Available at: http://www.ictp.trieste.it/~pub_off IC/2003/67

United Nations Educational Scientific and Cultural Organization and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SOLITON LIKE EXCITATIONS ON A DEFORMABLE SPIN MODEL

Jean–Pierre Nguenang1 Condensed Matter Laboratory, Department of Physics, Faculty of Science,

University of Douala, P.O. Box 24157, Douala, Cameroun2, Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I,

B.P. 812, Yaoundé, Cameroun and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,

Aurelien J. Kenfack and Timoleon C. Kofané Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I,

B.P. 812, Yaoundé, Cameroun.

MIRAMARE – TRIESTE

July 2003 1 Junior Associate of the Abdus Salam ICTP. 2 Permanent address. nguenang@yahoo.com

1

Abstract

We study numerically non-linear excitations on a one-dimensional deformable discrete classical ferromagnetic chain. In the continuum limits the equations of motion are reduced to a Klein-Gordon equation with a Remoissenet – Peyrard substrate potential. From a numerical computation of the discrete system with a suitable choice of the deformability parameters, the solitons solutions are shown to exist and move both with a monotonic oscillating (i.e. nanopteron) and a monotonic non- oscillating tails and also with a non- oscillating tails but with a splitting propagating shape. The stability of all these various solitons shape is confirmed numerically in a greater range of the reduced magnetic field 61.00 <≤b compared to the case of a rigid magnetic chain i.e. 33.00 ≤≤ b . From a kink- antikink and a kink-kink colliding simulation, we found various effects including a bound state of a kink and an antikink as well as a moving kink profile with higher topological charge that appears to be the bound state of two kinks. We also observed a three particles interaction that also arises from a kink-kink collision. The breather that intercalates between the two kinks has length that varies from its minimal value to the maximal one as far as the alternation between an attractive and a repulsive phenomenon is produced. From our results it appears that the value of the shape parameter of the substrate potential or the modified Zeeman energy is a factor of outmost importance when modelling magnetic chains.

2

I INTRODUCTION

It is commonly well acknowledged that the investigation of nonlinear waves and soliton-like excitations has provided fruitful and interesting outcomes in many areas of condensed matter physics [1]: dislocation in crystals [2-4], planar domain walls in ferromagnets [5] and ferroelectrics [6,7,8], nonlinear spin waves [9,10]: charge carriers in weakly pinned charge-density wave condensates [11], incommensurate systems [12,13] and bond-alternation domain wall in poly-acetylene [14]; to cite just a few examples. Great attention has been focussed on two-dimensional (2D) systems as well as one-dimensional (1D) models.

In the context of (2D) models that are applied to magnetic films and super -lattices, the interest has been driven by enormous experimental advances in the growth and characterization techniques of these materials, as well as the investigations, which have led to the realization of high quality samples and the discovery of many new and interesting phenomena, such as the oscillatory exchange coupling between ferromagnetic films separated by non magnetic spacers [15], the giant magneto resistance effect [16]. Owing to these advances, the determination of the ground state of the magnetic model through a uniaxial antiferromagnet in the presence of an applied magnetic field was formulated as a two-dimensional area-preserving map; the results were consistent with the experimental data on Fe/Cr(211) super lattices [17] and other theoretical works [18,19]. The advances done on the characterization of these magnetic materials help to set up good tools for the magnetic bubble propagation through solitary configuration with particle-like properties. In recent years these solitary magnetic bubbles have played a vital role in the rapidly growing of information technology. In the specific context of the one-dimensional (1D) model, nonlinear excitations in quasi-one-dimensional (1D) magnetic system have received much theoretical, experimental and computational attention in recent years [9, 20-24].

In most of the aforementioned studies related to the dynamics of magnetic chains, the part of the symmetry-breaking potential introduced in the system through the Zeeman energy contribution under the action of an external magnetic field leads to the sine-Gordon picture when the continuum approximation is used and the anisotropy is assumed to be extremely strong. The important question of relaxing the simplifying assumptions mentioned above has been widely investigated mostly by treating these assumptions as independent from each other: the investigation of the ferromagnetic chain with finite anisotropy in the classical continuum limit [25-28] revealed the existence of out of plane instabilities as well as a soliton spectrum quite different from the sine-Gordon results; similar studies of the quantum sine-Gordon equation [29 ] as well as the discrete sine-Gordon chain [30-31] have contributed to our understanding of the modelization of a very closed to a real quasi-one-dimensional solid.

However in real magnetic systems, whenever the above mentioned assumptions can be used or not in order to approach the macroscopic description of their dynamics: it remains the fact that, for a more accurate investigation, the possibility that the potential can deviate strongly from the sinusoidal one should be taken into account. This behavior can be induced in the system by the generation of the nonlinear harmonics in the sinusoidal potential that are due to microscopic structures and the influence of various interactions. The magneto- elastic interactions, for instance can arise either from the strain derivatives of the crystalline electric field, as a one – ion effect, eventually leading to a two –ions interaction, or from a distance

3

dependent exchange interaction or else from a local moment formation or change. Such a microscopic situation may lead to isotropic as well as anisotropic macroscopic deformations. Among the different effects, it is often customary to distinguish more particularly the magneto-volume effect, which is a spontaneous deformation at a magnetic ordering, the positive and negative Joule magnetostriction which is the field–induced parallel and perpendicular deformation of the material and the thermal expansion or contraction of magnetic origin. All these various effects lead to new shapes of the domain wall that are consequently due to the appearance of new harmonics on the substrate potential [32,33].

In that context, recently one of us [34] began by putting forward a novel model in which the Hamiltonian was characterized by the nearest neighbor exchange energy, the single ion anisotropy and a deformable Zeeman energy. In that paper, the attention was focussed on the situation where the Zeeman energy can be varied continuously as a function of the deformability parameter r. While taking the continuum limit, he showed that such a magnetic system could be mapped approximately onto the one-dimensional deformable sG equation at extremely low temperatures, from which implicit soliton solutions that depend on the parameter r could be derived. However, encouraging as that model was, it was nevertheless somewhat limited in its applicability for a wide range of real magnetic systems. Pursuing then the same idea with a spirit of generalisation, we have very recently introduced a one dimensional spin model with a modified Zeeman energy that leads to the nonlinear deformable periodic substrate potential, we have shown that the range of the deformability parameter and therefore the shape of the substrate potential is a factor of outmost importance when modelling magnetic physical system [35]. Henceforth, the range of the magnetic field for the stability of the nonlinear excitations propagating in the system is enhanced. The static critical magnetic field disappears for the shape potential parameters 0≠r . Moreover, the model shows very different phenomenologies that comprise: a ballistic, a diffusive as well as stochastic behaviour, respectively. Therefore, the deformability provides insight into unusual behaviours that may be typical of multivariate dynamical magnetic systems. Hence, important and interesting new features are then introduced in the dynamics of ferromagnetic chains through the propagation and the collision processes of the findings.

This paper aims at carrying out a theoretical analysis through numerical simulation of the topological soliton solution that can propagate in a discrete ferromagnetic chain under deformability effects. From the result obtained in [35], we also investigate the kink-kink and kink –antikink interactions in a larger range of the reduced magnetic field .i.e. 61.00 ≤≤ b , than the allowed range of the reduced magnetic field ( 33.00 ≤≤ b ) for the rigid magnetic chain without deformability effects.

The material of this paper is organized as follows. In section II, we describe the model and give its different limits and next we derive the discrete equations of motion. In the continuum limits, the implicit kink solutions are calculated exactly. Section III is devoted to the numerical studies of the soliton -like excitations in the discrete magnetic chain under deformability effects. We investigate their stability and particle-like interaction through collision processes. We consider both moving kink solution with monotonic and oscillating tails. In section IV, we give a summary and our concluding remarks.

4

II THE MODEL The model we considered in this paper is a discrete chain of classical spins

with planar single-ion anisotropy. It is also subjected to deformability that is introduced in the system through a deformable Zeeman energy. We take into account exchange interactions between nearest–neighbors’ spins only, and we write the Hamiltonian in the following form:

( ) ( ) ( )∑∑∑ −

−

+ ++−

−++−=i

xi

xi

xBi

zi

iii SrSr

SSSBrgSASSJH 12

122

1 211.1. µ

rr (2.1)

Here, S represents the modulus of the atomic spin ( iSSr

= in units of h ) and the

index x on the magnetic field component indicating that the magnetic field is applied in the X axis and r is the shape parameter that varies in the range –1<r<1 as previously introduced by Remoissenet and Peyrard [36,37]. J, A, and all other notations are standard [5]. The Z direction is the direction of the chain and the easy plane is the (XY) plane. In the last term of the Hamiltonian (Eq.(2.1)) that corresponds to the deformable Zeeman term, the presence of ( ) ( )ii

xiSS ϕθ coscos1 =−

in the denominator comes from the mathematical recombination of the Taylor expansion of the sinusoidal potential that is subject to the above-mentioned nonlinear harmonics generation. Before pursuing this study it is necessary, as we did in Ref [35], to notice the three important limits of this Hamiltonian. Case 1 When the deformability parameter r tends to 1, the deformable Zeeman energy tends to 0. The total Hamiltonian then reads: ( )∑∑ +−= +

i

zi

iii SASSJH 2

1.rr

. Consequently two situations can occur. We first have

an Ising-like system if the anisotropy constant A is negative (A<0), or secondly we have a planar Heisenberg system if A has a positive sign (A>0). In this latter case we notice that, the magnetic compound should behave as if it is not submitted to any magnetic field and therefore it may display a paramagnetic structure. For the deformability parameters values close to one, our model describes a magnetic compound for which, the deformability effects tend to erase that of the applied magnetic field.

For the values of r tending to –1, we recover once more a Heisenberg model Hamiltonian with a planar anisotropy but with the difference here that there is a presence of an additional term, which is noticed as defH and reads: SNBgH xBdef µ2= . This expression just shifts the energy of the system without any influence neither on the dynamics nor on the stability of the system under modelization. In the context of a system that displays an intrinsic magnetic gap, this Hamiltonian limit can be very useful because this gap can be absorbed or cancelled by the residual fundamental magnetic energy. This situation is always found in ferrimagnets with a great tendency to ferromagnetic order. Case 2 When r=0, the last term of Eq.(2.1) becomes ( )∑ −=

i

xixBdef SSBgH µ . In this last term

we notice the presence of the ground state energy and the Zeeman energy. Hence, from our deformable model, when the deformability parameter vanishes, we recover

5

the Heisenberg model with Zeeman energy and a planar anisotropy and ground state energy. Fundamentally the presence of the ground state energy here doesn’t matter because it has no influence, neither on the magnetic structure and nor on the spin dynamics stability. This Hamiltonian then, turns out to be more suitable because it allows to study the dynamics of a wide range of ferromagnetic and antiferromagnetic systems that belong to the class of the one-dimensional model. Hence, it is a sort of interpolation between an Ising and a planar Heisenberg model. A Equations of Motion

As mentioned above, the framework of our present considerations here is the

classical spin theory where the spins are usual three dimensional (3D) vectors →

iS of constant length S, i.e. neglecting quantum effects

<<

+24

)1(π

SJSA [5,20]. In

classical notations, we represent the spin field in spherical coordinates, i.e. ( ) ( ))sin(,)sin()cos(),cos()cos(, iiiiii StzS θϕθϕθ=

r, with πθ ≤≤ i0 and πϕ 20 ≤≤ i .

Then using the undamped Bloch equations for the spin vectors],

iii FS

dtSd rrr

h Λ= (2.2)

with the following relation for the effective field

ii S

HF rr

∂∂

−= (2.3)

It comes that

( ) ( )( )212

2

112112

xi

xxBz

ziiii

SrSrerSBgeASSSJF

−−+++

−+−+=

rrrrr µ (2.4)

Here, )( zx ee rr is the unit vector along the x axis (the z axis). While iFr

, represents the effective field acting on each spin, the expression ii FS

rrΛ represents the torque on the

spin at the lattice site i, respectively. Therefore, after some mathematical combinations between the spherical spin coordinate and Eqs (2.4) with Eqs (2.2), the equations of motions are [22, 24, 32-34];

( )[ ])cos()cos()cos()cos(sin)cos( 1111 iiiiiiiii

dtd

JSϕϕθϕϕθθθ

ϕ−+−= −−++

h ( )[ ])sin(sin)cos( 11 −+ +− iii θθθ

+ +)sin()cos(2iiJ

AS θθ( )

[ ]22

22

2 )cos()cos(21

)cos()sin(1

ii

iixB

rr

rJS

Bg

ϕθ

ϕθµ

++

− (2. 5a)

)sin()cos()sin()cos( 1111 iiiiiii

dtd

JSϕϕθϕϕθθ

−+−= −−++h

( )[ ]22

22

2 )cos()cos(21

)sin(1

ii

ixB

rr

rJS

Bg

ϕθ

ϕµ

++

−− (2.5b)

The set of coupled nonlinear differential–difference equations (2.5a) and (2.5b), defines the collective excitations for the in-plane angle iϕ and the out -of -plane angle

iθ of a discrete lattice. It is extremely difficult to perform a complete analysis of the

6

system of equation (2.5) analytically. Therefore, only a numerical simulation can allow a detailed study of the topological solitons travelling in the system. For this purpose, we first need to carry out some calculations in the continuum limit. B Analytical solution in the continuum limit

While the model under study is discrete, it is important to derive the continuum limit because it allows establishing the analytical calculation of the implicit soliton solutions that would be used as initial conditions for our numerical computation. Using then classical approximation at sufficiently low temperature

< 2

1

)(AJT , in the continuum limit (where the length scale

rotation 1>>xB Bg

JSµ ), equation (2.5a) and (2.5b) can be reduced to the following

partial nonlinear coupled differential equations:

( ) ( ) [ ]22

222

coscos21cossin1cossin1cos

ϕθϕθθθϕθθϕ ηηητ

rrrb

++−+−+−= (2.6a)

( ) [ ]22

22

coscos21sin1sin2cos

ϕθϕθϕθθϕθ ηηηητ

rrrb

++−−−−= (2.6b)

In these equations, we have introduced the dimensionless quantities aZ

JA2

=η

(where a is the lattice spacing), tAS

=

h

2τ and AS

Bgb xB

2µ

= . The sine-Gordon

approximation can then be obtained at the order 2ε if ( )1~, <<∂∂

∂∂ εε

τη(which

corresponds to the continuum approximation for permanent profile solutions), εθ ~ and 2~ εb .

Equations (2.6a) and (2.6b) then reduce to

( ) [ ] 0cos21

sin1 22

22 =++

−−−ϕ

ϕϕϕ ττηηrr

rb (2.7a)

τϕθ = (2.7b) Here b and r are the two constants needed to specify the time evolution of the spin excitations. It is therefore obvious that the last term of equation (2.7a) is the first derivative with respect to ϕ of the Remoissenet – Peyrard potential (RP) [36,37]

( ) ( )ϕ

ϕϕcos21

cos11,2

2

rrrrVRP ++

−−= , 1<r (2.7c)

Equation (2.7a) is the well-known deformable sG equation; it has solutions in the form of large amplitude travelling waves (kinks), low amplitude linear modes (spin wave or magnon) and breather [36, 37]. This equation also shows that in this approximation the ferromagnetic solitary excitations are composed of two families of implicit kink solution with the velocity v given in terms of moving coordinate

τηξ v−= which are travelling wave rotations of the spin through π2 within the easy-plane. Under this assumption, equation (2.7b) informs us on the fact that the spin

7

tilting out-of easy plane will be proportional to the kink translation velocity. The implicit solutions are [36,37]

( ) ( )

+

+

+

−−−= −−

21

22

1

21

22

21

212

12

)1(

2tan

tanh

2tan

1tan1)sgn(ϕα

α

ϕα

αααπϕγξ

d (2.8a)

with the rest energy

( )

−−= −− 2

12

121

2)1( 1tan1'8ααααbAE s

for 01 ≤<− r (2.8b)

and

( ) ( )

+

−

+

−−−= −−

21

22

1

21

22

21

212

12

)2(

2tan1

1tanh

2tan1

1tanh1)sgn(

ϕαϕα

ααϕπ

γξd

(2.8c)

and the rest energy is

( )

−−= −− 2

12

121

2)2( 1tanh1'8ααααbAEs for 10 <≤ r (2.8d)

and,

rr

+−

=11

α (2.8e)

αα /, 0)2(

0)1( dddd == ,

bd 1

0 = , Aa

A2

'2h

= and ( ) 21

21 −−= vγ (2.8f)

The out-of-plane remains defined by equation (2.7b) for both of the implicit in -plane solutions.

Let us mention that, since the Remoissenet-Peyrard potential presents more than two harmonics [36], such a model can be applied to a bi-layer film of Ni/Pd on which Tsukamoto [38] has shown, by molecular dynamic analysis, that the perpendicular magnetic anisotropy is related to the amount of stress in each Ni layer, and consequently the distribution potential between two sites on a fcc(111) and fcc(100) surface presents a shape with many harmonics. In the case of Fe/Cr (211) superlattice system, that is isomorphic to a classical two-sub lattice uniaxial antiferromagnet, the intrafilm interaction keeps all the spins belonging to the same film parallel to each other. Then, a single layer can represent each film. Hence, the determination of the ground state of the three dimensional super- lattices reduces to a one-dimensional problem in the direction normal to the film surfaces. Therefore, the Hamiltonian that is defined by equation (2.1) becomes suitable not only for the well-studied CsNiF3 material [5], but also for such a system.

Since the validity of the continuum limit and other approximations, however, can only be tested by comparison with numerical calculations, we are left with a central question of the physical problem described by the discrete non-integrable system of equations (2.5a) and (2.5b) for our magnetic chain, namely: what is the signature of the implicit soliton solution on the dynamics of the magnetic chain for different values of the parameter r.

8

III NUMERICAL EXPERIMENTS In this section, we present the results of a numerical experiment in a discrete

deformable magnetic chain. The validity of the analytical result obtained from the continuum approximation is discussed in detail. In order to investigate the dynamics of a single implicit kink soliton as well as its collision with other kink and antikink, we are going to solve numerically the set of coupled nonlinear differential-difference equations of motion (2.5a) and (2.5b).

A Computer - simulation details

Our numerical calculations have been done through a computer- simulation

program to study the dynamics of the implicit kink (antikink) soliton on a cyclic magnetic chain, which is constituted of 160 spins. In physical application minimum distance scale that leads to discreteness effects naturally occur (e.g. lattice constant). Therefore in these simulations, the degree of discreteness is controlled by the ratio

( )idγ (where, i=1,2), and ( )id is also a function of the reduced magnetic field and the

parameter r (see Eq.(2.8e) and Eq.(2.8f)). In order to control the discreteness effect arising in the system, for a given velocity, we chose particular values of the parameter ( )id . For instance, in some cases the physical parameters that can induce the discreteness effects were chosen in order to keep the implicit static soliton width in a range of 10 to 20 spins during the complete run. Let us introduce as an example, the following set of parameter of the CsNiF3 structure, namely [13]: J=23.6K, A=4.5K and S=1. In addition working with the cyclic magnetic chain, provided that we chose a periodic boundary conditions and an offset of π2 at chain ends, is needed.

As far as equations (2.5a) and (2.5b) are concerned with the description of the subsequent time evolution of the in- plane and the out-of- plane excitations, the origin of the energy scale of our discrete ferromagnetic chain is chosen from the uniform ferromagnetic state. Its dimensionless form is given in terms of the in- plane and the out-of-plane angles component by ∑=

iiEE , with

)1.3()))cos(2rcos(r(1

))cos(cos(-1r)-(1a 2

)(sina ))sin(sin( -

)-)cos()cos(cos( - 1))sin(sin( - )-)cos()cos(cos( - 1

2

23

2

21

11111

ii

iiiii

iiiiiiiiiiiE

ϕθϕθθθθ

ϕϕθθθθϕϕθθ

++++

+=

−

−−+++

where JAa 2

2 = , JS

Bga xBµ=3 , and the index (i) stands for the lattice sites. With a

suitable choice of the time step, this energy is a conserved quantity. Therefore, as we did in Ref [35], it was frequently monitored in our simulations to ensure an accuracy of about 0.01% for the fourth-order Runge–Kutta scheme.

The initial conditions, which are typically at time t=0, initial profiles of the in-plane and the out-of- plane component of the spin deviation are obtained by using the implicit solutions (see Eq.(2.8)).

For the numerical implementation of these implicit solutions, the Newton-Raphson scheme, which is the approach used by Remoissenet and Peyrard [36] or a pseudo spectral approach can be chosen [40]. However, theses approaches seem to be

9

limited because these methods remain just an approximation. We used a direct method instead and it turned out to represent a π2 -kink soliton when r=0 more accurately. To this end, once we have the velocity v, next, we define a set of discrete values of the in-plane in such a way that the interpolation points of lines are given by

Ni

iπϕ 2

= (3.2)

with ],..,0[ Ni∈ and N=160 is the spin lattice size. Using then this definition of iϕ in the discrete version of equations (2.8a) and (2.8c) given by:

( ) ( )

+

+

+

−−−= −−

21

22

1

21

22

21

212

12)1(

2tan

tanh

2tan

1tan1)sgn(

ii

iid

ϕα

α

ϕα

ααα

γπϕξ

(3.3a)

for 01 ≤<− r and

( ) ( )

+

−

+

−−−= −−

21

22

1

21

22

21

212

12

)2(

2tan1

1tanh

2tan1

1tanh1)sgn(

ii

iid

ϕαϕα

ααγ

ϕπξ

(3.3b)

for 10 <≤ r . Eqs.(3.2), Eqs.(3.3a) and Eqs.(3.3b) allow then to recover for each amplitude of iϕ , its corresponding localization on the lattice site iS , for different values of the deformability parameter. At this step we shall remind that the out- of- plane is defined by:

i

iii d

dvdt

dξϕϕθ −== (3.4a)

Therefore, it can be numerically extracted through the following relation

( )11

11

−+

−+

−−

−=ii

iii v

ξξϕϕθ (3.4b)

where v represents the reduced effective velocity of the soliton moving in the spin lattice. Therefore, in the present simulation, we are able to test the validity of the implicit solutions through their propagation in the discrete chain for different values of the parameter r and the reduced magnetic field b.

B Outcome of the analytical solutions in the discrete deformable Spin Chain

In this sub-section we present results of numerical treatment in the case of both strong and low coupling between neighbouring spin so that the continuum approximation is valid or not. When the continuum approximation is valid, thus the system exhibit a free -kink propagation scheme with a shape, which is very close to the initial profile. However, when it is not valid, the topological soliton involving in the spin lattice may then present either a splitting shape while moving or a kink moving with an oscillating background for the in-plane component, while the shape of the out –of-plane component displays a breather shape instead of a simple pulse of the initial conditions.

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B1 Free kink propagation

Let us first present Fig.1a and Fig.1b where we plotted the in-plane and the out-of-plane components motion for the deformability parameter r=0.5 and the reduced magnetic field b=0.5, respectively. As we can see in these figures the in-plane propagates for a very long time without a sensible change from its initial profile. Only the out-of-plane propagates with a slight radiation of magnon in the lattice. This tendency to long life of wave propagating in the chain is a good indication of a possible great stability of the kink in the magnetic chain. It is important to notice that for the reduced magnetic field b=0.5 it is not possible to find a stable propagating solitary wave in a rigid magnetic chain without deformability effects [5]. Here, in our deformable spin system, this is possible even for larger values of the magnetic fields. In Fig.2, we plotted the maximal value of the effective velocity of the propagating wave as a function of the deformability parameter (r) for all the reduced magnetic field (b) in the range 61.00 ≤≤ b . From the schematic plot of Fig.2 it follows that, the discrete system of Equation (2.5a) and (2.5b) admits moving kink solution with constant profile for each value of the deformability parameter (r), and the reduced applied magnetic field (b). These moving solitons display a finite interval for the effective velocity that vary from a minimal value of zero, which corresponds to the points on the abscise axis of the shape parameter (r), to a maximal speed with range [0.1, 0.5] given on the upper curve. Thus, in these conditions the implicit kink (see Eqs.(2.8)) solution used as initial condition seems to represents good solution of a discrete lattice. From Fig.2, we also found that the effective speed of the solitary waves propagating increases with increasing the shape parameter. It is also interesting to notice that if we introduce the ideal sine-Gordon kink as initial condition, we will obtain the same result as Peyrard and Remoissenet [37] in the case of a deformable atomic lattice. Here also, the shape of the kink will progressively tends to that of the implicit soliton solution (see Eq.2.8) with small amplitude oscillations but, as we mentioned in Ref [35], the highest values that can be attended by the out-of plane do alter the stability of this kind of kink.

B2 Splitting kink propagating

For all the deformability parameters, when the reduced applied magnetic field (b) is in the range 1.00 <≤b , then as soon as the initial condition is introduced in the chain it rapidly transform to a bi-kink shape that are really two linked kinks moving in opposite direction. It also follows from our simulations that, depending on the value of the deformability parameter; the size of the two kinks can be equal or different. When the sizes of two kinks are equal, the π2 bi-kink, while propagating, exhibit a two kink−π motion scenario, which is equivalent to an elastic collision that is produced in the middle of the chain and, due to a periodic boundary condition, also at the chain’s end. This can be seen in Fig.3a and Fig.3b where we plotted the in-plane component as well as the out-of-plane component of the spin wave propagating with a transparent collision motion, which is produced as a mutual crossing of each other. Figure 3c is a schematic that illustrate very well the mutual crossing of the two excitations engaged in the colliding process. The contour plot of the time against the position shows that it happens both at the chain’s end for the first time and the next time at its middle and the processes continue in the same order, while the entire π2 -kink moves progressively. However, when the sizes of the two sub-kinks are

11

different, the same transparent collision motion is produced but at different spin lattices sites situated between the middle and the end of the chain. We also noted in this latter case that, the kink with greater size was more rapid than the other one. Therefore it was possible to find colliding processes repeated many time in the same sigh of the chain before changing. Fig.4a and Fig.4b are shown to illustrate these colliding processes that happen when the in-plane component and the out-plane spin tilting are propagating in the lattice. This multiple collision in the same sight is well illustrated in Fig.4b where the out of plane components moves. Here the wave represented by the reversed pulse shape is faster than the wave represented by the upper pulse shape.

B3 Moving solitons on an oscillating background (nanopteron)

By moving solitons on an oscillating background we mean, waves that

propagate with a bound state of a self-trapped kink and a small amplitude wave with a non-linear dispersion law. When such waves move in our spin lattice, the scenario exhibited by the nonlinear excitation while propagating is that of a kink pinned to the (nonlinear) wave that decreases its velocity, and finally it turns out that this wave forces the kink to propagate in a coupled way, i.e., some kind of dynamical self trapping mechanism of the resulting kink with the nonlinear wave. Fig.5 is a schematic that illustrates kink profile of the in-plane component propagating with and oscillating tail, which is the so-called “nanopteron”. A nanopteron is a permanent but non-local and spatially extended soliton whose spatial radiation can only be minimized but not eliminated [39]. Here, the soliton motion occurs on a background of nonlinear modulated and oscillating wave train that can allow the kink to propagate in the discrete spin chain with any constant subsonic velocity, whenever the Peierls Nabarro barrier exist. It is also worth of mentioning that this nonlinear wave appears with a small amplitude of A=0.06 in a dimensionless units with a multi periodic shape constituted of a period for the rapid oscillation and a period for the bumps that map the wave train’s shape.

C Colliding process of the solitons in the discrete deformable Spin Chain

We are mainly interested by the properties of the π2 -kink (or a π2 -antikink)

when they collide in order to determine their stability in a ferromagnetic chain under deformability effects. In order to study the collisions of the in-plane and out- of- plane moving soliton component of the spin deviations, we simulate the system of equation (2.5) with periodic boundary condition for the chain consisting of N=200 spins. Using boundary conditions here is important because it helps avoiding ends effects. We still use the fourth order Runge-Kutta method with the same accuracy required for the single soliton propagation. We choose initial conditions in the same way as in the case of the single soliton i.e., here we take a pair of the implicit-kink (or antikink) for the in-plane and the derived pulses for the out- of-plane with opposite velocities from Eq.(2.8a) and Eq.(2.8c) and Eq.(3.4), respectively. In theses conditions the two solitary waves move towards each other both for two spin components.

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C1 Head on collision between a π2 -kink and a π2 -kink When the two solitary waves of π2 -kink type involve in such a magnetic chain with the same velocities; if they display opposite directions; then they start by moving towards each other. Then, we observed that the final state can be either dependent or not of the initial energy of the two excitations. Depending on the range of the shape parameter (r), each of the two π2 -kink faces either a repulsive interaction or an attractive one. Three processes allowed for these collisions are dictated by the parameter r. The two π2 -kink can pass through each other, or be reflected elastically or non- elastically, or else they can form a standing or a non standing bound state. For different values of the deformability parameter (r ) with range ]-1..0.4] and whatever the initial velocity chosen for a reduced magnetic field greater than b=0.33, the collision of the solitary waves of π2 -kink profile results in a strong inelastic interaction. The scenarios displayed here by the two excitations consist of a completely destroyed shape when they collide for the π2 -kink of the in-plane but, for the particular value of b=0.33 and r=0.4 while the π2 -kink shape is destroyed by the collision process the resulting breathers of the out-of- plane faces a mutual crossing with just a slight radiation. Therefore, in these range of parameters the system can support long time moving π2 -kink solitary waves but without particles-like properties. For us, this is the signature of the fact that within these parameters range the system does not admits π2 -kink soliton solution and therefore it is not integrable but may be nearly integrable. When the deformability parameters with range]-1, 1[are used, for all the reduced magnetic field in the range 33.00 ≤≤ b , the collision scenarios displayed by the two excitations corresponds to a mutual crossing of each other (see Fig.6a and Fig.6b). We notice from Fig.6b that the out-of-plane component moves with a breather shapes instead of a simple pulse of the initial condition. The mutual crossing phenomenon displayed by the two excitations is a good indication of the fact that, in this range of parameters, the system appears to be integrable since the soliton’s robustness is proved by the elastic colliding process whenever their final shape are different from those of the initial conditions. Moreover, for the reduced magnetic field greater than b=0.33, the mutual crossing happens only for 7.0≥r , while for the values of r with range ]0.4,..,0.7[, we observed most of the time an elastic reflection of the two excitations engaged in the colliding process. However, we found an unexpected scenario, for the deformability parameter r=0.6 and the reduced magnetic field b=0.21, which consist of a three particles interaction resulting from the collision process. Here we noted the presence of a breather between the two excitations of π2 -kinks type. It then engages itself in the colliding processes after the first collision between the two first excitations. Here the motion of the breather is equivalent to that of an elastic string that is embedded between the two kinks (antikink) and therefore it is submitted to compression or an extension of its length during the collision. In Fig 6c, which is shown to illustrate this new phenomenon, we observed that as soon the initial conditions representing the two excitations are introduced in the magnetic chain, they suddenly move toward each other and then rapidly the first collision happen at time of t=3000. After that, an interaction of three particles type is engaged as long as the simulation process will stay, with the domination of the compression and the extension of the breather, which is, intercalate between the resulting π2 -kink. For instance the second extension when the repulsive phenomenon dominates happens between t=12000 up to t= 20000.

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C2 Head on collision between a π2 -kink and an π2 -antikink The results obtained from the numerical computation of a single soliton

dynamics and that of the colliding processes of the soliton with the same topological charge described above, gave valuable qualitative description. However, for a quantitative description we still need to simulate kink–antikink head on collision at different range of magnetic field and deformability parameters, for allowed range of the initial velocities, through the discrete equation of motion (2.5a) and (2.5b). The lattice size is still N=200 and we work here with the same conditions of parameters and numerical scheme as in the case of the two π2 -kinks collision. The only difference comes from the opposite sign on the topological charge of the initial conditions. The initial positions of the two excitations are chosen with a distance of 100 spin lattice sites in order to avoid any possible interference between them at the beginning of the process. Notice that for all the simulations, the processes were more rapid for r>0 than in the case of r<0. This is in agreement with Fig.2 where we plotted the velocities against the deformability parameter. So, in order to accelerate the process for a while, for negative values of r it was necessary to chose higher order of initial velocities of 7.0≥v .

Let us first present Fig.7a and Fig.7b where we plotted the in-plane and the out-of-plane spin components colliding processes, for r =-0.3 and b=0.45 with initial velocity v=0.4, respectively. These figures display a reflective phenomenon after any collision for the in-plane spin component, whereas the out-of-plane component is flipped (see Fig.7b). Notice that this phenomenon is also possible with a faster process for r=0.6 and 65.03.0 ≤≤ b with v=0.34. We also observe some few radiations of small amplitude waves on the shapes of the out–of- plane component after each collision. This reflective phenomenon is a good indication of the fact that the soliton obtained in this range of parameters can propagate and survive for a long time whenever it interacts with another soliton. This is also an indication of the fact that the discrete system of equations (2.5) is at least nearly integrable. Figure.8a is a schematic obtained for r=-0.8, b=0.1 when the initial velocity is v=0.9 that illustrate the creation of a radiationlessly standing bound state of a π2 -kink and a π2 -antikink after their collision, this phenomenon can happen for the initial velocities with range 93.084.0 ≤≤ v . This can be understood by the fact that, as long as kink and antikink are sufficiently far apart there is no radiation visible. So in such condition, radiation losses are negligible. For the same parameters, Fig.8b also illustrates exactly at what time each of the excitations stop moving and remain static as long as the process is produced. This is a good indication of a non-elastic but attractive interaction of the two excitations in the deformable spin chain. Since there is no internal oscillation observed, we deduced that, this is a zero frequency translational mode formation. Needless to mention is the fact that, for the same value of the deformability parameter i.e. r=-0.8 and the reduced magnetic field, there is a window for the reduced velocities in the range 99.094.0 ≤≤ v where the colliding process of the two excitations results first in a breather formation that finally disintegrate in the chain. Thus, this velocities window’s for colliding process leads to an annihilation phenomenon. For the parameters r=0.9, b=0.5 and v=0.65, we observe in Fig.9a and Fig.9b that, with periodic boundary conditions, the first collision happens at chain’s end. After this first collision, the in-plane spin component comes out with a new shape, which is constituted of a new kink’s shape with a reduced size very closed to that of a π -kink. Then the resulting solitons continue the colliding process both at the middle and at the chain’s end with some slight radiations. But, these radiations do

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not alter the colliding processes because after any collision these solitons with new shape display a very robustness behavior. The out-of- plane also display a robust behavior through the colliding process with a scenario of mutual crossing without flipped shape while kipping the same size. Therefore this is a good indication that with these parameters solitons solutions exist for the discrete system of Eq. (2.5), whenever their shape may differ from those of the initial conditions. From Fig.10a and Fig.10b, we do observe a scenario of four particles interactions as a result colliding process engaged between the two excitations of the different spin components. It is clear from these figures that, when the in-plane excitations are introduced in the chain, some time later, while moving towards each other, each of them is divided into two excitations moving in opposite direction. Therefore, the colliding scenario becomes that of a kink and an antikink that are the first to collide at the middle of the chain. At the same time the two other excitations engaged in the collision process display opposite direction while moving and this would lead to their collision at the end of the chain. The same colliding scenario happens for the initially two excitations engaged for the out-of-plane component. The difference here comes from the fact that within the four particles interactions, the first couple of excitations that faced first the collision display a reversed pulse shape while the second couple of excitations display the ordinary pulse shape. We also notice for the two spin components that the first couple of excitation that faced first the collision at the middle of the chain does collide with the other excitations of the second couple before the second couple finally faced their first collision. Hence the first couple‘s motion is faster than that of the second couple. IV SUMMARY AND CONCLUDING REMARKS Since the previous studies on the numerical simulations of the soliton dynamics in a rigid Heisenberg chain [5, 41] was rather complete and that of a deformable magnetic chain [35] was involved a tremendous amount of numerical calculation, it is not our aim here to perform similar studies on the deformable spin model. For instance, Etrich et al [41] have shown that in the discrete ferromagnetic spin chain, two essentially different static in-plane soliton structures may occur. One with its center located on a lattice site, so-called central spin- configuration, and the other with its center located in the middle between two neighboring spin lattice sites, which is the so-called central-bon configuration. This kind of specification arises from a deep study on the discreteness effects on the soliton dynamics. We will address these problems in a future publication. Here, we only wanted to stress a few points that bring new features to the soliton dynamics in a ferromagnetic chain when the deformability effects are considered.

Finally to summarize, we have been able to examine the limit of the validity of the implicit soliton solution by a numerical computation in a classical deformable easy-plane ferromagnetic chain of the Heisenberg model, for the deformability parameter in the range of -1<r<1. Modifying the Zeeman energy of a Heisenberg chain, we have analyzed the nonlinear dynamics of the solitons structures taking the model of 3CsNiF material as a particular example. The different simulations of the single soliton’s propagation point out that, for some particular values of the deformability parameters, the motion of the Kink (in-plane component) on a double periodic oscillating back ground can happen for any subsonic velocity in a discrete ferromagnetic chain under deformability effects. From a physical point of view, this

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kind of motion that is conducted by the nanopterons wave type is the result of a nonlinear interaction of a kink with a wave propagating along the chain. From this interaction, it comes that a bound state of a kink and the wave is created in such a way that the wave pushes the kink, on the one hand, and the wave is finally pinned to the kink, on the other hand. Therefore, some kind of dynamical self-trapping of a kink and the wave takes place. This mechanism finally reduced the bound state velocity to that of a sub-sonic wave. It has also been possible to find a non-oscillating kink for the in-plane spin component angle and the corresponding pulse for the out-of-plane spin component angle as solution of the nonlinear discrete coupled equations of motion for the magnetic chain. This is possible only for some finite number of preferred velocity and suitable chosen deformability parameters. A similar results for moving kink concerning the discrete spectrum of the velocity has been obtained in [42] for a modified 4φ model where one explicit kink solution is known. We also notice from the kink –kink interaction that it was possible to find moving or standing bound state of the same topological charge 1±=Q to form a soliton of higher topological charge. This moving bound states display a free propagation behavior and it has been first discovered in the past by Peyrard and Kruskal [31]. They interpreted this as a resonance interaction. We have also observe a three particles interaction that manifest itself with and alternation between and attractive and a repulsive interaction that happens between the two excitations of kink type engaged in the collision while keeping a minimum distance between them whenever they face an attractive interaction. Within this minimum distance is located a breather that length‘s vary from a minimum to the maximal values. Moreover from a kink and antikink interaction simulations, we have obtained, on the one hand, a standing bound state of a kink and antikink, and on the other hand, a four particles interactions. This four particles interaction is obtained as result of the creation of a second pair of kink and antikink of reduced size (for the in-plane component) and a second pair of pulse moving towards each other for the out-of plane component. The final colliding scenario is that of four excitations of soliton type interacting in the magnetic chain for both of the spin component. For all the bound states described above, only the standing bound state of the kink –antikink can be found in the continuous limit of our magnetic system. Therefore the other entire one described above are features of deformability effects on discrete deformable ferromagnets. Thus the existing results demonstrate the importance and the potential of a simple, intuitive and physically appealing picture, which, even if it might not be quantitatively correct, reasonably describes the complex dynamics and certainly stimulates further experiments on magnetic model systems. The fundamental question for the existence of the implicit-soliton in one-dimensional magnet, as suggested by our theory, appears to have been answered positively by our numerical experiments. Further computational, experimental and theoretical efforts should now aim at the investigation of detail of these implicit soliton-like bearing systems. This in particular calls for more theoretical work to establish analytically expression of the critical magnetic field (which is actually very difficult to derived) for these implicit –solitons and compare to its corresponding numerical value. We also expect that and important role that this modified Zeeman energy in our magnetic model for future investigations, both theoretical and experimental, will be to clarify the transition between the ballistic, the stochastic and the diffusive behavior for the dynamics [35], and investigate a possible induction of phase transition for the thermodynamics in a none dimensional Heisenberg model.

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Acknowledgments

Dr. Nguenang Jean-Pierre would like to thank the Abdus Salam International Centre for Theoretical Physics (AS-ICTP), Trieste, Italy, and the Swedish International Development Agency (SIDA) for sponsoring his visit as a Junior Associate at the AS-ICTP. The authors would like to acknowledge Prof Alexander A. Nersesyan of the Condensed Matter group of the Abdus Salam ICTP for helpful suggestions and critical reading of the manuscript.

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[34] T.C.Kofané, J.Phys.Condensed.Matter11, 2481(1999). [35] J.P.Nguenang, A.J.Kenfack, and T.C.Kofané, submitted to Phys.Rev.B. [36] M. Remoissenet and M.Peyrard. J.Phys.C: Solid State.Phys 14L481 (1981). [37] M.Peyrard and M.Remoissenet, Phys.Rev.B26, 2886 (1982). [38] A.Tsukamoto, T.Fijita, K.Nakagawa and A.Itoh, Trans on Magnetics 35, 2998 (1999). [39] J.P Boyd, Nonlinearity 3, 177 (1990). [40] A.V.Savin, Y.Zolotaryuk, J.C.Eilbeck, Physica D 138, 267 (2000). [41] C.Etrich, H.J.Mikeska, E.Magyari,H.Thomas, amd R.Weber, Z.Phys.B: Condens.Matter 62,97(1985). [42 ] S. Flach, Y. Zolotaryuk, K. Klado, Phys.Rev. E 59, 6105(1999).

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Figure Captions Fig.1: Computer generated transient motion of (a) the in-plane and (b) the out-of-plane solitary waves governed by the original discrete system of Equation (2.5a) an d (2.5b) for the deformability parameter (r=0.5) and the reduced magnetic field b=0.25. Here the initial conditions are based on the analytical solution of the continuum approximation with the initial velocity of v=0.7. Fig.2: Maximum and minimum mean propagation velocity of the resulting soliton against the deformability parameter (r) for all the reduced magnetic field with range [0, 0.61]. Fig.3: Computer generated motion both for the in-plane (Fig.3a) and out-of-plane (Fig.3b) solitary waves governed by the original discrete system of Eq (2.5a) and Eq (2.5b) for the deformability parameter (r=-0.9) and the reduced magnetic field b=0.013 with the initial velocity v=0.52. Fig.3c : Is the contour plot of the position of the two excitations engaged in the colliding process against the time. Solid line corresponds to the trajectory of the excitation initially positioned on the left side of the center of the chain. While the corresponds to the second excitation initially positioned on the right side of the spin lattice. Fig.4: Plot of the time development of an exact solution of the continuum equation of motion in the discrete spin for b=0.03 and r=0.4 and the velocity of the initial condition v=0.5. a: In-plane component motion b: Out–of–plane component motion Fig.5: Plot of the shape the nanopteron for the in-plane component of the spin moving in a spin chain for the parameters r=-0.8, b=0.32,and the velocity v=0. 74. Fig.6: Head on collision for a π2 -kink and a - π2 kink (or equivalent to a π2 -antikink and a π2 -antikink collision) for the spin motion of the in-plane component (Fig.6a) and the derived pulse-pulse collision of the out-of–plane spin motion (Fig.6b) for the reduced magnetic filed of b=0.5 and the deformability parameter of r=0.5. Fig.6c: Illustrate a three particles interaction as a result of a head on collision between a π2 -kink and a - π2 kink (or equivalent to a π2 -antikink and a π2 -antikink collision) for the reduced magnetic field of b=0.21, and the deformability parameter r =0.6, with an initial velocity of v=0.52 . Fig.7: Head on collision for a π2 -kink and a - π2 antikink for the spin motion for the parameters r=-0.3, and b= 0.45 with the initial velocity of v=0.4: a: In-plane component motion where we observe and elastic collision that is produced as a reflective process of each other. b: Out–of–plane component motion, here we observe and elastic collision that is produced as of a reflective process of each other but, with a reversion of the orientation of their amplitude after each collision. Fig.8: a} Creation of a standing bound state of π2 -kink and a π2 -antikink after their collision for deformability parameter r=-0.8 and the reduced magnetic field b= 0.1 with initial velocity v=0.9 (Fig.8a). b} Contour plot of the potions against time of the two excitations engaged in

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the colliding process. This schematic illustrate exactly at what time each of the excitations stop moving and remain static as long the process is

produced. (see Fig.8b). Fig.9: a} Creation of a new kink and antikink solitons with reduced shape after the first collision of the initial conditions for deformability parameter r=0.9 and the reduced magnetic field b=0.5 with initial velocity v=0.6

(Fig.9a). b} Mutual crossing of the out of plane spin component during the colliding processes of the two excitations with size conserved and no flipping shape. (see Fig.8b). Fig.10 Colliding process for the in-plane and the out of plane component for the parameters r=0.8, b=0.02 and the initial velocity v=0.52. a} From the two initial excitations we came out with a four particles interactions constituted of two couples of kink and antikink. b} From the two initial excitations we came out with a four particles interactions constituted of two couples of pulses with flipped shapes.

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