefore, the following kink energy sum ruleholds between the different types of MSTB kinks:

E N x v v u u

E K x E x

; 2 d 1 2 d 1

q q

static

12 2

static,

static

ò òg = - + -= +

s s

s

l

-[ ( )] ( ) ( )

[ ( )] [ ( )]( ) ( )

that is, the total energy of a nontopological kink is the sum ofthe total energies of the two classes of topological kinks.From the graphical representation of the N x;static g

( )-kinkenergy density in figure 6, it can be understood that thisrelation is not accidental. Two energy lumps can be visualizedin this graphics, which correspond to a K xqstatic

,l ( )( ) lumptogether with a xqstatic ( )

( ) lump. In other words, theN x;static g

( )-kinks describe a nonlinear combination of the twobasic extended particles of the system. The parameter γ setsthe separation between these particles, a kind of relativecoordinate. For γ=0 the lumps are exactly overlappedwhereas for large values of γ the lumps of energy density areincreasingly separated.

3. Kink dynamics

In this section the kink dynamics in the MSTB model will benumerically addressed. This study will be restricted to theregime A where several types of kinks coexist. Two types ofbasic extended particles were identified in this regime, whichare described by the topological K xqstatic

,l ( )( ) and xqstatic ( )( ) kinks.

Note, however, that the second of these solutions is unstable,so the question about the fate of this unstable topologicaldefect naturally arises. This particle carries a non-null topo-logical charge and cannot decay to the vacuum sector.

Figure 4.Graphical representations of the field components (a), energy density (b), and the orbits in the Cartesian (c) and elliptic (d) plane forthe K xqstatic

,l ( )( ) -kink.

Figure 5. Spectrum of the small kink fluctuation operatorK xqstatic

, l[ ( )]( ) as a function of the coupling constant σ.

8

Phys. Scr. 94 (2019) 085302 A A Izquierdo

Therefore, the only possibility is that the unstable xqstatic ( )( )

kink decays into the stable K xqstatic,l ( )( ) kink. This matter will be

discussed in section 3.1.The scattering between two stable K xqstatic

,l ( )( ) kinks whereq, λ=±1 will also be investigated in this section. Dueto topological constraints the scattering processes mustinvolve kinks with opposite topological charges. Thisallows the construction of a continuous initial multikinkconfiguration, whose evolution is studied later. If symmetryconsiderations are also included in this framework, allthe possible scattering events fall into one of the followingtwo classes:

(a) K x K xq qstatic,

static,-l l-( ) ( )( ) ( ) scattering processes. The colli-

sions between a kink–antikink pair are encompassed inthis category. This kind of phenomena will be discussedin section 3.2.

(b) K x K xq qstatic,

static,-l l- -( ) ( )( ) ( ) scattering processes. These

events comprise the collisions between a kink and theantikink of the other kink existing in this model. Thissituation will be described in section 3.3.

As previously mentioned, numerical analysis is applied on theevolution equations (3) and (4) to determine the behavior ofthe scattering solutions. The numerical scheme that has beenemployed in this paper follows the algorithm introduced in[69] by Kassam and Trefethen, which is spectral in space andfourth order in time. As a complement to the previousscheme, an energy conservative second-order finite differencealgorithm [67, 70] implemented with Mur boundary condi-tions [71] has also been used. This algorithm lets to controlthe effect of radiation in the simulation because it absorbs thelinear plane waves at the boundaries. The two previousnumerical schemes provide identical results.

3.1. Disintegration of the ðqÞ ðx Þ-kink

The linear stability study of the xq ( )( ) -kink, given insection 2, concludes that the application of an infinitesimalfluctuation following the form of the eigenmode F x2 =

w ( )x0, sech t( ) of (15) causes the instability of this solution. The

evolution of the xq ( )( ) -kink under these circumstances isinvestigated in this section. Topological arguments suggeststhat this kink decays to the K( q,λ)(x) kink, which belongs tothe same topological sector but is less energetic than theprevious one.

In figure 7 the evolution of the xq ( )( ) -kink when slightlyperturbed by a F x2

w ( )-fluctuation is displayed for the caseσ=0.7. The two first graphics in figure 7 illustrate thebehavior of the field components. Globally, the initial con-figuration xq ( )( ) evolves to the K( q,λ)(x)-kink although someinternal vibration modes of this last solution are excited. Note,for example, the periodic oscillations of the maximum valuesof f2 which are reached at x=0. A strong radiation emissionis also apparent, mainly through the second field channel.

In figure 7 the total energy of the evolving topologicaldefect in the simulation interval is plotted by using a solidline. A large amount of the x t,q ( )( ) -kink energy is lost dueto radiation emission. This fact implies that E x t,q[ ( )]( ) is adecreasing function as we can see in figure 7(c). For the sakeof comparison, the total energies of the static topologicalkinks xq ( )( ) and K( q,λ)(x) have been drawn by means ofdashed lines in figure 7. Observe that the x t,q ( )( ) -kink totalenergy asymptotically approaches to the K( q,λ)(x)-kink totalenergy, although a small amount of energy seems to be savedin internal vibrational eigenmodes. In figure 8 we haverepresented the parts x t,1 ( ) and x t,2 ( ) of the energy density

Figure 6.Graphical representations of the field components (a), energy density (b), and the orbits in the Cartesian (c) and elliptic (d) plane forthe N x; 6static

+ ( )-kink.

Figure 7. xq ( )( ) -kink disintegration: Evolution of (a) the first field component, (b) the second field component and (c) the total energy of theevolving kink configuration (solid curve). The total energies of the static xq ( )( ) and K( q,λ)(x) kinks are depicted as dashed lines.

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Phys. Scr. 94 (2019) 085302 A A Izquierdo

x t,( ) corresponding to the first and second field,

x tt x

x tt x

,1

2

1

21

21

1

2,

,1

2

1

21

22

1

2,

11

21

2

12 2

12

22

22

22

2

22

22 2

12

22

f f

f f f

f f

f f s f f

=¶¶

+¶¶

+ - +

=¶¶

+¶¶

+ - + +

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

( )

( )

( )

where the interaction energy has been equally split betweenthe two contributions. Notice that x t,2 ( ) initially vanishesbut becomes negative due to the coupling terms. This clearlyimplies that the original kink xq ( )( ) is unstable when anorthogonal perturbation is applied. As observed in figures 7and 8 the radiation energy loss rate is small.

This process can be represented as

x F x K x radiation.q q2,sign* + +w( ) ( ) ( )( ) ( )

The λ-charge of the emerging two-component kink isselected by the sign of the second fluctuation field component.If the small perturbation F x2

w ( ) is positive the unstable

xq ( )( ) -kink will evolve towards the K( q,1)-kink, which livesin the f2>0 half-plane whereas if the perturbation is nega-tive we will find the K( q,−1)-kink as the resulting topologicaldefect in this process.

The asterisk superscript used in the previous relationemphasizes the fact that the resulting kink has excited internalvibration modes. An explicit study of this fact can be visua-lized in figure 9, where a spectral analysis of the behavior ofthe magnitude t0,2f ( ) is shown for values of the couplingconstant σ ä [0.5, 1]. The Fourier transform shows that sev-eral frequencies are excited with strengths that have beenrepresented using a gray scale. It can be checked that one ofthese frequencies is dominant over the other ones. Infigure 9(b) the square root of the eigenvalue 1

2n of the secondorder small fluctuation operator (15) has been plotted forseveral values of σ overlapped with the previous spectralgraphics. These frequencies of the operator K xq, l[ ( )]( ) arerepresented by small red circles. It can be checked the con-cordance between these values and the excited frequenciesextracted from the spectral analysis. This fact allows us toconclude that the xq ( )( ) -kink decays to the K( q,λ)(x)-kinkand that this process excites the discrete eigenfluctuation ofthis stable kink described in section 2. This justifies the

Figure 8. Evolution of the energy density x t,( ) (a) and the partial contributions x t,1 ( ) (b) and x t,2 ( ) (c) associated with each scalar fieldcomponent for the xq ( )( ) -kink disintegration illustrated in figure 7.

Figure 9. Spectral analysis of the second field component valued at the spatial origin f2(0, t) for the evolution of the static xq ( )( ) -kink whenperturbed by the fluctuation F x2

w ( ). In the second graphics the frequencies ν1 of the operator K xq, l[ ( )]( ) for several values of σ areoverlapping and highlighted with red small circles.

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Phys. Scr. 94 (2019) 085302 A A Izquierdo

internal shape oscillations which suffer the K( q,λ)(x)-kinkafter the disintegration, see figure 7. To visualize the evol-ution of this internal wobbling mode the maximum value ofthe energy density has been plotted in figure 10 as a functionof time. We can observe that the amplitude of this modedecreases very slowly.

3.2. K ðq;λÞ ðx Þ�K ð�q;λÞ ðx Þ scattering processes

The scattering between a stable kink K( q,λ) and its antikinkK(− q,λ) is the topic numerically investigated in this section.We are interested in classifying all the possible scatteringevents arising in this scenario, which depend on the initialvelocity v0 and the value of the coupling constant σ. Theidentities of the emerging topological defects and theirseparation velocities vf are the significant variables of thisproblem. The initial configuration for these numerical studiesconsists of two well separated boosted static kinks

K x x t v K x x t v, ; , ; , 18q q, 0 0 , 0 0È- + -l l-( ) ( ) ( )( ) ( )which are pushed together with speed v0. Here

K x t v K x v t v, ; 1q q, 0 static,

0 02= - -l l( ) [( ) ]( ) ( ) . The trajectory

of the multikink configuration (18) describes a semi-ellipticalcurve, which is traversed twice. For q=±1, the curvedefined by (18) goes from the vacuum Am to the oppositevacuum A± and later the same path is traveled in the reversedirection arriving at the point Am again. If λ=1 the multi-kink orbit (18) lives in the half-plane 02 f whereas ifλ=−1 the second component of the concatenation (18) isnegative.

The final velocity vf of the scattered kinks is plotted infigure 11 (left) as a function of the collision velocity v0 andthe coupling constant σ where σ ä [0.5, 1). Plane sections ofthis 3D graphics for three fixed values of the parameter σ aredisplayed in figure 11 (right). The behaviors found in thesethree plots are considered representative of the scatteringevents arising in this framework, which are described in thefollowing points:

– For large enough values of the initial velocity v0 the kinkK( q,λ)(x) and its antikink K(− q,λ)(x) collide and emergemutated into their f2-mirror symmetric partners

K( q,−λ)(x) and K(− q,−λ)(x), which travel away with acertain velocity vf

Figure 13. Evolution of the first (left) and second component (right) in the K( q,λ)–K(− q,λ) collision with impact velocity v0=0.65 for themodel parameter σ=0.94. Here, the excitation of the f2-fluctuations after the kink collision is so large that provokes the successive changebetween each topological defects and its f2-mirror partner.

Figure 11. Final kink velocity as a function of the initial velocity v0 ä (0, 1) and the model parameter σä(0.5, 1) for the K( q,λ)–K(− q,λ)collisions (left, top view). Plane sections of the 3D graphics for the values σ=0.6,0.71 and 0.94 (right). Zero final velocity indicates theformation of a kink-antikink bound state.

Figure 12. Evolution of the first (left) and second (right) component in the K( q,λ)–K(− q,λ) collision with impact velocity v0=0.5 for themodel parameter σ=0.72. The kink and its own antikink collide and mutate into its f2-mirror symmetric partners (one-bouncetransmutation regime).

12

Phys. Scr. 94 (2019) 085302 A A Izquierdo

they attract each other again. The transformed kinksapproach, collide and change into the original pair ofsolutions, the kink K( q,λ)(x) and the antikink K(− q,λ)(x).The reborn kinks bounce back and move apart until theyattract again. This process is repeated over and overemitting a decreasing amount of radiation in everyimpact. In figure 15 this scattering event has been plottedfor the values σ=0.72 and v0=15. The region (σ, v0)where this behavior manifests will be referred to as thebion formation regime.

– For 0.68s the resonant energy transfer mechanismarises in this type of scattering events. An energyexchange between the kink translational mode and the

internal vibrational mode is now possible in eachcollision and so the chance that the kink and antikinkescape after colliding a finite number of times. Thepresence of resonance windows starts off gradually forparameter values close to 0.68 but the effect isaccentuated for greater values of σ. For example, acomplex pattern of resonance windows can be observedfor σ=0.94, see figure 16. The number of bouncessuffered by the kinks before escaping is pointed out inthis figure.

It is assumed that other n-bounce windows with narrowerthan our search stepΔv0=10

−4 widths can exist. Indeed, thefinal scattered kink configuration depends on the number of

Figure 15. Evolution of the first (left) and second component (right) in the K( q,λ)–K(− q,λ) collision with impact velocity v0=0.15 for themodel parameter σ=0.72. The kink and its own antikink collide and form a bound state where a transmutation occurs after every kinkcollision (bion formation regime).

Figure 16. Final kink velocity as a function of the initial velocity v0 ä (0.58, 0.64) for the coupling constant σ=0.94 showing the resonancewindow structure.

Figure 14. Evolution of the multikink orbit for the scattering process illustrated in figure 13.

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Phys. Scr. 94 (2019) 085302 A A Izquierdo

collisions. In every collision the kinks are transmuted into itsf2-reflected kinks. As a result, if N is odd the scatteringprocess is characterized by the process (19) whereas if N iseven a global kink reflection

K v K v

K v K v radiation

q q

qf

q

,0

,0

, ,1* *

ÈÈ-

- +

l l

l l

-

-

( ) ( )( ) ( )

( ) ( )

( ) ( )

is found. These phenomena are illustrated in figure 17, for theparameter 0.72s = and v0=0.016 03.

The presence of this resonance scheme can be justified bythe existence of the discrete eigenvalue 1

2n in the small kinkfluctuation operator spectrum. The isolation of this discreteeigenvalue in the spectrum strengthens the resonance mech-anism for large values of σ. On the other hand, for smallvalues of σ the continuous spectrum dominates the behaviourof the kink collision and the resonance windows disappear.The MSTB model with σ=1 restores the f4 model reso-nance window arrangement because in this case the secondcomponent of the K( q,λ)(x)-kinks vanishes and the evolutionequations (3) and (4) do not change this circumstance.

3.3. K ðq;λÞ ðx Þ�K ð�q;�λÞ ðx Þ scattering processes

In this section the study of the collisions between a kinkK( q,λ) and the antikink K(− q,−λ) will be addressed. TheK( q,1)(x)-solution describes a semi-elliptical orbit confined inthe f2>0-half-plane whereas the K

(− q,−1)-solitary wavesurvives in the half-plane f2

configuration governs the behaviour of the K( q,λ)−K(−q,−λ)

scattering processes.The data found in the numerical analysis are displayed in

figure 18 (left), where the dependence of the final velocity vfof the scattered kinks on the initial velocity v0 and the para-meter σ is plotted. Three plane sections of this 3D graphicsare included in figure 18 (right). The particular behavior of vfwith respect to v0 is shown for the values σ=0.60, 0.72 and0.84. The pattern found in figure 18 (left) allows us to dis-tinguish five types of initial velocity regimes, which aredescribed in the following points:

– For low enough collision velocities v0 the kink scatteringis almost elastic for any value of the coupling constant σ.This process, symbolically represented as

K v K v

K v K v ,

q q

q q

,0

,0

,0

,0

ÈÈ

- -

l l

l l

- -

- -

( ) ( )( ) ( )

( ) ( )

( ) ( )

is illustrated in figure 19, where the evolution of the kinkcomponents is displayed. Here, the kinks approach eachother with initial velocity v0, collide, bounce back andmove away with approximately the same speed. This setof initial velocities and coupling constants, where the twoinvolved kinks are reflected, will be named the elastic

reflection regime. The upper boundary of this regimeinvolves a practically linear dependence on the parameterσ. Indeed, if we denote v1(σ)=1.160 73 – 1.153 25 σthen the region 0

For these events the f1-perturbations provoked by thecollision are strong enough to make the composite kinkorbit (20) jump the potential peak located at (f1,f2)=(0, 0), see figure 21. This process involves akinetic energy loss in form of radiation emission andinternal mode excitations. This energy loss prevents theevolving solution from returning to the original loopconfiguration. Consequently, kink annihilation takesplace and a radiation vestige remains. Here, the orbit isconfined to a region close to the vacuum A−, seefigure 21. The set of collision velocity windows wherethese events happen will be referred to as the annihilationregime. It is approximately confined in the bandv1(σ)

tails of these curves arises. This fact can be explained byassuming that the magnitude ρ depends on the modelparameter σ in this part of the graphics. Besides, this curveis the upper boundary of the transmutation regime, definedby the condition v q v vq2 0 s( ) ( ). For this regime theK(q,λ) and K(−q,−λ) kinks collide and emerge as itscorresponding antikinks after the impact, see figure 22.This type of event is symbolically represented as

K v K v

K v K v radiation 21

q q

q q

,0

,0

,1

,1* *

ÈÈ-

- +

l l

l l

- -

- -

( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

with v v1 0< . The excitation of internal modes has beensymbolized by the asterisk superscript in (21). Otherpossible interpretation of the previous event is that the kinkK(q,λ) and the antikink K(−q,−λ) collide and reflect whileexchanging the charge λ. This is the main difference withrespect to the kink reflection processes where the λ-chargeis preserved by the colliding kinks. The collision velocityv0 in this regime is large enough to make theK(q,λ) – K(−q,−λ) solution overcome the potential barriertwice, returning to the original loop configuration. How-ever, the f2-fluctuations produced by the kink collision flipthe elliptical orbit branches of the original loop configura-tion (20) with respect to the f1 axis. This flip transforms theoriginal kinks into the K v K vq q, 1 , 1* *È-l l- -( ) ( )( ) ( )configuration.

– The following distinguished domain corresponds toinitial velocities greater than the values of the rest ofregimes, that is, for v0>vq(σ) and v0>v2(σ). Thescattering processes in this case can be described by therelation

K v K v

K v K v radiation

q q

q q

,0

,0

,1

,1* *

ÈÈ

- - +

l l

l l

- -

- -

( ) ( )( ) ( )

( ) ( )

( ) ( )

with v1

whereas if N is even the scattering events

K v K v

K v K v radiation

q q

q q

,0

,0

,1

,1* *

ÈÈ

- - +

l l

l l

- -

- -

( ) ( )( ) ( )

( ) ( )

( ) ( )

are found. These phenomena are illustrated in figure 23.

Finally, the different regions in the plane (σ, v0) delimitedby the curves vi(q) (i=1, 2, q), where the previously men-tioned regimes arise, are outlined in figure 24.

4. Conclusions and further comments

Kink dynamics in the one-parametric family of relativistic(1+1)-dimensional two scalar field theories known asMSTB models has been thoroughly explored. Increasing thenumber of fields in the theory, henceforth enriching thenumber and types of kinks in a model, enlarges the variety ofpossible scattering processes. In particular, the presence oftwo different stable topological kinks doubles the numberof kink–antikink scattering events. The kink can be obliged tocollide with its own antikink or with the antikink of the otherstable topological kink. The output of these processesdepends on the collision velocity v0 and the coupling constantσ of the model. The domains in the (v0, σ)-plane where dif-ferent types of scattering events take place have beenestablished.

In the K( q,λ)−K(− q,λ) scattering a one-bounce trans-mutation regime (where the kinks are converted into itsf2-mirror symmetric partners K

( q,−λ)−K(− q,−λ) after thecollision) and a bion formation regime (where the kink andthe antikink collide and bounce back over and over again,mutating into its symmetric solutions in every impact) arefound. The previous pattern is general except for tiny regionswhere the resonant energy transfer mechanism is turned on. Inthis situation the bion state is broken after a finite number Nof collisions. The final scattered kinks depends on the parityof N.

The second class of scattering events in this frameworkcorresponds to the K Kq q, ,-l l- -( ) ( ) collisions. Now, therange of events is wider. An elastic reflection regime (wherekink and antikink elastically reflect), an annihilation regime(where kink and antikink mutually annihilate), a transmuta-tion regime (where kink and antikink mutate into its f2-mirrorpartners) and an ineslastic reflection regime (where thesolutions reflect with energy loss) are found. Resonant win-dows are also found in this context.

The large variety of kink scattering events found in theone-parametric family of MSTB models shows that the studyof the kink collisions in N 2 scalar field theory models canprovide new insights in the broad topic of one-dimensionaltopological defect (domain walls in 3D) dynamics. It wouldbe worthwhile, thus, to analyse kink collisions in othermodels involving several scalar fields. Continuous models ofspin chains describing ferromagnetic or antiferromagneticphases are particularly interesting in this direction. In [72]Haldane showed that kinks breaking the O(3) symmetry in thenonlinear sigma-model with target space a 2D-sphere are

important in characterizing some topological phase in anti-ferromagnetic materials. The full variety of kinks in thiseffective model was analyzed by myself and my collaboratorsin [73]). The surprising result is that the kink variety in themassive nonlinear sigma model is identical to the kink varietyin the MSTB model. There exists a discrete set of stable andunstable topological kinks and one-parametric families ofunstable nontopological kinks. Moreover, in [74] the semi-classical corrections to the masses of topological kinks werecomputed using heat kernel/zeta function regularizationmethods. It is therefore tantalizing to study kink scatteringprocesses in the massive nonlinear sigma model in the hopeof grasping a deeper understanding of topological phases inantiferromagnetic materials.

Acknowledgments

The author acknowledges the Spanish Ministerio de Econo-mía y Competitividad for financial support under grantMTM2014-57129-C2-1-P. They are also grateful to the Juntade Castilla y León for financial help under grant VA057U16and BU229P18.

ORCID iDs

A Alonso Izquierdo https://orcid.org/0000-0002-3882-3702

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1. Introduction2. The MSTB model and its static kink variety3. Kink dynamics3.1. Disintegration of the K(q)(x)-kink3.2. K(q,λ)(x)-K(-q,λ)(x) scattering processes3.3. K(q,λ)(x)-K(-q,-λ)(x) scattering processes

4. Conclusions and further commentsAcknowledgmentsReferences