Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified exp(-Ω(η))-expansion function method

Tuluce Demiray, S.; Bayrakci, U.; Tuluce Demiray, S.; Bayrakci, U.

doi:10.31349/revmexfis.67.393

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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.67 no.3 México may./jun. 2021 Epub 21-Feb-2022

https://doi.org/10.31349/revmexfis.67.393

Research

Gravitation, Mathematical Physics and Field Theory

Soliton solutions for space-time fractional Heisenberg ferromagnetic spin chain equation by generalized Kudryashov method and modified exp(-Ω(η))-expansion function method

S. Tuluce Demiray¹

U. Bayrakci¹

^¹Department of Mathematics, Osmaniye Korkut Ata University, Osmaniye, Turkey. e-mail: seymatuluce@gmail.com; ubayrakci42@gmail.com

Abstract

This paper addresses the Heisenberg ferromagnetic spin chain equation with beta derivative. Initially, beta derivatives and their features are presented. Then, by submitting the generalized Kudryashov method and modified exp(-Ω(η))-expansion function method, dark, bright, and dark-bright soliton solutions of this equation, which can be explained with beta derivative, are procured. Thus, it seems that these methods can supply significant outcomes in finding the exact solutions fractional differential equations with beta time derivative.

Keywords: Heisenberg ferromagnetic spin chain equation; generalized Kudryashov method; modified exp(-Ω(η))-expansion function method; dark soliton; bright soliton; dark-bright soliton; beta time derivative

PACS: 35C08-35N05-68N15

1.Introduction

In recent years, fractional differential equations (FDEs) shed light on the science environment because of their important position in many areas of complicated physical events, from fluid dynamics and optical fiber to quantum field theory. In fact, the soliton solutions of such equations have been one of the most remarkable solutions due to their more clear view of the nonlinear physical properties and then guide to the next aims. As a result, several methods have been used by many authors to calculate such solutions for better insight into the main properties of physical constructions in different media ¹^-⁵.

A lot of kinds of fractional derivative operators have been described by scientists. Some of them are Caputo derivative, Riemann-Liouville derivative, Caputo-Fabrizio, Jumarie’s modified Riemann-Liouville derivative, Atangana-Baleanu derivative ⁶^-⁹. By using these derivative operators, many methods have been submitted to provide solutions of FDEs ¹⁰^-¹⁵.

Then, the conformable derivative has been identified by Khalil et al.¹⁶. Also, exact solutions of the FDE have been found by using this derivative ¹⁷. Then, some theorems, definitions, and properties related to conformable derivative have been presented by Atangana et al. ¹⁸. Consequently, a new fractional derivative called beta-derivative has been given by Atangana et al. ¹⁹. Then, solutions of FDEs with conformable derivatives have been considered by a lot of authors ²⁰^-²⁵.

Heisenberg’s ferromagnetic spin chain equation has been highly considered for its significance from different aspects ²⁶^-²⁹. This paper will address Heisenberg ferromagnetic spin chain equation to get its soliton solutions by a strong algorithm that was lately submitted. The methods are the generalized Kudryashov method (GKM) ³⁰^-³³ and modified -expansion function method (MEFM) ³⁴^-³⁷. The algorithms supply optical solitons such as dark, bright, and dark-bright. The results are thus found after the extensive experience of the algorithmic operation.

2.Beta Derivatives and its features

Definition 1. The conformable derivative has been identified by Khalil et al. ¹⁶. Let $w:[0,∞)$ be a function β-th order, the conformable derivative of w(t) for all is given as follows:

$Fβ(w(t))=dβw(t)dtβ=lime→0w(t+et1-β)-w(t)e,0<β≤1.$

Also, if w is β-differentiable in (0,a), $a>0$ , and $lime→0+w(β)(t)$ exists, then it can be written as $w(β)(0)=lime→0+(t)$ .

Definition 2. Let w(t) be a function qualified for all non-negative t. Then, the beta derivative of w(t) is defined by ¹⁹

$Fβ(w(t))=dβw(t)dtβ=lime→0wt+et+1Γ(β)1-β-w(t)e,0<β≤1.$

Although the conformable fractional derivative presented by Khalil et al. supplies some basic properties such as the chain rule, Atangana’s fractional derivative is presented because it can yield the maximum features of the basic derivatives.

Such derivatives can not only be assumed as fractional derivatives but also considered as a natural extension of the classical derivative. There is a significant theorem for beta-derivatives ¹⁹:

Theorem. Let w(t) and v(t) be β-differentiable functions for all $t>0$ and $β∈(0,1[$ . Then

$Fβ(mw(t)+nv(t))=mFβ(w(t))+nFβ(v(t)),∀m,n∈R.Fβ(w(t)v(t))=v(t)Fβ(w(t))+w(t)Fβ(v(t)),Fβw(t)v(t)=v(t)Fβ(w(t))-w(t)Fβ(v(t))(v(t))2,Fβ(w(t))=t+1Γ(β)1-βdw(t)dt.$

3.General structure of GKM

We survey the following FDE with beta derivative for a function of two real variables, space x, and time t:

$P(k,Fβk,kt,kx,kxx,⋯)=0$ (1)

Step 1. Initially, we should perform the traveling wave solution of Eq.(1) as follows;

$k(x,y,t)=H(η)eiλ(x,y,t),$ (2)

Where

$η=cos⁡ϕβx+1Γ(β)β+sin⁡ϕβy+1Γ(β)β+pβt+1Γ(β)β,$ (3)

$λ(x,y,t)=-(cos⁡ϕβx+1Γ(β)β+sin⁡ϕβy+1Γ(β)β)+σβt+1Γ(β)β,$ (4)

where p and σ arbitrary constants. Then, by substituting Eqs.(2-4) to Eq.(1), a nonlinear ordinary differential equation can be obtained as:

$N(H,H',H″,H‴,⋯)=0,$ (5)

where the prime displays differentiation about η.

Step 2. Assume that the exact solutions of Eq. (5) can be considered in the form

$k(η)=∑i=0Naiγi(η)∑j=0Mbjγj(η)=A(γ[η])B(γ[η]),$ (6)

where $γ(η)=1/1±eη$ . We highlight that the function γ is the solution of the equation:

$γη=γ'=γ2-γ.$ (7)

Taking account of Eq. (6), we supply

$k'(η)=A'γ'B-AB'γ'B2=γ'A'B-AB'B2=(γ2-γ)A'B-AB'B2,$ (8)

$k″(η)=γ2-γB2([2γ-1][A'B-AB']+γ2-γB×[B{A″B-AB″}-2B'A'B+2A{B'}2]),$ (9)

Step 3. The solution of Eq.(5) can be defined as follows:

$k(η)=a0+a1γ+a2γ2+⋯+aNγN+⋯b0+b1γ+b2γ2+⋯+bMγM+⋯.$ (10)

To compute the values M and N in Eq. (10) that is the pole order for the general solution of Eq. (5), we procure comparably as in the classical Kudryashov method on balancing the highest-order nonlinear terms in Eq. (5), and we can find a relation of M and N. We can get values of M and N.

Step 4. Substituting Eq. (6) into Eq. (5) ensures a polynomial R(γ) of γ. Extracting the coefficients of R(γ) to zero, we get a system of algebraic equations. Solving this system, we can identify p and the variable coefficients of a ₀, a ₁, a ₂, …a _N, b₀, b₁, b₂,…b_M. Thus, we get the exact solutions to Eq. (5).

4.Soliton solutions for Heisenberg ferromagnetic spin chain equation by GKM

In this section, we look for exact solutions of the Heisenberg ferromagnetic spin chain equation with beta time derivative by using GKM.

It is helpful to use the spin to lie in a planet right angles to the chain axis. Thus, we consider Heisenberg ferromagnetic spin chain equation with beta time derivative ³⁸

$i∂βk∂tβ-i∂βk∂xβ+∂2βk∂x2β+∂2βk∂y2β-2∂2βk∂xβ∂yβ+2|k|2k=0.$ (11)

Firstly, we take wave variable transformations as follows

$k(x,y,t)=H(η)eiλ(x,y,t),$ (12)

where

$η=cos⁡ϕβx+1Γ(β)β+sin⁡ϕβy+1Γ(β)β+pβt+1Γ(β)β,$ (13)

$λ(x,y,t)=-cos⁡ϕβx+1Γ(β)β+sin⁡ϕβy+1Γ(β)β+σβt+1Γ(β)β.$ (14)

Putting (12-14) into (11) provides

$(cos⁡ϕ-sin⁡ϕ)2H″(η)-[(cos⁡ϕ-sin⁡ϕ)2+cos⁡ϕ+σ]H(η)+2H3(η)=0,$ (15)

where $p=2sin2ϕ-cosϕ-2.$

For the balance principle between higher-order derivative H´´ and highest power nonlinear terms H³ in Eq. (15), one can be procured

$N-M+2=3N-3M⇒N=M+1.$ (16)

By using GKM, the solution of Eq. (11) can be given as

$H(η)=a0+a1γ+a2γ2b0+b1γ$ (17)

where a ₀, a ₁, and a ₂ are found later and $γ(η)=(1/1±eη).$ The function $γ(η)$ provides as

$γη=γ'=γ2-γ.$ (18)

Thus, the exact solutions of Eq. (11) are accessed as the following;

Case 1.

$a0=12i(cos⁡ϕ-sin⁡ϕ)b0, a1=-a22-i(cos⁡ϕ-sin⁡ϕ)b0, b1=-a2i(cos⁡ϕ-sin⁡ϕ), σ=-12cos⁡ϕ(2+3cos⁡ϕ)+3cos⁡ϕsin⁡ϕ-3sin2ϕ2.$ (19)

Replacing Eq. (19) into Eq. (17), dark soliton solutions of Eq. (11) can reached as

$κ1(x,y,t)=12i cos⁡ϕ-sin⁡φtanh⁡(12[cos⁡ϕβ{x+1Γ(β)}β+sin⁡ϕβ{y+1Γ(β)}β+pβ{t+1Γ(β)}β])× exp⁡(i[-{cos⁡ϕβx+1Γ(β)β+y+1Γ(β)2}+{-12βcos⁡ϕ(2+3cos⁡ϕ)+3βcos⁡ϕsin⁡ϕ-3sin2⁡ϕ2β}{t+1Γ(β)}β])$ (20)

$k2(x,y,t)=12i cos⁡ϕ-sin⁡ϕcoth⁡(12[cos⁡ϕβ{x+1Γβ}β+sin⁡ϕβy+1Γβ}β+pβt+1Γβ}β×exp⁡(i[-{cos⁡ϕβx+1Γ(β)β+sin⁡ϕβy+1Γ(β)β}+{-12βcos⁡ϕ(2+3cos⁡ϕ)+3βcos⁡ϕsin⁡ϕ-3sin2⁡ϕ2β}{t+\gab}β])$ (21)

Case 2.

$a0=0,a1=-i(cos⁡ϕ-sin⁡ϕ)b1,a2=i(cos⁡ϕ-sin⁡ϕ)b1,b0=-b12,σ=-cos⁡ϕ.$ (22)

Replacing Eq. (22) into Eq. (17), bright soliton solutions of Eq. (11) can be determined as

$κ3(x,y,t)=i(cos⁡ϕ-sin⁡ϕ)csc⁡h(cos⁡ϕβ[x+1Γβ]β+sin⁡ϕβ[y+1Γβ]β+pβ[t+1Γβ]β)× exp⁡(i[-{cos⁡ϕβx+1Γβb+sin⁡ϕβy+1Γββ}-cos⁡ϕβ{t+1Γ(β)}β])$ (23)

$κ4(x,y,t)=i(cos⁡ϕ-sin⁡ϕ)sec⁡h(cos⁡ϕβ[x+1Γβ]β+sin⁡ϕβ[y+1Γβ]β+pβ[t+1Γβ]β)× exp⁡(i[-{cos⁡ϕβx+1Γββ+1Γβy+1Γββ}-cos⁡ϕβ{t+1Γβ}β])$ (24)

Case 3.

$a0=12icos⁡ϕ-sin⁡ϕb1,a1=-icos⁡ϕ-sin⁡ϕb1,a2=icos⁡ϕ-sin⁡ϕb1,b0=-b12, σ=-cos⁡ϕ1+3cos⁡ϕ+6cos⁡ϕsin⁡ϕ-3sin2⁡ϕ.$ (25)

Replacing Eq. (25) into Eq. (17), dark soliton solutions of Eq. (11) can be ascertained as

$κ5(x,y,t)=-icos⁡ϕ-sin⁡ϕcoth⁡(cos⁡ϕβ[x+1Γβ]β+sin⁡ϕβ[y+1Γβ]β+pβ[t+1Γβ]β)×exp⁡(i[-{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ}+{-cos⁡ϕ(1+3cos⁡ϕ)+6cos⁡ϕsin⁡ϕ-3sin2⁡ϕβ}{t+1Γβ}β])$ (26)

$κ6(x,y,t)=icos⁡ϕ-sin⁡ϕtanh⁡(cos⁡ϕβ[x+1Γβ]β+sin⁡ϕβ[y+1Γβ]β+pβ[t+1Γβ]β)×exp⁡(i[-{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γβ}+{-cos⁡ϕ(1+3cos⁡ϕ)+6cos⁡ϕsin⁡ϕ-3sin2⁡ϕβ}{t+1Γβ}β])$ (27)

Case 4.

$a0=-14icos⁡ϕ-sin⁡ϕb1,a1=icos⁡ϕ-sin⁡ϕb1,a2=-icos⁡ϕ-sin⁡ϕb1,b0=-b12, σ=-12cos⁡ϕ(2+3cos⁡ϕ)+3cos⁡ϕsin⁡ϕ-3sin2⁡ϕ2.$ (28)

Putting Eq. (28) into Eq. (17), dark-bright soliton solutions of Eq. (11) can be provided as

$κ7(x,y,t)=12icos⁡ϕ-sin⁡ϕ([-2+3coth⁡{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ}+3csch{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ}]-a1b1[-1+coth⁡{12(cos⁡ϕβ×\bracix+1Γβ+sin⁡ϕβy+1Γββ+pβt+1Γββ)}])exp⁡(i[-{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ}+{-12βcos⁡ϕ(2+3cos⁡ϕ)+3βcos⁡ϕ-3sin2ϕ2β}{t+1Γβ}β])$ (29)

$κ8(x,y,t)=12icos⁡ϕ-sin⁡ϕ([-2+3tanh⁡{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ}+3 sech{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ}]-a1b1[-1+tanh⁡{12(cos⁡ϕβ×x+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ)}])exp⁡(i[-{cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ}+{-12βcos⁡ϕ(2+3cos⁡ϕ)+3βcos⁡ϕsin⁡ϕ-3sin2⁡ϕ2β}{t+1Γβ}β])$ (30)

In Fig. 1, 3D and 2D graphs are investigated to illustrate the influence of the parameter β on the dynamics of the first dark soliton solution. Clearly, the physical behavior of the dark soliton solution is altered when the parameter β gets different values.

Figure 1 3D image of |κ₁(x; y; t)| for φ = 60^o , β = 0:5, y = 1 and 2D image of |κ₁(x; y; t)| for these values and t = 0:2.

In Fig. 2, 3D and 2D graphs are examined to show the influence of the parameter β on the dynamics of the bright soliton solution. Unquestionably, the physical behavior of the bright soliton solution is changed when the parameter β takes different values.

Figure 2 3D image of |κ₃(x; y; t)| for φ = 30^o, β = 1:5, y = 2 and 2D image of |κ₃(x; y; t)| for these values and t = 0:4.

5.General structure of MEFM

We survey the following FDE with beta derivative for a function of two real variables, space x, and time t:

$P(κ,Fβκ,κt,κx,κxx,⋯)=0.$ (31)

Step 1. Initially, we should perform the traveling wave solution of Eq. (31) as follows;

$κ(x,y,t)=H(η)eiλ(x,y,t),$ (32)

Where

$η=cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+pβt+1Γββ,$ (33)

$λ(x,y,t)=-(cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+σβt+1Γββ),$ (34)

where p and σ arbitrary constants. Then, by substituting Eqs. (32-34) to Eq.(31), a nonlinear ordinary differential equation can be obtained as:

$N(H,H',H″,H‴,⋯)=0,$ (35)

where the prime displays differentiation about η.

Step 2: Presume the traveling wave solution of (4) can be indicated as follows:

$κ(η)=∑i=0NFi(exp⁡[-Ω{η}])i∑j=0MGj(exp⁡[-Ω{η}])j=F0+F1exp⁡(-Ω)+⋯+FNexp⁡(N(-Ω))G0+G1exp⁡(-Ω)+⋯+GMexp⁡(M(-Ω)),$ (36)

where F _i, G _j, $(0≤i≤N, 0≤j≤M)$ are constants to be described later, such that $FN≠0$ , $GM≠0$ , and $Ω=Ω(η)$ is the solution of the following ordinary differential equation:

$Ω'(η)=exp(-Ω[η])+nexp(Ω[η])+m.$ (37)

The solution families of (37) can be shown as follows:

Family1: If $n≠0$ , $m2-4n>0$ ,

$Ω(η)=ln⁡(-m2-4n2ntanh)×[m2-4n2n{η+E}]-m2n).$ (38)

Family 2: If $n≠0$ , $m2-4n<0$ ,

$Ω(η)=ln⁡(-m2+4n2ntan)×[-m2+4n2{η+E}]-m2n).$ (39)

Family 3: If n = 0, $m≠0$ , and $m2-4n>0$ ,

$Ω(η)=-ln⁡mexp⁡[m{η+E}]-1.$ (40)

Family 4: If $n≠0$ , $m≠0$ , and $m2-4n=0$ ,

$Ω(η)=ln⁡-2m[η+E]+4m2[η+E].$ (41)

Family5: If n = 0, m = 0, and $m2-4n=0$ ,

$Ω(η)=ln(η+E).$ (42)

The positive integers M and M can be determined attending the homogeneous balance principle in Eq. (36).

Step 3: Replacing Eqs. (37) and (38-42) into Eq.(36), we ascertain a polynomial of exp(-Ω(η)). We stabilize all the coefficients of same power of exp(-Ω(η)) to zero. This operation determines a system of equations that can be unfastened to reach F ₀, F ₁, F ₂,…F _N, G ₀, G ₁, G ₂,…G _M, E, m, n by the way of Wolfram Mathematica 12. Inserting the values of these constants into Eq. (36), the general solutions of (36) supplies the determination of the solution of Eq.(31).

6.Soliton solutions for Heisenberg ferromagnetic spin chain equation by MEFM

In this section, we seek exact solutions of the Heisenberg ferromagnetic spin chain equation with beta time derivative by using MEFM.

For the balance principle between higher-order derivative H´´ and highest power nonlinear terms H³ in Eq. (15), one can procure

$N=M+1.$ (43)

By using MEFM, the solution of Eq. (11) can be given as

$H=F0+F1exp(-Ω)+F2exp(2[-Ω])+G0+G1exp(-Ω)=Zτ,$ (44)

And

$H'=Z'τ-τ'Zτ2,$ (45)

$H″=Z″τ3-τ2Z'τ'-(τ″Z+τ'Z')τ2+2(τ')2Zττ4,$ (46)

where $F2≠0$ and $G1≠0$ . The function $Ω=Ω(η)$ provides as

$Ω'(η)=exp(-Ω[η])+nexp(Ω[η])+m.$ (47)

Thus, the exact solutions of (11) are accessed as the following;

Case 1:

$F0=12icos⁡ϕ-sin⁡ϕmG0, F1=12icos⁡ϕ-sin⁡ϕ(2G0+mG1), F2=icos⁡ϕ-sin⁡ϕG1, bσ=cos⁡ϕ-1+sin⁡ϕ2+m2-4n-12(2+m2-4n).$ (48)

Replacing (48) into (44), dark soliton solutions of (11) can be reached as

$κ9=12icos⁡ϕ-sin⁡ϕJ+mJtanh⁡J\lefccos⁡ϕx+1Γββ+sin⁡ϕy+1Γββ+pt+1Γββ+Eβ2βm+Jtanh⁡Jcos⁡ϕx+1Γββ+sin⁡ϕy+1Γββ+pt+1Γββ+Eβ2β×exp⁡(i[-cos⁡ϕβx+1Γββ+sin⁡ϕβy+1Γββ+cos⁡ϕβ-1+sin⁡ϕ[2+m2-4n ]-12β[2+m2-4n]t+1Γββ]),$ (49)

where $n≠0$ , and $J=m2-4n>0$ .

In 3, 3D and 2D graphs are considered to indicate the influence of the parameter on the dynamics of the dark soliton solution. Explicitly, the physical behavior of the dark soliton solution is shifted when the parameter β receives different values.

Case 2:

$G0=r-s2F12-2F0F2-F1F12-4F0F2, G1=F1-F12-4F0F222F0F12-2F0F2-F1F12-4F0F2r-s R=F1-F12-4F0F2F2,r=cos⁡ϕ,s=sin⁡ϕ, σ=-(r-s)2F12+2(r-s)2F0F2+(-r-r2+2rs-s2+2(r-s)2S)F22+(r-s)2F1F12-4F0F2F22.$ (50)

Replacing (50) into (44), dark soliton solutions of (11) can be reached as

$κ10(x,y,t)=A(F2+12n-F1-F12-4F0F2F2-χtanh⁡f(x,y,t)F1+F0-F1-F12-4F0F2F2-χtanh⁡[f{x,y,t}]2nB(C-Dtanh⁡[f{x,y,t}])F1-F12-4F0F2F2+χtanh⁡[f{x,y,t}]$ (51)

where,

$A=4i2(r-s)2n2F0F1F2,B=F1F12-4F0F2+(F12-2F0F2)(r-s)2,C=-(r-s)2F12(F0-nF2)+(F0-SF2)(r-s)2F1F12-4F0F2,D=-2(r-s)2F0F1F2F12-2F2(F0+nF2)-F1F12-4F0F2F22f(x,y,t)=χcos⁡ϕx+1Γββ+sin⁡ϕy+1Γββ+pt+1ΓβB+Eβ2βχ=-4n+(F12-F1F12-4F0F2)2F12F12,$

and

$-4n+(F12-F1F12-4F0F2)2F12F12>0.$

7.Conclusion

In this work, the soliton characters of the Heisenberg ferromagnetic spin chain equation with beta time derivative were investigated by using GKM and MEFM. Dark, bright, and dark-bright soliton solutions of this equation have been accomplished found. Then, 3D and 2D images were presented for some solutions, which display the vitality of the solutions with proper values. Numerical results, together with the graphical demonstrations, have exhibited the reliability of these methods. Also, these solutions have been reported to the literature with novel substantial physical properties. These methods can be applied to other FDEs with beta time derivatives.

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Received: November 12, 2020; Accepted: December 17, 2020

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