A spin rotator model for Heisenberg helimagnet

Physica A 491 (2018) 1–12

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Physica A journal homepage: www.elsevier.com/locate/physa

A spin rotator model for Heisenberg helimagnet A. Ludvin Felcy a , M.M. Latha b, * a b

Department of Physics, St. Xavier’s Catholic College of Engineering, Nagercoil-629 003, India Department of Physics, Women’s Christian College, Nagercoil-629 001, India

highlights • The dynamics of a helimagnetic spin system has been studied by proposing a spin rotator model. • The effect of inhomogeneity is also analysed. • Similar studies are carried out for the system including biquadratic type interactions.

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Article history: Received 24 November 2016 Received in revised form 24 April 2017 Available online 25 September 2017 Keywords: Helimagnet Spin rotator model

a b s t r a c t We study the dynamics of a helimagnetic spin system by proposing a spin rotator model taking into account bilinear, twist interplane and anisotropic interactions in the continuum limit. The dynamical equations of motion are obtained and studied numerically. The influence of different types of inhomogeneities is also analysed. Similar studies are carried out for the system including biquadratic type interactions. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Over the last few decades, there has been considerable interest in the study of low-dimensional helimagnetic spin models [1–4]. Many experimental and theoretical studies have been carried out in the recent past on helimagnetic systems by proposing different models. Theoretical studies include quantum properties of helimagnetic thin films [5,6], low temperature thermodynamics of Heisenberg helimagnets [7], phase transition properties [8–11], effect of nonmagnetic impurities [12– 14] etc. Experiments to measure heat capacity, temperature derivative of resistivity, thermal expansion etc. [15,16] and studies on quantum phase transitions [17–19] have also been carried out recently. In a recent experimental study on spin waves and magnetic interactions in LiCu2 O2 helimagnet, using single crystal inelastic neutron scattering, Masuda et al. [20] have measured the dispersion of spin wave excitations in the vicinity of magnetic Bragg reflection. In addition, investigations on nonlinear spin excitations have been done [21,22] both in discrete and continuum limits using the Heisenberg helimagnetic model. Extension of the above studies to higher dimensions has been carried out by the present authors [23,24] by proposing a square lattice model of Heisenberg helimagnet taking into account bilinear, biquadratic and anisotropic interactions. In the studies on the nonlinear excitations of Heisenberg helimagnet, the dynamics is represented by nonlinear partial differential equations. These equations are derived after bosonizing the Hamiltonian using the Holstein– Primakoff (H–P) representation of spin operators. But the angle of rotation of the spins in the XY and XZ-planes (interplane twist interaction along the helical axis) has not been taken into account in the earlier studies. Hence in this work, we analyse the dynamics of a Heisenberg helimagnet incorporating interplane twist interaction in addition to bilinear, biquadratic and

*

Corresponding author. E-mail address: [email protected] (M.M. Latha).

https://doi.org/10.1016/j.physa.2017.08.059 0378-4371/© 2017 Elsevier B.V. All rights reserved.

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A.L. Felcy, M.M. Latha / Physica A 491 (2018) 1–12

Fig. 1. A horizontal projection of the nth spin in the xy and xz plane.

anisotropic interactions. Moreover, in real helimagnets, the interplane twist interaction may vary and hence is expected to be inhomogeneous. Most of the studies on the nonlinear spin models of helimagnet are based on the homogeneous Heisenberg Hamiltonian. Recently Saravanan [25] has reported a study on the nonlinear spin dynamics of the Heisenberg helimagnet under the perturbation of electromagnetic wave propagation in the system. In the present study, we also analyse the effect of site dependent inhomogeneity by considering a model incorporating the interplane twist interaction along the helical axis. The paper is organized as follows: In Section 2, we propose a spin rotator model Hamiltonian with bilinear and anisotropic interactions in a one dimensional lattice of Heisenberg helimagnet and construct the equations of motion by using the continuum approximation. We analyse the soliton excitations in the system graphically. In Section 3, we consider an inhomogeneous helimagnetic model with bilinear, anisotropic and twist interactions and investigate the effect of nonlinear type inhomogeneities such as localized, cubic, biquadratic and periodic inhomogeneities. In Section 4, we study the nonlinear dynamics of Heisenberg helimagnetic spin chain by taking into account the effect of biquadratic exchange interactions in the continuum limit. The nonlinear solitary excitations in the inhomogeneous helimagnetic spin system is studied in Section 5 and the results are concluded in Section 6. 2. Model and equations of motion We consider a single helix spin model of a helimagnet with helical axis taken in the z-direction. The exact helix spin model is schematically shown in Fig. 1 where the spin is depicted by an arrow with unit length. Let the coordinate of Pn , which is the point where the nth spin is attached to the helix be (R cos nφ0 , R sin nφ0 , Zn ) with φ0 = 2pπ . Here R is the radius of the circle and p is the number of spin per turn. Also let (θn , φn ) be the angle of rotation of the nth spin around the point Pn . The quantities (θn , φn , Zn ) measure the deviations of the position of the nth spin. The Heisenberg model of the Hamiltonian for the above system is written as H = H0 + H1 + H2 .

(1)

∑ ∑ Here H0 = − n J(Sn .Sn+1 ) represents the isotropic exchange interaction. H1 = − n 0 [k.(Sn × Sn+1 )]2 is responsible for the helical arrangement of spins in the magnetic system. k is the unit vector along the z-axis which is perpendicular to the ∑ plane. H2 = − n − A(Szn )2 corresponds to the uniaxial anisotropy energy with anisotropy axis parallel to the z- axis which y tends to rotate the spin in planes perpendicular to the helical axis. In terms of the components (Snx , Sn , Snz ), Hamiltonian (1) is rewritten as H = a1 + a2 + H2 ,

(2)

where a1 = −

∑

y

J(Snx Snx+1 + Sny Sn+1 + Snz Snz+1 ),

(3)

n

a2 = −

∑

y

y

y

0 (Snx Sn+1 Snx Sn+1 + Sny Snx+1 Sny Snx+1 − 2Snx Sn+1 Sny Snx+1 ).

(4)

n

In order to incorporate the twist angles, we define Sn ≡ (Snx , Sny , Snz ) = (sin θn cos φn , sin θn sin φn , cos θn )

(5)

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for the nth spin. Using Eq. (5) we write Eq. (2) as H = b1 + b2 + b3 ,

(6)

where b1 = −

∑

J(sin θn cos φn sin θn+1 cos φn+1 , + sin θn sin φn sin θn+1 sin φn+1 + cos θn cos θn+1 ),

(7)

n

b2 = −

∑

0 (sin2 θn cos2 φn sin2 θn+1 sin2 φn+1 + sin2 θn sin2 φn sin2 θn+1

n

× cos2 φn+1 − 2sin2 θn cos φn sin2 θn+1 sin φn sin φn+1 cos φn+1 ), b3 = −

∑

Acos2 θn .

(8) (9)

n

Having constructed the Hamiltonian for our model, we now derive the equations of motion using the relations

θn =

∂H

1

sin θn ∂φn

φn = −

,

(10)

∂H . sin θn ∂θn 1

(11)

Using Eq. (6) in Eqs. (10) and (11), we obtain

∂θn = a11 + a12 + a13 + a14 , ∂t

(12)

∂φn = b11 + b12 + b13 + b14 + b15 + b16 ∂t

(13)

where a11 , . . . a14 , b11 , . . . b16 are given in Appendix A. Eqs. (12) and (13) establish a coupled relation between the dynamical variables θ and φ . Here we assume that θn and φn are very small and hence rewrite Eqs. (12) and (13) as

∂θn = a21 + a22 + a23 + a24 , ∂t

(14)

∂φn = b21 + b22 + b23 + b24 + b25 + b26 + b27 + b28 + b29 , ∂t

(15)

where a21 , . . . a24 , b21 , . . . b29 are given in Appendix B. In studying the properties of the dynamical system governed by Eqs. (14) and (15), we limit our discussion to the specific case by employing a continuum approximation after assuming φn and θn as φ (x, t) and θ (x, t) respectively and using the following Taylor expansion

θn±1 = θ ± γ θx +

γ2

φn±1 = φ ± γ φx +

2

θxx ± · · · ,

γ2 2

φxx ± · · · ,

(16)

where γ is the lattice spacing between neighbouring spins in a particular chain and suffix x represents partial derivative with respect to x. Using Eq. (16), the equations of motion (14) and (15) can be written as

θt + a31 + a32 = 0,

(17)

φt − b29 + b31 + b32 + b33 = 0,

(18)

where a31 , a32 , b31 , b32 , b33 are given in Appendix C. Eqs. (17) and (18) represent the dynamics of a helimagnetic spin system with bilinear, anisotropic and twist interactions. The above set of coupled equations are solved numerically using Mathematica and the solutions for θ and φ are plotted in Fig. 2. A close inspection of Fig. 2 reveals that the magnetic medium with bilinear, anisotropic and twist interactions supports soliton-like excitations. Figure shows that a small disturbance in the angular displacement of the spins propagates in the form of a soliton. θ and φ variations are almost the same except for a small change in amplitude.

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Fig. 2. Solitonic profiles for θ and φ with γ = 1.25, J = 12, A = 0.6 and 0 = 1.5.

3. Inhomogeneous helimagnetic spin system Most of the studies on helimagnetic spin systems are based on the homogeneous Heisenberg Hamiltonian where the exchange interaction coupling between the nearest-neighbour spins is a single constant J or at most two constants as in the case of uniaxial anisotropy. But the presence of the magnetic defects introduces inhomogeneity in the exchange interaction. Let us consider a one-dimensional Heisenberg helimagnetic spin system having inhomogeneities. The Hamiltonian associated with the system is given by HI = fn H0 + gn H1 + H2 ,

(19)

where the parameters fn and gn represent the variation of the bilinear and twist interactions at the lattice sites along the spin chain due to inhomogeneities. The Hamiltonian in the component form is HI = a1 fn + a2 gn + H2 .

(20)

Now, we rewrite the spin Hamiltonian (20) by introducing Eq. (5) as HI = b1 fn + b2 gn + b3 .

(21)

To understand the spin dynamics of the inhomogeneous Helimagnetic system, we construct the equations of motion corresponding to the spin Hamiltonian using Eqs. (10) and (11). They are:

∂θn = a11 fn−1 + a12 fn + a13 gn−1 + a14 gn , ∂t ∂φn = (b11 + b12 )fn−1 + b13 fn + b14 gn−1 + b15 gn + b16 . ∂t Using the approximations as in Section 2 and assuming θn and φn to be very small, we rewrite Eqs. (22) and (23) as

(22)

(23)

∂θn = a21 fn + a22 gn + a23 fn−1 + a24 gn−1 , ∂t

(24)

∂φn = b21 fn + b22 gn + b23 fn−1 + b24 gn−1 + b25 fn−1 + b26 fn + b27 gn−1 + b28 gn + b29 . ∂t

(25)

Using Taylor series expansions given by Eq. (16) and also the following equations

fn±1 = f ± γ fx +

γ2

gn±1 = g ± γ gx +

2

γ

fxx ± · · · 2

2

gxx ± · · · .

(26)

Eqs. (24) and (25) can be written in the continuum form as

θt + a31 f + a32 g + a33 fx + a34 gx = 0,

(27)

φt − b29 + (b31 + b32 )f + b33 g + (b34 + b35 )fx + b36 fxx + b37 gx = 0

(28)

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Fig. 3. The undeformed propagation of soliton with f = g= 1, c = 0.002, γ = 1.25, J = 12, 0 = 1.5, A = 0.6.

φ2 θ

φ2

where a33 = γ 2 J θ φx ; a34 = −γ 2 2J(θ 3 φx +θ 3 φ 2 φx ); b34 = γ J φ 2 ; b35 = −γ 2 J( θθx + θ x +2θφx +φφx ); b36 = −γ 2 J( 2 ); b37 = γ 2 0 (4θ 3 φx + 4θ 3 φ 2 φx ). In recent years, various powerful methods for obtaining explicit travelling solitary wave solutions to nonlinear equations have been proposed. Among them Sine–Cosine function method [26] is the simplest one to identify an explicit soliton solution because of its simple assumption of an ansatz to the given nonlinear PDEs. Here, the above set of coupled equations are solved by using the Sine–Cosine function method. To apply this method, we use the wave variable ξ = x − ct in Eqs. (27) and (28) and they become

− c θξ + a41 f + a42 g + a43 fx + a44 gx = 0,

(29)

− c φξ − b29 + (b31 + b41 )f + b42 g + (b34 + b43 )fξ + b36 fξ ξ + b44 gξ = 0,

(30)

where a51 ...a44 , b41 ...b44 are given in Appendix D. We ascertain that Eqs. (29) and (30) admit solutions of the following form:

θ (ξ ) = λ1 cosβ1 (µξ ),

(31)

φ (ξ ) = λ2 cosβ2 (µξ ),

(32)

where λ1 , λ2 , µ, β1 and β2 are constant parameters to be determined. We obtain the values of the parameters β1 and β2 by balancing the higher order derivative term with nonlinear term of the evolution equations that gives β1 = −1 and β2 = −1. After substituting the values, β1 = −1 and β2 = −1 in Eqs. (29) and (30), we obtain a system of algebraic equations. Solving these equations with the aid of symbolic computation we obtain

√ µ=

−4A + 4fJ λ22 + J γ 2 fξ ξ λ22 −2fJ γ 2 − 4fJ λ22 γ 2

and

(33)

√

λ1 = √

i fJ

−2g 0 − 2g 0 λ22

.

(34)

We now analyse Eqs. (33) and (34) under the influence of various nonlinear inhomogeneities like (i) cubic (ii) biquadratic (iii) periodic and (iv) localized inhomogeneities. 3.1. Effect of inhomogeneity on the stability of soliton When f = g = 1, the system represents a homogeneous Heisenberg helimagnetic spin chain. The soliton propagation corresponding to this is represented graphically in Fig. 3. From Fig. 3 we observe that the over all robust nature of soliton resembles Fig. 2. Fig. 4 displays the effect of localized inhomogeneity of the form f = 1 + P tan h(x) and g = 1 + R tan h(x). We notice an asymmetric multi fluctuation in the soliton structure at the localized region and a phase shift in the profiles of θ and φ when the strength of the inhomogeneities exceeds the negative limiting values P = −0.93 and R = −0.991. Thus the soliton is found to be stable when P < −0.93 and R < −0.991. The other parameters are chosen as γ = 1.25, J = 12, 0 = 1.5. The influence of cubic inhomogeneity of the form f = 1 + P1 x3 + P2 x2 and g = 1 + R1 x3 + R2 x2 is viewed in Fig. 5. When the inhomogeneity parameters P1 < 0.311, P2 < 3.3, R1 < 0.1 and R2 < 1.1, soliton is found to be stable and above these limiting values, it is unstable which is noticed from a fluctuation in the left tail region with double humps.

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Fig. 4. Deformed propagation of soliton with localized inhomogeneity for θ and φ , P = −0.93, R = −0.991.

Fig. 5. Perturbed soliton with cubic inhomogeneity for θ and φ , P1 = 0.311, P2 = 3.3, R1 = 0.1, R2 = 1.1.

Fig. 6. Perturbed soliton with biquadratic inhomogeneity for θ and φ , P3 = 4.7, P4 = −2.3, R3 = 0.52, R4 = 0.6.

Fig. 6 represents the effect of biquadratic inhomogeneity of the form f = 1 + P3 x4 + P4 x2 and g = 1 + R3 x4 + R4 x2 . From Fig. 6 we observe that multihumps are formed in the localized region when P3 = 4.7, P4 = −2.3, R3 = 0.52, R4 = 0.6. These are the limiting values of the inhomogeneity parameters that determine the stability of soliton. Next, we consider periodic type inhomogeneity f = 1 + P5 sin(x) and g = 1 + R5 sin(x). Its effect is portrayed in Fig. 7 similar distortions as in the case of biquadratic inhomogeneity is noticed in this case also. The limiting values are P5 = −0.99 and R5 = −0.77. Thus the soliton stability is affected by the presence of inhomogeneities. 4. Spin rotator model of helimagnet with biquadratic interactions In the previous section, we discussed soliton excitations in homogeneous and in inhomogeneous helimagnetic system with bilinear, anisotropic and twist interactions. Now a days, magnetic systems with different kinds of interaction have acted as important dynamical models exhibiting interesting nonlinear phenomena [27]. Among them, the biquadratic exchange interaction plays an important role and therefore in recent years there has been a considerable interest in the study of ferromagnetic spin chains with competing bilinear and biquadratic exchange interactions [28–31]. Hence in this section, we investigate the dynamics of a Heisenberg helimagnet incorporating interplane twist interaction in addition to biquadratic and anisotropic interactions. we consider a helimagnetic spin system with biquadratic interactions for which

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Fig. 7. Perturbed soliton with periodic inhomogeneity for θ and φ , P5 = −0.99, R5 = −0.77.

the Hamiltonian is written as H ′ = H + H3 .

(35)

∑ Here H3 = − n J1 (Sn .Sn+1 )2 represents the biquadratic interaction between the adjacent spins. In terms of the y x components (Sn , Sn , Snz ), Hamiltonian (35) is rewritten as H ′ = a1 + a2 + a3 + H 2 ,

(36)

where a3 = −

∑

y

y

y

J1 ((Snx )2 (Snx+1 )2 + (Sny )2 (Sn+1 )2 + (Snz )2 (Snz+1 )2 + 2Snx Snx+1 Sny Sn+1 + 2Sny Sn+1 Snz Snz+1 + 2Snz Snz+1 Snx Snx+1 ).

(37)

n

Using Eq. (5) we write Eq. (36) as H ′ = b1 + b2 + b3 + b4 ,

(38)

where b4 = −

∑

J1 ((sin θn cos φn sin θn+1 cos φn+1 )2 + (sin θn sin φn sin θn+1 sin φn+1 )2 + (cos θn cos θn+1 )2

n

+ 2 sin θn cos φn sin θn+1 cos φn+1 sin θn sin φn sin θn+1 sin φn+1 + 2 sin θn sin φn sin θn+1 sin φn+1 cos θn cos θn+1 + 2 cos θn cos θn+1 sin θn cos φn sin θn+1 cos φn+1 ).

(39)

Having constructed the Hamiltonian for our model, we now derive the equations of motion using the relations Eqs. (10) and (11) and are given by

∂θn = a11 + a12 + a13 + a14 + a+ + a− , ∂t

(40)

∂φn = b11 + b12 + b13 + b14 + b15 + b16 + b17 + b18 , ∂t

(41)

where a± = 2J1 (sin2 θn±1 sin θn sin φn cos φn (cos2 φn±1 − sin2 φn±1 ) − sin2 θn±1 sin θn cos φn±1 sin φn±1 (cos2 φn − sin2 φn )

+ sin θn±1 cos θn cos θn±1 sin(φn − φn±1 )),

(42)

b17 = 2J1 (sin2 θn+1 cos θn (cos2 φn+1 cos2 φn + sin2 φn+1 sin2 φn ) − cos θn (cos2 θn+1 + cos2 θn−1 )

+ 2 cos φn cos φn+1 sin φn sin φn+1 cos θn sin2 θn+1 +

1 sin θn

(sin θn+1 cos θn+1 cos(φn − φn+1 )(cos2 θn − sin2 θn ))),

b18 = 2J1 (2 cos φn−1 cos φn sin φn sin φn−1 cos θn sin2 θn−1 + sin θn−1 cos θn−1 cos(φn − φn−1 )(cos2 θn − sin2 θn )).

(43)

(44)

Using the approximations given in Section 2, the equations of motion (40) and (41) can be written in the continuum form as

θt + a31 + a32 + a35 = 0,

(45)

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Fig. 8. Soliton profile of θ and φ for γ = 0.08, J = 12, 0 = 2, J1 = 6, A = 0.01.

φt − b29 + b31 + b32 + b38 + b39 + b40 = 0,

(46)

where a35 = 2γ 2 J1 (2θx φx + 4θ 2 θx φx + 4θ 2 φ 2 θx φx + 2θ 3 φφx2 + θφxx + θ 3 φxx + θ 3 φ 2 φxx ),

(47)

b38 = −4J1(φ 2 + θ 2 φ 2 + θ 2 φ 4 ),

(48)

b39 = −2γ 2 J1

+

(

2θx2 + 4φ 2 θx2 + 2φ 4 θx2 +

2φθx φx

θ

θxx φ 2 θxx + θθxx + + 3θφ 2 θxx + 2θ 5 θxx θ θ

)

+ 6θφθx φx + 8θφ θx φx + 2θ φ φ + θ φφxx + 2θ φ φxx + φφxx , 3

2

2

2 x

2

2

3

(49)

b40 = −γ 2 40 θ 2 φx2 .

(50)

We now try to see whether the non linear spin excitations of helimagnetic model can be expressed in terms of solitons by solving Eqs. (45) and (46). Fig. 8 represents the profile which shows that the biquadratic interactions do not affect the soliton nature. 5. Inhomogeneous helimagnetic system with biquadratic interactions Let us consider a one-dimensional Heisenberg helimagnetic spin system with biquadratic interactions and having inhomogeneities. The Hamiltonian associated with the system incorporating biquadratic interactions is given by H ′ = HI + fn′ H3

(51) ′

where the parameter fn represents site dependent inhomogeneity associated with biquadratic interactions. The corresponding equations of motion are:

θt + a31 f + a32 g + a33 fx + a34 gx + a35 f ′ + a36 fx′ = 0,

(52)

φt − b29 + (b31 + b32 )f + (b34 + b35 )fx + b36 fxx + (b38 + b39 )f ′ + b40 g + (b51 + b52 )fx′ + b53 fxx′ = 0, 2φ 2 θx

(53)

where a36 = 2γ 2 J1 (θ φx +θ 3 φx +θ 3 φ 2 φx ), b51 = 2γ J1 (φ 2 +θ 2 φ 2 +θ 2 φ 2 ), b52 = −γ 2 J1 ( 2θθx + 2θθx + θ + 6θφ 2 θx + 4θφ 4 θx + 2φφx + 2θ 2 φφx + 4θ 2 φ 3 φx ); b53 = −γ 2 J1 (φ 2 + θ 2 φ 2 + θ 2 φ 4 ). Thus the dynamics of inhomogeneous anisotropic continuum Heisenberg helimagnet spin chain with varying bilinear and biquadratic exchange interactions in the semi-classical limit is governed by a set of generalized higher order nonlinear equations and its solution is constructed numerically which is depicted in Fig. 9 for f = f ′ = g = 1. The choice of parameters are γ = 0.08, J = 12, 0 = 2, J1 = 6, A = 0.01. Fig. 10 displays the effect of localized inhomogeneity of the form f = 1 + P tan h(x), f ′ = 1 + Q tan h(x) and g = 1 + R tan h(x). The profile of φ shows a gradual decrease in soliton amplitude as it propagates in time for values

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Fig. 9. Soliton evolution of θ and φ with f = f ′ = g = 1, γ = 0.08, J = 12, 0 = 2, J1 = 6, A = 0.01.

Fig. 10. Soliton evolution of θ and φ with localized inhomogeneity P = 26, Q = 12 and R = 9.

Fig. 11. Soliton evolution of θ and φ with cubic inhomogeneity P1 = 0.35, P2 = 0.2, R1 = 0.02, R2 = 0.03, Q1 = 0.04, Q2 = 0.08.

P = 26, Q = 12 and R = 9 and above. For values of P , Q and R less than the above said values, soliton is found to be stable. The other parameters are chosen as γ = 0.08, J = 12, J1 = 6, A = 0.01, 0 = 2. In the case of cubic inhomogeneity, we choose the deformity functions as f = 1 + P1 x3 + P2 x2 , F = 1 + Q1 x3 + Q2 x2 and g = 1 + R1 x3 + R2 x2 . The solitonic profile for cubic inhomogeneity with P1 = 0.35, P2 = 0.2, R1 = 0.02, R2 = 0.03, Q1 = 0.04, Q2 = 0.08 is portrayed in Fig. 11. A small downward fluctuation occurs in left tail region and upward fluctuation occur in the right tail region and also the amplitude of the soliton decreases as it propagates in time. Below these values, the shape of the soliton is found to be stable and robust. Next, we consider biquadratic inhomogeneity of the form f = 1 + P3 (x4 ) + P4 (x2 ), F = 1 + Q3 (x4 ) + Q4 (x2 and g = 1 + R3 (x4 ) + R4 (x2 ). For P3 < 0.03, P4 < 0.007, R3 < 0.005, R4 < 0.001, Q3 < 0.006, Q4 < 0.008 stable propagation of soliton is observed. It is clear from Fig. 12 that the deformity creates tail-humped excitations on both ends for θ when P3 , P4 , R3 , R4 , Q3 and Q4 exceeds the above values. We plot Eqs. (52) and (53) by including periodic inhomogeneity f = 1 + P5 sin(x), F = 1 + Q5 sin(x) and g = 1 + R5 sin(x) in Fig. 13. It demonstrates that when P5 = −48.05, R5 = 0.1, Q5 = −2.2, fluctuation in θ occurs in the right localized region with no variation in amplitude. The amplitude of the soliton associated with φ decreases as it propagates in time. Thus the periodic inhomogeneity in the twist produces a damping effect on the progressing soliton without altering the velocity of it.

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Fig. 12. Soliton evolution for θ and φ with biquadratic inhomogeneity P3 = 0.03, P4 = 0.007, R3 = 0.005, R4 = 0.001, Q3 = 0.006, Q4 = 0.008.

Fig. 13. Soliton evolution for θ and φ with periodic inhomogeneity P5 = −48.05, R5 = 0.1, Q5 = −2.2.

6. Conclusion In this paper we investigate the existence of solitary excitations in a spin rotator model of a helimagnet with bilinear, interplane twist and anisotropic interactions. The dynamics is found to be governed by a set of nonlinear partial differential equations in the continuum limit. These equations are solved numerically and the results are illustrated graphically. Plots show stable evolution of bell-shaped solitary waves without any broadening or compression in the amplitude. Also, a model for inhomogeneous helimagnetic system is constructed and the effect of inhomogeneity is studied by using the Sine–Cosine function method. We investigate the behaviour of soliton propagation in the inhomogeneous helimagnetic system for various types of nonlinear inhomogeneities and the results of the analysis show a symmetric or asymmetric fluctuation either in the localized or tail region of the soliton when the amount of inhomogeneity exceeds a limiting value. In addition, we investigate the dynamics of a spin rotator model of a helimagnetic system by including bilinear, biquadratic, interplane twist and anisotropic interactions. The dynamics is found to be governed by a set of higher order nonlinear partial differential equations. Solutions for various parameters are analysed graphically for nonlinear type inhomogeneities such as cubic, biquadratic, periodic and localized types. Our results indicate a distortion in soliton propagation when the strength of inhomogeneity crosses a limiting value that indicates the instability in soliton propagation. Appendix A The coefficients of Eqs. (12) and (13) are: a11 = J sin θn−1 sin(φn − φn−1 ); a12 = J sin θn+1 sin(φn − φn+1 ); a13 = −20 (sin2 θn−1 sin θn sin φn cos φn (cos2 φn−1 − sin2 φn−1 ) − sin2 θn−1 sin θn cos φn−1 sin φn−1 (cos2 φn − sin2 φn )); a14 = −20 (sin2 θn+1 sin θn sin φn cos φn (cos2 φn+1 − sin2 φn+1 ) − sin2 θn+1 sin θn cos φn+1 sin φn+1 (cos2 φn − sin2 φn )); b11 = J sin1θ (sin θn−1 cos θn cos(φn − φn−1 )); n b12 = −J(cos θn−1 + cos θn+1 ); b13 = J sin1θ sin θn+1 cos θn cos(φn − φn+1 ); n

b14 = 20 (cos2 φn−1 sin2 φn sin2 θn−1 cos θn +sin2 φn−1 cos2 φn sin2 θn−1 cos θn −2 cos φn−1 sin φn sin φn−1 cos φn sin2 θn−1 cos θn ); b15 = 20 (cos2 φn sin2 φn+1 sin2 θn+1 cos θn +sin2 φn cos2 φn+1 sin2 θn+1 cos θn −2 cos φn sin φn+1 sin φn cos φn+1 sin2 θn+1 cos θn ); b16 = −2A cos θn .

A.L. Felcy, M.M. Latha / Physica A 491 (2018) 1–12

Appendix B The coefficients of Eqs. (14) and (15) are: a21 = θn+1 J(φn − φn+1 ); a22 = −20 θn+1 (θn θn+1 φn − θn θn+1 φn φn2+1 − θn θn+1 φn+1 + θn θn+1 φn+1 φn2 ); a23 = θn−1 J(φn − φn−1 ); a24 = −20 θn−1 (θn−1 θn φn − θn θn−1 φn φn2−1 − θn θn−1 φn−1 + θn θn−1 φn−1 φn2 ); Jθ

φ

Jθ

φ

b21 = φn+1 n+θ 1 n ; n b22 = φn+1 20 (φn+1 θn2+1 − 2φn θn2+1 );

b23 = φn−1 n−θ 1 n ; n b24 = φn−1 20 (φn−1 θn2−1 − 2φn θn2−1 ); b25 =

J θn−1

θn

− 1; b26 =

J θn+1

θn

− 1; b27 = 20 φn2 θn2−1 ; b28 = φn2 θn2+1 ; b29 = −2A.

Appendix C The coefficients of Eqs. (17) and (18) are: a31 = γ 2 J(2θx φx + θ φxx ); a32 = −γ 2 0 (8θ 2 θx φx + 8θ 2 φ 2 θx φx + 4θ 3 φφx2 + 2θ 3 φxx + 2θ 3 φ 2 φxx ); b31 = −2J φ 2 ; φ2 θ 2φθ φ b32 = −γ 2 J( θθxx + θ xx + 4θx φx + θx x + 2θφxx + φφxx ); 2 2 2 2 b33 = γ 0 (16θ θx φx + 16θ φ θx φx − 4θ 2 φx2 + 8θ 3 φφx2 + 4θ 3 φxx + 4θ 3 φ 2 φxx ). Appendix D The coefficients of Eqs. (29) and (30) are: a41 = γ 2 J(2θξ φξ + θφξ ξ ); a42 = −γ 2 0 (8θ 2 θξ φξ + 8θ 2 φ 2 θξ φξ + 4θ 3 φφξ2 + 2θ 3 φξ ξ + 2θ 3 φ 2 φξ ξ ); a43 = γ 2 J θ φξ ; a44 = −γ 2 2J(θ 3 φξ + θ 3 φ 2 φξ ); θ

φ2 θ

2φθ φ

b41 = −γ 2 J( ξθξ + θ ξ ξ + 4θξ φξ + θξ ξ + 2θφξ ξ + φφξ ξ ); b42 = γ 2 0 (16θ 2 θξ φξ + 16θ 2 φ 2 θξ φξ − 4θ 2 φξ2 + 8θ 3 φφξ2 + 4θ 3 φξ ξ + 4θ 3 φ 2 φξ ξ ); θ

φ2 θ

b43 = −γ 2 J( θξ + θ ξ + 2θφξ + φφξ ); b44 = γ 2 0 (4θ 3 φξ + 4θ 3 φ 2 φξ ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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