Dynamics of soliton collision phenomena on classical discrete Heisenberg weak ferromagnetic spin chain

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

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Dynamics of soliton collision phenomena on classical discrete Heisenberg weak ferromagnetic spin chain

T

E. Parasuraman Department of Physics, Indian Academy Degree College – Autonomous, Hennur Cross, Bangalore 560 043, Karnataka, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Discrete spin dynamics Collision of soliton Dzyaloshinskii-Moriya interaction

We study the collision dynamics of soliton in discrete classical ferromagnetic spin chain with DzyaloshinskiiMoriya (DM) interaction in the classical limit. The discrete spin dynamics are found to be governed by discrete nonlinear Schrödinger (DNLS) equation. One soliton and two soliton solutions of DNLS equation are obtained by employing the discrete Hirota bilinearization method. Through the graphical illustrations for the one soliton and two soliton solution, we discuss the effect of DM interaction in ferromagnetic spin chain. Inelastic collision phenomenon occurring in this system and both amplitudes of each soliton get suppressed or enhanced after collision. We also find the discreteness effect on the soliton excitation in ferromagnetic spin chain. The modulation in the soliton excitation induced by DM interaction and discreteness is also discussed briefly.

1. Introduction The idea of discrete nature has much better to understand the effect of discreteness on topological solitons and non-topological solitons, modulational instabilities, collision of wave phenomena, intrinsic localized vibrational states, and diffusion in discrete nonlinear dynamical systems [1–7]. There are many discrete models which are very useful to practical applications, such as, systems of coupled optical waveguides, models for energy transport in biophysical systems; systems that model the dynamics of DNA, discrete soliton equations related to cellular automata, discrete quantum motors, discrete ferromagnetic spin chain [8–10]. In this paper, we also consider the discrete model of weak ferromagnetic spin chain and to recognize the effect of discreteness and Dzyaloshinskii-Moriya interaction (DMI) on the collision of soliton phenomena. The phenomenon of soliton exists in the system due to the compensation of dispersion effect and nonlinear effect. Due to this compensation effect, soliton can travel over long distances with neither attenuation nor change of shape which has the prominent applications in high-rate telecommunication with optical fibers and magnetic data storage devices. This is the main reason; the investigation of soliton has received significant attention in many fields such as particle physics, nonlinear optics and condensed matter physics. Taking into the account of the Heisenberg ferromagnetic spin system, spin–spin interaction has acted as nonlinear interaction which is a significant cause to create nonlinear excitations such as spin-wave excitation and magnetic solitons [11–14]. The dynamics of nonlinear excitation is of great significance in the ferromagnetic spin chain for its potential applications

which mainly focus on the communication systems and nonlinear signal processing devices [15,16]. Recently, there are many theoretical and experimental methods to examine the nonlinear excitations in one-dimensional Heisenberg ferromagnetic spin chain with different kind of significant magnetic interactions such as biquadratic exchange interaction, anisotropic interaction, weak interaction, octupole-dipole interaction, dipole-diploe interaction and site-dependent exchange interaction [17–21]. Among all magnetic interactions, DMI plays a significant role for describing insulators, spin glasses, low temperature phases of copper oxide superconductors, phase transitions etc. DMI is basically an antisymmetric spin coupling which occurs when the symmetry around the magnetic ions is not high enough, thus leading to the mechanism of weak ferromagnetism which is due to the combined effect of spin-orbit coupling and spin-spin exchange interactions [22,23]. DMI act as natural candidates for the realization of entanglement basis and it is important for the construction of a quantum computer. References [24,25] have shown experimentally that DMI plays a sigBaCu2 nificant role on materials such as M 2O7 (M = Si, Ge ), La2 CuO4 , Yb4 As3 , YVO3 − SrVO3 and CuGeO3 by investigating the high field neutron scattering measurements and electron paramagnetic resonance. For the past few years, theoretically many magnetic models with DMI interaction have been proposed and studied [26–28,28–31]. Zaspel [28] has investigated the topological solitons in the ferromagnetic spin with DMI by the use of the classical approach. He has shown that antisymmetric exchange interaction increases the stability of soliton. Later, Daniel et al. investigated the nonlinear spin dynamics of an anisotropic Heisenberg ferromagnetic

E-mail address: [email protected]. https://doi.org/10.1016/j.jmmm.2019.165403 Received 28 January 2019; Received in revised form 1 May 2019; Accepted 2 June 2019 Available online 04 June 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

spin chain with DMI in the semiclassical limit. They were found that the addition of discreteness and DMI does not alter the velocity and amplitude of the envelope soliton [29]. Recently, Kavitha et al. established the collision of electromagnetic wave soliton along weak ferromagnetic spin chain with weak DMI in the continuum limit [30]. Kavitha et al. also studied analytically and numerically the nonlinear localized discrete breather modes through modulational instability in one-dimensional discrete weak ferromagnetic spin chain in the semiclassical limit. We found that DMI and the on-site anisotropy affect significantly the excitation of intrinsic localized modes, and the canted ferromagnetic system with the antisymmetric nature can produce the long-lived localized excitation [31]. Very recently, Huang et al. investigated the dynamics of quantum-memory-assisted uncertainty relation and Bell non-locality under a two-qubit Heisenberg XX model with an inhomogeneous magnetic field [32]. They analyzed the entropy uncertainty relation changes when choose different strength of inhomogeneous or homogeneous magnetic field and also found that Clauser-Horne-Shimony-Holt inequality has an opposite variation compared to that of the uncertainty. Afterward, Ming et al. studied the dynamics of uncertainty through entropy in the Heisenberg XXZ spin model under direction-oriented parameter of the DMI [33]. From this work, they showed that the dynamical evolution of the entropic uncertainty relation in Heisenberg spin models can control the DMI and it can be useful for the measurement based on quantum information processing. Recently in Ref. [34], the dynamical characteristic of the measurement of uncertainty through entropy in the one-dimensional Heisenberg XYZ model in the presence of an inhomogeneous magnetic field and DMI has been studied. This shows that DMI is desirably working to effectively reduce the magnitude of the measurement of uncertainty in the region of high-temperature. Nevertheless, the objective of present work is to study the collision of soliton phenomenon in discrete ferromagnetic spin with DMI by classical approach. In discrete ferromagnetic spin chain, the spatial size of the soliton excitation can be comparable to the lattice spacing and consequently the discreteness of the underlying physical systems is expected to have a significant effect on the properties of soliton excitation. There are two major reasons for studying discreteness and DMI on collision of soliton in ferromagnetic spin system. First reason, evaluate the effect of discreteness which might stimulate symmetry fluctuation in the region of soliton excitation. Second reason; explore the DMI impact on soliton collision in the discrete ferromagnetic spin chain. Therefore, in this paper, we study the collision of soliton in one-dimensional discrete Heisenberg ferromagnetic spin with DMI through a classical approach. The plan of this paper is organized as follows. In Section 2, we present the mathematical model for the discrete weak ferromagnetic system and construct the DNLS equation of motion using classical spin vectors. In Section 3, we employ the discrete Hirota bilinearization method on DNLS equation and formulate a discrete Hirota bilinear equation. We solve the discrete Hirota bilinear equation, attempt to construct a one soliton and two soliton solutions which is used to explore the role of DMI and discreteness on the same. The results are presented in Section 4.

J > 0 is the short-range nearest neighbor exchange coupling constant. The second term is the DMI term which is proportional to the spin–spin interactions. Dzyaloshinsky first suggested that antisymmetric spin coupling to study the weak ferromagnetism in some antiferromagnetic materials of crystals from a purely symmetry ground state and later Moriya was derived by theoretically [37–39]. This interaction is also initiate to enhance the fluctuation of the spin components in the plane perpendicular to D. The vector D represents the intensity of DMI imposed along the chain. The third term expresses the single ion uniaxial anisotropy energy due to the crystalline field. It constrains the spin to lie in a plane perpendicular to the chain axis. A is the uniaxial crystalfield anisotropy parameter. The nonlinear equation motion for the spin operators Sn+ can be constructed from

iℏ

i

dSn+ dt = − ∑ [J1 [Snz Snx+ 1 + Snz Snx− 1 + iSnz Sny+ 1 n

+ iSnz Sny− 1 − iSny Snz+ 1 − iSny Snz− 1 − Snx Snz+ 1 − Snx Snz− 1] + D z [iSnz Snx− 1 + Snz Sny+ 1 − iSnz Snx+ 1 − Snz Sny− 1]+A [2iSny Snz + 2Snx Snx ]]. (3) In the quasi-classical approximation, where the components of the spin operators Snz S

i

=

are

Sn

complex

amplitudes,

un =

Sn+ , S

un∗ =

Sn− S

and

|2

1 − |un , we obtain

1 1 dun = − ∑ ⎡J1 [(un + 1 + un − 1 − 2un ) − |un |2 (un + 1 + un − 1) + (|un + 1 2 2 dt n ⎣ 1 |2 + |un − 1 |2 )] + iD z [ |un |2 (un + 1 − un − 1) 2 1 −(un + 1 − un − 1)] + A [un − |un |2 un]⎤. (4) 2 ⎦

Above equation is in the form of generalized DNLS equation which describes the complete dynamics of weak ferromagnetic spin chain. The formation of Eq. (4) also resembles that of the DNLS equation representing the dynamics of ferromagnetic spin chain is obtained through quasi-classical approximation [4,31]. When the Heisenberg ferromagnetic spin chain acts as a continuum, soliton solution for continuum spin system has been explored to understand the peculiarities of slow neutron scattering on magnets, dynamics of switching in magnets, dynamic structures from factors, and so on [40–42]. However, discreteness of the system makes the properties of the system as periodic in which interplay between the nonlinearity and discreteness gives rise to several interesting class of features which are absent in the continuum limit. To the best of our knowledge, collision of soliton phenomena in a discrete weak ferromagnetic spin chain has not been reported in the literature. Therefore, in the next section, we derive the one soliton and two soliton solutions for DNLS equation through the discrete Hirota bilinearization method.

The model which we consider in this section is a chain of N classical spins interacting through nearest neighbours. It is also subject to DMI and an anisotropic field perpendicular to the chain direction. Hence, we write the following Heisenberg Hamiltonian for a ferromagnetic system as [35,36]

n

(2)

In order to express the spin Hamiltonian in dimensionless form, we S write S n = hn and introduce S±n = Snx ± iSny . The vector D is to be chosen ± x D = D ± iD y in analogy to the spin vectors. For our convenience, the Planck constant has been chosen as unit. The nonlinear equation of motion for Sn+ are

2. Model and discrete nonlinear spin dynamics

H = − ∑ [J1 (S n . S n + 1) + D. (S n × S n + 1) − A (Snz )2].

dSn+ = [Sn+, H ]. dt

3. The discrete Hirota bilinearization method for DNLS equation To solve the nonlinear Schrödinger (NLS) type of equation is difficult and tedious, but the solution of the equation has been always in a great demand to explain the complex dynamics of the underlying physical systems. In the past decades, several powerful methods have been established and obtain the exact solution of NLS equation

(1)

where S = (Snx , Sny, Snz )(n = 1, 2, ...,N ) represents the classical component of spin vectors on the nth site. The first term in the Hamiltonian (Eq. (1)) represents the nearest-neighbor exchange interaction, where 2

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

[43–47,47,48]. However, it is more difficult for finding and studying the soliton solution of DNLS equation than that of the continuum nonlinear system. The soltion solution of DNLS equation plays a significant role in the study of the discrete nonlinear physical system. Recently, many powerful methods has been presented to find the exact solution of DNLS equation, such as the multi-linear variable separation approach [49], tanh function expansion method [50], the hyperbolic function approach [51], the Jacobi elliptic function expansion method, etc. These methods have been used for constructing one soliton solution of DNLS equation, but it is very difficult to construct a multi-soliton solution of DNLS equation by the same methods. In 1990s, Narita used discrete Hirota bilinearization method to construct a n-soliton solution to DNLS equation [52,53]. Hence, the same method has been used for constructing two soliton solutions to describe the dynamics of discrete ferromagnetic spin chain in the presence of DMI. We assume the Hirota bilinear transformation [52–57],

un =

gn fn

where ∊ is the form a power series expansion parameter, the above expansion as an insert into Eqs. (6)–(9) and solve to construct the one soliton and two soliton solution in the next section. 3.1. One soliton solution of DNLS equation In order to get the one soliton solution, the power series expansions are terminated as

gn = ∊ gn(1) ,

fn = 1 +

(18)

Substituting Eqs. ((17) and (18)) into Eqs. (6)–(9) and collecting the terms with the same power of ∊, we get

A1 (gn(1) . 1) = 0, f n(2)

+

A1 (gn(1) . f n(2) )

= 0,

(5)

where gn and fn are complex and real functions respectively. Substituting the above transformation into Eq. (4) and decoupling of resultant equations lead to bilinear equations

(19)

f n(2) )

A2 (1.

.

(17)

∊2 f n(2) .

=

−gn(1) .

gn(1) *,

(20) (21)

and

A3 (gn(1) . 1) = 0,

(22)

(7)

A 4 (gn(1) . gn(1) *) = 0,

(23)

A3 (gn . fn ) = 0,

(8)

A3 (gn(1) . f n(2) ) = 0.

(24)

A 4 (fn2 . gn . gn∗) = 0.

(9)

A1 (gn . fn ) = 0,

(6)

A2 (fn . fn ) = −gn . gn∗,

We set that

gn(1) = e θ ,

With

A1 = iDnt + 2J1 (coshDn − 1) + 2D z sinhDn + 2A, A2 =

J1 (coshDn − 1), A

A3 = J1coshDn +

(25) ∗

f n(2) = ae θ + θ , θ = kn + ωt .

(10)

(26)

Then Eq. (19) and Eq. (20) successively gives (11)

D z sinhDn ,

ω = 2i [J1 (coshk − 1) + D z sinhk + A],

(12)

a=−

(13)

A 4 = J1cosh2Dn. The Dt and Dn are defined as m

A 1 . 4J1 sinh2 k + k∗

(

2

)

(28)

Substituting Eqs. ((27) and (28)) into Eqs. ((17) and (18)) and set the ∊ = 1, then the one-soliton solution of Eq. (4) can be obtained as follows

q

∂ ∂ ⎞ ⎛ ∂ ∂ ⎞ Dtm Dnq gn (t ) fn (t ) = ⎛ − − g (t ) fn′ (t ′)|t = t ′, n = n′ . ∂t ′ ⎠ ⎝ ∂n ∂n′ ⎠ n ⎝ ∂t

(27)

(14)

4J1sinh2

(

k + k∗ 2

) exp[kn + 2i [J (coshk − 1) + D sinhk + A]] t . 4J sinh ( ) − Aexp[k + k ] z

1

The above set of equations can be solved recursively by introducing power series expansion, the following power series expansion for the fn and gn as

un =

gn = ∊ gn(1) + ∊3 gn(3) + ∊5 gn(5) + …,

Solution of Eq. (29) describes the dynamics of one-soliton in a discrete ferromagnetic spin chain in the presence of DMI as shown in Fig. 1. Under the fluctuations of the value of exchange interaction J1, the

fn = 1 +

∊2

f n(2)

+

∊4

f n(4)

+

∊6

f n(6)

1

∗

2

(29)

(15)

+ …,

2

k + k∗

(16)

Fig. 1. Profile of the one soliton represented by solution of Eq. (29) with the choice of parameters k = 0.9, k ∗ = 1.2, J1 = 0.7, A = 5.1 and D z = 5.2 . 3

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

ω2 = i [2J1 (cosk2 − 1) + 2D z sink2 + 2A].

(47)

Eq. (32) successively gives

a (1, 1∗) = −

k1 + k1∗ ⎞ A sinh−2 ⎛ , J1 ⎝ 2 ⎠

(48)

a (1, 2∗) = −

k1 + k 2∗ ⎞ A , sinh−2 ⎛ J1 ⎝ 2 ⎠

(49)

a (2, 1∗) = −

Fig. 2. Profile of the one soliton represented by solution of Eq. (29) with various values of J1. The choice of parameters k = 0.9, k ∗ = 1.2, A = 5.1 and D z = 5.2 .

a (2, 2∗) = −

changes occur in the amplitude and phase of one soliton periodically with the evolution of time as depicted in Fig. 2. According 2-D plot, when DMI as low, an increases the strength of exchange interaction which abruptly modifies the soliton amplitude and phase during the propagation and widths are invariable.

1∗)

a (1, 2,

gn = ∊

+

gn(3) ,

k2 + A sinh−2 ⎛ J1 ⎝ 2 ⎜

⎟

k1∗ ⎞ ⎟

,

(50)

⎠

k2 + k 2∗ ⎞ A . sinh−2 ⎛ J1 ⎝ 2 ⎠ ⎜

⎟

(51)

(

(

) ( (

k −k ⎡ sinh 1 2 2 A⎢ J1 ⎢ sinh k1 + k2∗ sinh ⎢ 2 ⎣

(

2

) k1 + k1∗ 2

)

(52)

2

)

) (

⎤ ⎥ , ⎥ ⎥ ⎦

k2 + k2∗ 2

)

⎤ ⎥ . ⎥ ⎥ ⎦

(53)

Eq. (34) gives

(30)

fn = 1 + ∊2 f n(2) + ∊4 f n(4) .

⎜

k −k ⎡ sinh 1 2 2 A = ⎢ J1 ⎢ sinh k2 + k1∗ sinh ⎢ 2 ⎣

a (1, 2, 2∗) =

In order to obtain the two soliton solution, the power series expansion Eqs. ((17) and (18)) is terminated on the higher-order terms

∊3

⎟

Eq. (33) gives

3.2. Two soliton solution of DNLS equation

gn(1)

⎜

a (1, 2, 1∗ , 2∗) = a (1, 2) a (1, 1∗) a (1, 2∗) a (2, 1∗) a (2, 2∗) a (1∗ , 2∗),

(31)

where

a (1∗ ,

2∗)

(54)

is defined as

Substituting Eqs. ((30) and (31)) into Eqs. (6)–(9) and collecting the terms with the same power of ∊, we obtain the following bilinear equations

a (1∗ , 2∗) =

A1 (gn(1) .

Substituting the Eqs. (48)–(54) into Eqs. ((30) and (31)) and obtain the explicit form of two soliton solution of Eq. (4) with ∊ = 1

1) = 0,

(32)

A2 (1. f n(2) + f n(2) ) = −gn(1) . gn(1) *,

(33) (34)

A2 (f n(4) + f n(2) . f n(2) + f n(4) ) = −gn(1) . gn(3) * − gn(1) *. gn(3) ,

(35)

(55)

∗

e θ1 + e θ2 + a (1, 2, 1∗) e θ1+ θ2 + θ1 + a (1, 2, 2∗) e θ1+ θ2 + θ2 ∗ 1∗) e θ1+ θ1

∗ 2∗) e θ1+ θ2

∗ 1∗) e θ2 + θ1 ⎞

⎟ ⎠

. (56)

3.3. Discussion on the two soliton solution

A3 (gn(1) . 1) = 0,

(36)

A 4 (gn(1) . gn(1) * + gn(1) . gn(1) *) = 0,

(37)

A3 (gn(1) . f n(2) + gn(3) ) = 0,

(38)

A 4 (gn(1) . gn(3) * + gn(3) . gn(1) * + 2gn(1) . gn(1) *. f n(2) + gn(1) . gn(1) *) = 0,

(39)

In this section, we investigate the dynamics of the collision of soliton through the two soliton solution of Eq. (56) and analyse the effect of exchange interaction and DMI on the collision of soliton evolution in the discrete ferromagnetic spin chain. The main motive of the work is to show the role of DMI on the evolution of the collision of soliton in ferromagnetic spin chain. Hence, we decided to analyse the two soliton solution in the following ways; (i) tune the value of the exchange interaction parameter J1 while keeping the DMI parameter as zero (ii) The exchange parameter as low value while varying the DMI parameter and find the effect of evolution of the soliton in ferromagnetic spin chain. In the first case, we have chosen parameters k1 = 1.0 + 4.7i, k2 = 1.2 − 4.1i, A = 0.101 and D z = 0 and plotted the inelastic collision of soliton as depicted in Fig. 3. From Fig. 3a, the value of J1 as 1.0, two soliton S1 and S2 pass each other and they collide each other at t = −1. After the collision both solitons amplitude have increased and induced more spikes on the top of two soliton which indicated the presence of perturbation on soliton evolution. Then, we change the value of parameter J1 = 1.5 in Fig. 3b and see that after collision of two solitons intensity and phase as changing (see corresponding counter plot of Fig. 3a) abruptly and induced more spikes on the top of two solitons. The intensity distribution among two solitons are changing during the collision and it turns out that an inelastic effect can be obtained. As a result of this case, we observed that before and after collision of soliton intensity and phase as increasing as the value of exchange interaction parameter J1 increases with D z = 0 .

We assume that

gn(1) = e θ1 + e θ2, = a (1, 2,

f n(2) = a (1,

(40)

∗ 1∗) e θ1+ θ2 + θ1

∗ 1∗) e θ1+ θ1

+ a (1, 2,

+ a (1,

∗ 2∗) e θ1+ θ2 + θ2 ,

∗ 2∗) e θ1+ θ2

+ a (2,

∗ 1∗) e θ2 + θ1

(41) ∗

+ a (2, 2∗) e θ2 + θ2 , (42)

f n(4)

⎟

+ a (1, + a (2, ⎛1 + a (1, ⎜ ∗ θ2 + θ2∗ + a (1, 2, 1∗ , 2∗) e θ1+ θ2 + θ1∗+ θ2∗ ⎝ + a (2, 2 ) e

and

gn(3)

⎜

∗

un (t ) =

A1 (gn(1) . f n(2) + gn(3) ) = 0,

k ∗ − k 2∗ ⎞ J1 sinh ⎛ 1 . A 2 ⎝ ⎠

= a (1, 2,

1∗ ,

∗ ∗ 2∗) e θ1+ θ2 + θ1 + θ2 ,

(43)

and

x1 = k1 n + ω1 t + x10 ,

(44)

x2 = k2 n + ω2 t + x 20.

(45)

Insert the above equations in the set of Eqs. (32)–(38) and solve the Eq. (32) and obtain the dispersion relations

ω1 = i [2J1 (cosk1 − 1) + 2D z sink1 + 2A],

(46) 4

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

Fig. 3. Profile of the inelastic collisions of two solitons expressed by solution of Eq. (56), parameters adopted are k1 = 1.0 + 4.7i , k2 = 1.2 − 4.1i , A = 0.101 and Dz = 0.

solitons S1 and S2 amplitude and phase as changing by increasing the value of J1 with absence of DMI. Fig. 6 (left and right)) shows that three different combinations of collision of soliton S1 and S2 for three different values of DMI parameter with J1 as 0.1. From this figure, we observed that before the collision of soliton amplitude as high and after the collision of soliton amplitude decreases abruptly. Thus, after the inelastic collision of solitons S1 and S2 amplitudes are suppressed under the weak DMI in ferromagnetic spin chain. We have also plotted another case of 2D soliton collision plot as shown in Figs. 7 and 8 with choice parameters k1 = 1.0 + i, k2 = 1.0 − i . We also observed the similar effect from Figs. 7 and 8, but no spikes on the soliton. Therefore, the interesting property of soliton in discrete ferromagnetic spin chain with DMI is that they exhibit shape changing collision characterized by intensity redistribution, amplitude-dependent phase shift, and relative separation distances. Therefore, this study will be more useful to construct the collision-based logic gates and all optical computations.

The second case, we vary the DMI parameter in Fig. 4, in which parameters are chosen as k1 = 1.0 + 4.7i, k2 = 1.2 − 4.1i, A = 0.101 and J1 = 0.01. In Fig. 4a, the low intensity of collision of soliton are observed, from which we found that before collision of two soliton amplitudes are low and after collision of two solitons S1 and S2 amplitudes are increasing for the value of D z = 1.0 . When we increase the value of DMI as D z = 1.5, amplitude and phase of two solitons are changing again while more breather spike are induced on top of soliton S1 compared with S2 as shown in Fig. 4b. From Fig. 4, it is evident that the antisymmetric spin coupling in the form of DMI induces the low intensity of soliton excitation (see J1 of case (i) Fig. 3 as high intensity of solitons) and intensity redistribution of inelastic collision of soliton as dependent on DMI. A similar low intensity breather like soliton excitation and inelastic collision of soliton in ferromagnetic chain was observed by Saravanan [26,27,30] and he has investigated the effect of DMI on the dynamics of soliton on the basis of continuum limit. However, we studied the effect of DMI on the evolution of the collision of soliton in the discrete limit and analyzed the role of exchange interaction with DMI and without DMI on the dynamics of inelastic collision of soliton in a discrete ferromagnetic spin chain. Figs. 5–8 shows that two typical moments of before (t = −2.5) and after (t = 2.5) collision of soliton for different values of J1 and D z . We also have plotted for different values of k1 = 1.0 + 4.7i, k2 = 1.2 − 4.1i (for Figs. 5 and 6) and k1 = 1.0 + i, k2 = 1.0 − i (for Figs. 7 and 8). Figs. 5–8 confirm our earlier discussion of 3D collision plots (see in Figs. 3 and 4). In the absence of DMI, we fix the different values of exchange interaction parameter J1 = 1.0, 1.5 and 2.0 , obtain three different types of two solitons (S1) and (S2 ) (before the collision plot of Fig. 5). After the collision, all three different types of two solitons (S1) and (S2 ) amplitude and phase as changing is exploited in Fig. 5 (Right side of figure). Therefore, we understood that after the collision of

4. Conclusions The soliton excitation in ferromagnetic spin chain with different kinds of interaction have been investigated in numerous literatures. But in this paper, we have investigated the dynamics of collision of soliton in ferromagnetic spin chain based on the discrete approximation. Under the quasi-classical approximation, the dynamics is found to be governed by a DNLS equation. With the help of Hirota Bilinearization method, one soliton and two soliton solutions of DNLS equation are derived. We found inelastic collision of soliton in ferromagnetic spin chain with choice of parameters k1 = 1.0 + 4.7i, k2 = 1.2 − 4.1i, A = 0.101 and D z = 0 (as seen in Fig. 3 and Fig. 4). We have analyzed the inelastic collision of soliton in ferromagnetic spin chain and found that both amplitudes of each soliton get suppressed or enhanced after the 5

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

Fig. 4. Profile of the Inelastic collisions of two solitons expressed by solution of Eq. (56), parameters are the same as those in Fig. (3) except for J1 = 0.01.

Fig. 5. Intensity profile of two soliton collision in discrete ferromagnetic spin chain with absence of DM interaction (D z = 0 ). Parameters adopted are k1 = 1.0 + 4.7i , k2 = 1.2 − 4.1i , A = 0.101 and D z = 0 .

Fig. 6. Intensity profile of two soliton collision in discrete ferromagnetic spin chain with presence of DM interaction. Parameters adopted are k1 = 1.0 + 4.7i , k2 = 1.2 − 4.1i , A = 0.101 and J1 = 0.01.

interaction, DMI, anisotropy along the easy axis of magnetization and magnetic field component of the propagation of electromagnetic waves. Whereas in our work, we considered the same model except magnetic field component. We note that the Ref. [30] has performed the analysis

collision due to the existence of DMI. Our results are almost close to the Ref.[30]. According to Ref. [30], the collision of electromagnetic soliton in a ferromagnetic spin chain with DMI is in the continuum limit. They used Heisenberg model with the incorporation of exchange 6

Journal of Magnetism and Magnetic Materials 489 (2019) 165403

E. Parasuraman

Fig. 7. Intensity profile of two soliton collision in discrete ferromagnetic spin chain with absence of DM interaction (D z = 0 ). Parameters adopted are k1 = 1.0 + i , k2 = 1.0 − i , A = 0.101 and D z = 0 .

Fig. 8. Intensity profile of two soliton collision in discrete ferromagnetic spin chain with presence of DM interaction. Parameters adopted are k1 = 1.0 + i , k2 = 1.0 − i , A = 0.101 and J1 = 0.01.

Appendix A. Supplementary data

in the framework of the Landau-Lifshitz-Maxwell equations governing the spin evolution in ferromagnet and also used a reductive perturbation method to obtain the perturbed derivative nonlinear Schrödinger equation which describes the spin excitations under the influence of electromagnetic wave. However, in our work, the dynamics of ferromagnetic spin have described as DNLS equation using quasi-classical approximation. In Ref. [30] authors have solved perturbed derivative nonlinear Schrödinger equation by the use of tangent-hyperbolic method and Hirota bilinerization method. They also constructed the one soliton and two soliton solutions through which demonstrated the collision scenario between the electromagnetic soliton and also established the role DMI in ferromagnetic spin chain. Whereas in our work, we used discrete Hirota bilinearization method to solved the DNLS equation. The dynamics of collision of soliton are analyzed by the presence and absence of DMI in ferromagnetic spin. We found that presence of DMI induces the low intensity of soliton excitation which exactly similar to the results as found in [30]. Moreover, we found that soliton which is perturbed due to discreteness effect and DMI effect. The effect of presence of DMI on the soliton collision profile has been compared with the absence of DMI on the soliton collision profile. Thus, we conclude that intensity redistribution of inelastic of collision of soliton depends on the value of DMI. In this work, mainly we addressed the effect of discreteness and DMI on collision of soliton in a discrete ferromagnetic spin chain.

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Acknowledgments I wish to thank the anonymous referees for helpful suggestions. The author thank Prof. A.J. Sievers, Prof. P. Nagaraju and M. Saravanan for useful discussions and suggestions. The author gratefully acknowledges the International Centre for Theoretical Sciences (ICTS) for having provided an opportunity to discuss this work with the senior Professors during my visit for participating in the program of Integrable systems in Mathematics, Condensed Matter and Statistical Physics-2018.

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