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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.67 no.5 México sep./oct. 2021  Epub 28-Mar-2022

https://doi.org/10.31349/revmexfis.67.050703 

Research

Gravitation, Mathematical Physics and Field Theory

Optical electromagnetic radiation density spherical geometric electric and magnetic phase by spherical antiferromagnetic model with fractional system

E. M. Khalil a  

T. Körpinar b  

Z. Körpinar c  

M. Inc d   e   f    

aDepartment of Mathematics, College of Science, POBox 11099, Taif University, Taif 21944, Saudi Arabia.

bMus Alparslan University, Department of Mathematics, 49250, Mus, Turkey.

cMus Alparslan University, Department of Administration, 49250 Mus, Turkey.

dDepartment of Computer Engineering, Biruni University, Istanbul, Turkey.

eFırat University, Science Faculty, Department of Mathematics, 23119 Elazig, Turkey.

fDepartment of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.


Abstract

In this article, we firstly consider a new theory of spherical electromagnetic radiation density with an antiferromagnetic spin of timelike spherical t-magnetic flows by the spherical Sitter frame in de Sitter space. Thus, we construct the new relationship between the new type electric and magnetic phases and spherical timelike magnetic flows de Sitter space S 1 2 Also, we give the applied geometric characterization for spherical electromagnetic radiation density. This concept also boosts to discover of some physical and geometrical characterizations belonging to the particle. Moreover, the solution of the fractional-order systems is considered for the submitted mathematical designs. Graphical demonstrations for fractional solutions are presented to an expression of the approach. The collected results illustrate that mechanism is a relevant and decisive approach to recover numerical solutions of our new fractional equations. Components of performed equations are demonstrated by using approximately explicit values of physical assertions on received solutions. Finally, we construct that electromagnetic fluid propagation along fractional optical fiber indicates a fascinating family of fractional evolution equations with diverse physical and applied geometric modeling in de Sitter space S 1 2.

Keywords: t-magnetic particle; optical fiber; geometric phase; evolution equations; traveling wave hypothesis; antiferromagnetic model

PACS: 02.40.Hw; 03.65.Vf; 05.45.Yv; 03.50.De; 42.15.-i. MSC 2010: 35C08; 78A05; 53A35

1. Introduction

An ideal optical fiber has perfect circular symmetry. The polarizations are completely degenerate. Perturbations and imperfections during the fabrication process may introduce anisotropic, which are mostly of a linear or Cartesian type. Sometimes, large linear anisotropic are introduced on purpose, either by modified core geometry or by mechanical stress, to get linear polarizationmaintaining fibers, also called high-birefringence or hi-bi fibers. Bending and squeezing optical fibers does also introduce linear birefringence. Rotational effects of polarization, however, are in general less common and more difficult to produce and to understand.

In recent times, the advancement of glowing lasers and the utilization of optical fiber mechanics have attended to immense importance on flow propagation by curled fluid flows and space-curved. Exclusively torque forces of the geometric phase of isolated light anholonomy with some optical fibers have been investigated; the substance of numerous papers insensitive of their compositions are efficient or analytical. For illustration, Smith [1] examined that the torque of divergence of light generate on monochromatic optical fiber immerse bundle the supervisor is convinced by the electromagnetic fields thanks to the magnetic particle present flows in some optical fiber present transformer. It was still completed that the present is commensurate to torque, which is regular of the order of numerous scopes. Another preliminary substantiation of the geometric effect of the torque of magnetic divergence for light propagating in a magnetic optical fiber detecting a magnetic trajectory was extended [2]. Ross improved a totally geometric system to investigate the reversion in the coiled optical fiber with a fixed-torsion and endorsed its effects with several measurements in the fiber angle into a magnetic fiber. Tomita and Chiao [3] summarized the previous review of Ross for more universal fiber shapes. Also, Chiao and Wu [4] obtained a new important theoretical phase of the results of geometric phase torque. Apart from preceding researches [6-10], we request that a new electromagnetic phase with an antiferromagnetic chain.

Optical fiber investigates, phase is mostly observed by a carrier of new electromagnetic particles and their features. Some nonlinear evolution structures are frequently committed as design to establish substantially complicated scientific developments in diverse provinces of disciplines, particularly in genuine-state physics, chemical physics, plasma physics, optical physics, fluid mechanics, etcetera. Having an advanced perception of the brief substances, likewise their gradual operations in analytical operations and genuine administration in a constructive generation, it is excessively fascinating to satisfaction explicit solutions of approximately systems. Thus there endures no comprehending and unified approach to demonstrate exact solutions of all nonlinear transformation system analysts operate a diversity of diverse concepts [11-23].

It is remarkable that in numerous fields of physics, the utilizations of approximately wave interpretation and its reactions are of comprehensive importance. Traveling any wave solutions are a consequential variety of representative with some partial differential aspect and distinct nonlinear fractional differential equations (FDEs) have been established to a selection of traveling some wave improvements. Thus, flood waves are commonly immensely significant of all-instinctive phenomena; they have a superordinary elegant mathematical construction.

Fractional geometry perturbs the operations of derivatives and integrals of optional order. Over the latest several decades, it obtained enormous recognition because of its numerous modeling in distinct scientific competitions. Arbitrary-order designs are softer integer-order designs. FDEs appear in various mathematical and modeling regions comparatively physics, geophysics, polymer rheology, biophysics, capacitor theory, aerodynamics, medicine, nonlinear vibration of earthquake, supervision theory, vital fluid flow phenomena, superelasticity, and magnetical districts. For the intensive study of its utilization, we introduce comprehensive works [24-28].

The outline of the paper is organized as follows: Firstly, we consider a new theory of spherical electromagnetic radiation density with an antiferromagnetic spin of timelike spherical t-magnetic flows by the spherical Sitter frame in de Sitter space. Thus, we construct the new relationship between the new type electric and magnetic phases and spherical timelike magnetic flows de Sitter space S 1 2. Also, we give the applied geometric characterization for spherical electromagnetic radiation density. This concept also boosts to discover of some physical and geometrical characterizations belonging to the particle. Moreover, the solution of the fractionalorder systems is considered for the submitted mathematical designs. Graphical demonstrations for fractional solutions are presented to the expression of the approach. The collected results illustrate that mechanism is a relevant and decisive approach to recover numerical solutions of our new fractional equations. Components of performed equations are demonstrated by using approximately explicit values of physical assertions on received solutions. Finally, we construct that electromagnetic fluid propagation along fractional optical fiber indicates a fascinating family of fractional evolution equations with diverse physical and applied geometric modeling in de Sitter space S 1 2.

2. Timelike spherical t − magnetic particle in

S 1 2

In this section, the orthonormal frame design is explained by the orthonormal Lorentzian new spherical Sitter frame and the particle γ : IS 1 2 refreshing this spherical Sitter frame equation is defined as Lorentzian spherical particle.

[SγStsn]=[01010ε0ε0][γtn],

where ∇ is a Levi-Civita form and ε = det(Ψ, t ,, t) is curvature of particle [19]. Then products of spherical vector fields are presented by

γ=Tn,t=γn,n=γT.

Assume that γ : IS 1 2 be a timelike spherical t-magnetic particle and G be the magnetic field in S 1 2 Timelike spherical t-magnetic particle is defined by

St=ϕt=×Gt.

*Lorentz force ϕ of a t-magnetic particle with the magnetic field G is presented by

ϕγ=t+βn

ϕ(γ) = t + βn, ϕ(t) = γ + εn,

ϕ(n) = -βγ + εt,

Gt= εγ - βt - n,

where β = h(ϕ(γ), n).

Putting force equation

mst=qt×B+qE,

where

E = ξγ + εξn,

ξ=m+qq.

Let γ(s,t) is the evolution of t-magnetic particle in de Sitter space. The flow of t-magnetic particle is given by

tγ=σ1t+σ2n,

where χ 1 2 are potentials.

Time derivatives of the spherical frame are produced by

tt=σ1γ+σ2s+σ1εn,

tγ=σ1t+σ2n,

tn=σ2s+σ1εt-σ2γ.

Condition of Lorentz forces ϕ(γ), ϕ(t), ϕ(n) and magnetic field G t for a rotational equilibrium of the timelike t-magnetic particle

tϕγ=σ1-σ2βγ+βσ2s+σ1ε+σ2ϑ+εσ1+βtn,

tϕt=-εσ2γ+σ1+εσ2s+σ1εt+εt+σ2n,

tϕn=σ1ε-βtγ+εt-σ1βt +εσ2s+σ1ε-βσ2n,

tGt=εt-βσ1+σ2γ+εσ1-βt-σϑ+σ1εt +σ2ε-βσ2s+σ1εn,

tE=-εξσ2γ+ξσ1+εξσ2s+σ1εt+ξσ2+εtn,

where β = h(ϕ(γ), n).

In this way, the equation for Lorentz forces ϕ(γ), ϕ(T), ϕ(N) reads

sϕγ=γ+βεt+ε+βsn,

sϕt=1+ε2t+sεn,

sϕn=ε-βsγ+εs-βt+ε2n,

sGT=εs-βγ-βst-βεn,

where

3. New geometric results with physical applications

The geometric density theory, or the theory of density systems, has had and has an enormous impact in applied physical mathematics and an extensive diversity of nonlinear phenomena in mathematical physics. This framework combines questions in nonlinear flux optics, field designs and sigma models, fluid dynamics, relativity, electromagnetic wave theory. In this section, we obtain spherical electric and magnetic radiation density conditions by using Heisenberg antiferromagnetic model.

♣ Spherical electric and magnetic radiation density are given by

ΦΠEt=tEtGt×tΠ,ΦΠGt=tGtGt×tΠ.

♣ By using density, a new type of spherical electric and magnetic phase is given by

ΩΠEt=(ΠEt)dς,  ΩΠGt=(ΠGt)dς.

New type spherical electric and magnetic phase of ϕ(γ)

Theorem 1. The spherical electric and magnetic phase of ϕ(γ) are given by

Proof. Definition of magnetic antiferromagnetic spin is given by

tϕγ=ϕγ×sϕγ.

Then, it is easy to see that

ϕγ×sϕγ=ε+βs-β2εγ-βt-n.

Straightforward computations lead to

ε+βs-β2ε=σ1-σ2β,  σ2s+σ1ε=-1,σ2s+εσ1+βt=-1.

On the other hand, the magnetic-total phase along Lorentz force ϕ(γ) is indicated by

ΛϕγGt=ϕΨGt×tϕγdρ.

With some calculations, we have

Gt×tϕγ=βσ2s+σ1ε-βσ2s+εσ1+βtγ+εσ2s+εσ1+βt+σ1-σ2βt+βσ1-σ2β+εβσ2s+σ1εn.

Magnetic anholonomy density of ϕ(γ) is given by

δϕγGt=-εσ2s+εσ1+βt+σ1+σ2β+ββσ1-σ2β+εβσ2s+σ1ε.

Also, we present

ΛϕγGt=-εσ2s+εσ1+βt+σ1+σ2β+ββσ1-σ2β+εβσ2s+σ1εdρ.

Spherical electric and magnetic radiation density are given by

By magnetic anholonomy density with an antiferromagnetic model is obtained

Gt×ϕγ×σϕγ=2ε+βs-β2εt+ββs-βεn.

After antiferromagnetic condition, we obtain

GtAFRδϕγGt=ϕγGt×ϕγ×σϕγ.

By this way, we conclude

δAFRϕγGt=2ε+βs-β2ε+β2βs-β2ε.

Similarly, we can easily obtain that

ΛAFRϕγGt=-2ε+βs-β2ε+β2βs-β2εdρ.

As a reaction, we get the following impressive results.

Theorem 2. Electromagnetic radiation density of ϕ(γ) is

Theorem 3. The antiferromagnetic spherical electric and magnetic phase of ϕ(γ) with antiferromagnetic spin are given by

Proof. Spherical electric and magnetic radiation density with an antiferromagnetic model are given by

If we use above theorem, we may give the following corollary:

Corollary 1. Electromagnetic radiation density with an antiferromagnetic model is

Spherical magnetic anholonomy density and time like t-magnetic particle flows in kernel of magnetic quadrupole for timelike spherical evolution with Lorentz force ϕ(γ). Spherical antiferromagnetic density is provided by the antiferromagnetic density algorithm in Fig. 1.

FIGURE 1 Spherical electromagnetic radiation density with ϕ(t)  

Magnetic antiferromagnetic spin for ϕ(t)

Theorem 4. Spherical electric and magnetic phase of ϕ(t) are given by

Proof. Magnetic antiferromagnetic spin is given by

tϕt=ϕt×sϕt.

By definition, we get

ϕt×sϕt=-ε1+ε2γ+εst-1-ε2n.

Furthermore, we have that

-εσ2=-ε1+ε2,εs=σ1+εσ2s+σ1ε,εt+σ2=-1-ε2.

The previous system becomes

σ2=1+ε2,εs=σ1+εσ2s+σ1ε,εt=-2σ2.

Magnetic total spherical phase for force ϕ(t) is obtained by

ΛϕtGt=ϕtGt×tϕtdρ.

We can immediately check that

Hence, we compute

δϕ(t)Gt=(σ1+ε(σ2s+σ1ε)-β(εt+σ2))+ε(ε(σ1+ε(σ2s+σ1ε))-βεσ2)).

From the magnetic total phase, we obtain

ΛϕtGt=σ1+εσ2s+σ1ε-βεt+σ2+εεσ1+εσ2s+σ1ε-βεσ2dρ.

It follows that

Gt×ϕt×sϕt=εs-β1+ε2γ+εεs-εβ1+ε2n.

Similar to the above, we calculate

AFRδϕtGΨ=εs-β1+ε2+εεεs-εβ1+ε2.

This implies that

AFRΛϕtGΨ=εs-β1+ε2+εεεs-εβ1+ε2dρ.

As a reaction, we get the following impressive results.

Theorem 5. The electromagnetic radiation density of ϕ(t) is

Theorem 6. Antiferromagnetic spherical electric and magnetic phase of ϕ(t) are given by

Proof. Spherical electric and magnetic radiation density with an antiferromagnetic model are given by

Under the antiferromagnetic condition, then we obtain that

Gt×ϕt×s×ϕt=εs-β1+ε2γ+εεs-εβ1+ε2n.

Corollary 2. Electromagnetic radiation density with an antiferromagnetic model is

Spherical magnetic anholonomy density and timelike t-magnetic particle flow in the kernel of magnetic quadrupole for timelike spherical evolution with Lorentz force ϕ(t). Spherical antiferromagnetic density is provided by the antiferromagnetic density algorithm in Fig. 2.

FIGURE 2 Spherical electromagnetic radiation density with ϕ(n).  

Magnetic antiferromagnetic spin for ϕ(n)

Theorem 7. Spherical electric and magnetic phase of ϕ(n) are given by

Proof. From spherical Sitter frame, we obtain

ϕ(n)×sϕ(n)=ε3γ-βε2t-((ε-βsε+β(εs-β))n.

We instantly calculate

tϕn=σ1ε-βtγ+εt-σ1βt+εσ2s+σ1ε-βσ2n.

Combining the above relations, we get

σ1ε-βt=ε3,εt-σ1β=-βε2,εσ2s+σ1ε-βσ2=ε-βsε+βεs-β.

Magnetic total spherical phase for spherical Lorentz force ϕ(n) is presented by

ΦϕnGt=ϕnGΨ×tϕndρ.

Consequently, we have

Gt×tϕ(n)=((εt-σ1β)-β(ε(σ2s+σ1ε)-βσ2))γ+((σ1ε-βt)+ε(ε(σ2s+σ1ε)-βσ2))t+(ε(σ2s+σ1ε)-βσ2))t+(ε(εt-σ1β)+(σ1ε-βt)β)n.

By further calculation, we have that

These equations are equivalent to

δϕ(n)Gt=-β((εt-σ1β)-β(ε(σ2s+σ1ε)-βσ2))-ε((σ1ε-βt+ε(ε(σ2s+σ1ε)-βσ2)).

We immediately compute

ΛϕnGt=--βεt-σ1β-βεσ2s+σ1ε-βσ2-εσ1ε-βt+εεσ2s+σ1ε-βσ2dρ.

By using a spherical Sitter frame, we give

Gt×ϕn×sϕn=βε-βsε+βεs-β-βε2γ+ε3-εε-βsε+βεs-βt.

We immediately compute the density, namely

δAFRϕnGt=-ββε-βsε+βεs-β-βε2-εε3-εε-βsε+βεs-β.

Since, we instantly arrive at

AFRΛϕ(n)Gt=-(β(β((ε-βs)ε+β(εs-β))-βε2)+ε(ε3-ε((ε-βs)ε+β(εs-β)))dρ.

As a reaction, we get the following impressive results.

Theorem 8. The electromagnetic radiation density of ϕ(n) is

EMRϕ(n)=(ξσ2+εtξ)(ε(εt-σ1β)+(σ1ε-βt)β)+((εt-σ1β)-β(ε(σ2s+σ1ε)-βσ2))(εt-βσ1+σ2)-βt)+ε(ε(σ2s+σ1ε)-βσ2))(εσ1-βt-(σ2ϑ+σ1ε))+(σ2ε-β(σ2s+σ1ε))(ε(εt-σ1β)-((σ1εNa+(σ1ε-βt)β)-εξσ2((εt-σ1β)-β(ε(σ2s+σ1ε)-βσ2))-(ξσ1+εξ(σ2s+σ1ε))×((σ1ε-βt+ε(ε(σ2s+σ1ε)-βσ2)).

Theorem 9. Antiferromagnetic spherical electric and magnetic phase of ϕ(n) are given by

ΩAFRϕnGt=εt-βσ1+σ2εs-β1+ε2+σ2ε-βσ2s+σ1εεεs-εβ1+ε2dζ,AFRΩϕnEt=-εs-β1+ε2εξσ2+εεs-εβ1+ε2ξσ2+εtξdζ.

Proof. By antiferromagnetic model, we get

Gt×ϕn×sϕn=βε-βsε+βεs-β-βε2γ+ε3-εε-βsε+βεs-βt.

Spherical electric and magnetic radiation density with an antiferromagnetic model are given by

Corollary 1. Electromagnetic radiation density with an antiferromagnetic model is

Spherical magnetic anholonomy density and timelike t-magnetic particle flow in the kernel of magnetic quadrupole for timelike spherical spherical evolution with Lorentz force ϕ(n). Spherical antiferromagnetic density is provided by antiferromagnetic density algorithm in Fig. 3.

FIGURE 3 Spherical electromagnetic radiation with ϕ(n).  

4. Optical Soliton for Extended direct algebraic method with Fractional Equation in S 1 2

In this section, we design perturbed fractional solutions of the nonlinear evolution equation governing the propagation of solitons by magnetic fields of the polarized light ray traveling in spherical fractional optical fiber. The traveling assumption concept is operated to gauge analytical soliton solutions. Then numerical duplications are also contributed to complement the rational outcomes. Also, we deal with the following evolution equations of Lorentz force ϕ(n).

εtη=-2σ2,εs=σ1+εσ2s+σ1ε,t>0,0<η1.

where εtη is the conformable derivative operator (ε = ε(s,t)) and σ 2 = σ 2(s,t). We consider σ 1 = σ 2 ,

εtη=-2σ2,εs=σ2+εσ2s+σ2ε,t>0,0<η1. (4.1)

The conformable derivative of order η ∈ (0,1] is defined as the subsequent interpretation [23]

tDηft=limϑ0ζt+ϑt1-η-ζtϑ,ζ:0,R. (4.2)

Some of the components of conformable derivatives are in such a way [29,30].

tDηtα=αtα-ηη  R,

Dtηζχ=ζtDηχ+χtDηζ,

Dtηζoχ=t1-ηχ'tζ'χt,

tDηζχ=χtDηζ-ζtDηχχ2

• Assume that the traveling wave variable is

εs,t=uϕ,   ϕ=s-Qtηη,   σ2s,t=vϕ, (4.3)

By using u(ϕ) and v(ϕ), we have

A ( u, u_ϕ, u_{ϕϕ}, u_{ϕϕϕ},    ) = 0,    B (v, v_ϕ, v_{ϕϕ}, v_{ϕϕϕ}, ) = 0. (4.4)

• Recognize the solution of Eqs. (4.4),

uϕ=i=0NαiGiϕ,\qqvϕ=i=0NβiGiϕ, (4.5)

where α n 0, β n 0 and G(ϕ) can be defined as:

G'ϕ=lnAfG2ϕ+χGϕ+ϑ,   A0,1, (4.6)

where ϑ,χ,f are arbitrary fixeds.

Some solutions of Eq. (4.6) are given by:

1) If χ 2 − 4ϑf < 0 and f≠0, then

G1ϕ=-χ2f+-χ2-4ϑf2fTanA-χ2-4ϑf2ϕ,  G2ϕ=-χ2f+-χ2-4ϑf2fcotA-χ2-4ϑf2ϕ.

2) If χ 2 − 4ϑf > 0 and f≠0, then

G3ϕ=-χ2f+χ2-4ϑf4ftanhAχ2-4ϑf4ϕ+cothAχ2-4ϑf4ϕ.

3) If ϑf > 0 and f = 0, then

G4ϕ=ϑftanA2ϑfϕ±ΔΩsecA2ϑfϕ.

4) If ϑf < 0 and χ = 0, then

G5ϕ=-ϑfcothA2-ϑfϕ±ΔΩcschA2-ϑfϕ.

5) If ϑ = f and χ = 0, then

G6ϕ=-tanAϑ2ϕ-cotAϑ2ϕ.

6) If ϑ = −f and χ = 0, then

G7ϕ=-tanhAϑϕ.

7) If χ 2 = 4ϑf, then

G8ϕ=-2ϑχϕlnA+2χ2ϕlnA.

8) If χ = k, ϑ = mk, (m≠0) and f = 0, then

G9ϕ=Akϕ-m.

9) If χ = f = 0, then

G10ϕ=ϑϕlnA.

10) If χ = ϑ = 0, then

G11ϕ=-1fϕlnA.

11) If ϑ = 0, and χ ≠ 0 , then

G12ϕ=-ΔχfcoshAχϕ-sinhAχϕ+Δ.

12) If χ=k, ϑ=0 and f=mk m0 , then

G13ϕ=ΔAkϕΩ-mΔAkϕ.

Remark. The generalized hyperbolic and triangular functions are defined [29];

sinAϕ=ΔAiϕ-ΩA-iϕ2i,   cosAϕ=ΔAiϕ+ΩA-iϕ2,

tanAϕ=-iΔAiϕ-ΩA-iϕΔAiϕ+ΩA-iϕ,  cotAϕ=iΔAiϕ+ΩA-iϕΔAiϕ-ΩA-iϕ,

secAϕ=2ΔAiϕ+ΩA-iϕ,  cscAϕ=2iΔAiϕ-ΩA-iϕ,

secAϕ=2ΔAiϕ+ΩA-iϕ,  cscAϕ=2iΔAiϕ-ΩA-iϕ,

sinhAϕ=ΔAϕ-ΩA-ϕ2,  coshAϕ=ΔAϕ+ΩA-ϕ2,

tanhAϕ=ΔAϕ-ΩA-ϕΔAϕ+ΩA-ϕ,  cothAϕ=ΔAϕ+ΩA-ϕΔAϕ-ΩA-ϕ,

sechAϕ=2ΔAϕ+ΩA-ϕ,  cschAϕ=2ΔAϕ-ΩA-ϕ,

By placing the above equations, we have

2vϕ-Qu'ϕ=0u'ϕ-uϕv'ϕ-vϕuϕ2-vϕ=0 (4.7)

The solutions of the above equation be a finite series as follows:

uϕ=j=0NαjGjϕ,   vϕ=j=0MβjGjϕ, (4.8)

where G(ϕ) satisfies Eq. (4.6) ϕ=σ-Qtη/η and αj, βj for j=1,N¯are values to be definited.

Putting

uϕ=α0+α1Gϕ,   ϕ=β0+β1Gϕ+β2Gϕ2, (4.9)

where G(ϕ) satisfied Eq. (4.6).

Also, solving the above system, we get

α0=-χlnA,   α1=-2flnA,   β0=-2fϑlnA2,   β1=-2fχlnA2,β2=-2f2lnA2,    Q=2.

New solutions of Eq. (4.1) are presented by:

1) If χ 2 − 4ϑf < 0 and f≠0, then

ε1s,t=-χlnA-2flnA-χ2f+-χ2-4ϑf2ftanA-χ2-4ϑf2ϕ

σ21s,t=-2fϑlnA2-2fχlnA2-χ2f+-χ2-4ϑhf2ftanA-χ2-4ϑf2ϕ

-2f2lnA2-χ2f+-χ2-4ϑf2ftanA-χ2-4ϑf2ϕ2

ε2s,t=-χlnA-2flnA-χ2f+-χ2-4ϑf2fcotA-χ2-4ϑf2ϕ,

σ22s,t=-2fϑlnA2-2fχlnA2-χ2f+-χ2-4ϑf2fcotA-χ2-4ϑf2ϕ,

-2f2lnA2-χ2f+-χ2-4ϑf2fcotA-χ2-4ϑf2ϕ2.

2) If χ 2 − 4ϑf > 0 and f≠0, then singular soliton and dark solutions are presented by

ε3s,t=-χlnA-2flnA-χ2f+χ2-4ϑf4ftanhAχ2-4ϑf4ϕ+cothAχ2-4ϑf4ϕ,

σ23s,t=-2fϑlnA2-2fχlnA2-χ2f+χ2-4ϑf4ftanhAχ2-4ϑf4ϕ+cothAχ2-4ϑf4ϕ

-2f2lnA2-χ2f+χ2-4ϑf4ftanhAχ2-4ϑf4ϕ+cothAχ2-4ϑf4ϕ2.

3) If ϑf > 0 and χ = 0, then the singular new periodic solutions are presented by

ε4s,t=-glnA-2flnAϑftanA2ϑfϕ±ΔΩsecA2ϑfϕ,

σ24s,t=-2fϑlnA2+-2fχlnA2ϑftanA2ϑfϕ±ΔΩsecA2ϑfϕ

-2f2lnA2ϑftanA2ϑfϕ±ΔΩsecA2ϑfϕ2.
4) If ϑf < 0 and χ = 0, then the dark, bright, and singular soliton solutions are presented by

ε5 (s,t) = - χln(A) - 2 f ln (A) ( - ϑ f [- cothA { 2 - ϑf ϕ} ±ΔΩ \cschA { 2 - ϑf ϕ } ])

σ25(s,t)=-2fϑln(A)2-2fχln(A)2(-ϑf[-cothA{2-ϑfϕ}±ΔΩcschA{2-ϑfϕ}])

-2f2ln(A)2(-ϑf[-cothA{2-ϑfϕ}±ΔΩ cschA{2-ϑfϕ}])2.

5) If ϑ = f and χ = 0, then singular new periodic solutions are presented by

ε6(s,t)=-χln(A)-2fln(A)(12[tanA{ϑ2ϕ}-cotA{ϑ2ϕ}]),

σ26(s,t)=-2fϑln(A)2+(-2)fχln(A)2(12[tanA{ϑ2ϕ}-cotA{ϑ2ϕ}])

-2f2ln(A)2(12[tanA{ϑ2ϕ}-cotA{ϑ2ϕ}])2.

6) If ϑ = −f and χ = 0, then dark and singular soliton new solutions are presented by

ε7s,t=-χlnA-2flnA-tanhAϑϕ,

σ27=-2fϑlnA2+-2fχlnA2-tanhAhϕ-2f2lnA2-tanhAϑϕ2.

7) If χ 2 = 4ϑf, then rational new solution is presented by

ε8s,t=-χlnA-2flnA-2ϑχϕlnA+2χ2ϕlnA,Nbσ28s,t=-2fϑlnA2+-2fχlnA2-2ϑχϕlnA+2χ2ϕlnA-2f2lnA2-2ϑχϕlnA+2χ2ϕlnA2.

8) If χ = k, ϑ = mk (m≠0) and f = 0, then the rational solution is presented by

ε9s,t=-χlnA-2flnAAkϕ-m,

σ29s,t=-2fϑlnA2+-2fχlnA2Akϕ-m-2f2lnA2Akϕ-m2.

9) If χ = f = 0, then rational new solution is presented by

ε10s,t=-χlnA-2flnAhϕlnA,

σ210s,t=-2fϑlnA2+-2fχlnA2ϑϕlnA-2f2lnA2ϑϕlnA2.

10) If χ = ϑ = 0, then rational new solution is presented by

ε11s,t=-χlnA-2flnA-1fϕlnA,

σ211s,t=-2fϑlnA2+-2fχlnA2-1fϕlnA-2f2lnA2-1fϕlnA2

11) If ϑ = 0, and χ≠0, then dark like new solitons are presented by

ε12(s,t)=-χln(A)-2fln(A)(-Δχf[coshA{χϕ}-sinhA{χϕ}+Δ]),

σ212(s,t)=-2fϑln(A)2+(-2)fχln(A)2(-Δgf[coshA{χϕ}-sinhA{χϕ}+Δ])

-2f2ln(A)2(-Δχf[coshA{χϕ}-sinhA{χϕ}+Δ])2.

12) If χ = k, and ϑ = 0 and f = mk (m≠0), then rational new solution is presented by

ε13s,t=-χlnA-2fAΔAkϕΩ-mΔAkϕ,

σ213s,t=-2fϑlnA2+-2fχlnA2ΔAkϕΩ-mΔAkϕ-2f2lnA2ΔAkϕΩ-mΔAkϕ2.

5. Graphical representation of the solutions

The magnetic flow graphics of the performed solutions are demonstrated in the illustrates by applying Mathematica. In Figs. 4-5, we produce some numerical simulations of ε(s,t) and σ 2(s,t) in 3D plots when 0 ≤ s ≤ 5 and 0 ≤ t ≤ 5.

FIGURE 4 The magnetic flow graphics for the analitical solutions of the fractional (4.1) equations (h = f = 2, χ = 1) a) ε 1(s,t), b) σ 21(s,t).  

FIGURE 5 The magnetic flow graphics for the analitical solutions of the fractional (4.1) equations (h = −2, f = 1, χ = 0) a) ε 5(s,t), b) σ 25(s,t).  

We displayed the some of solutions recovered for the presented fractional (4.1) equations by a conformable derivative operator. Besides, we showed 3D graphics for some of the solutions in Figs. 4-5. The above figures were drawn for A = 1.7, η = 0.8, ∆ = Ω = 1.

6. Conclusion

Geometrical models that describe a relativistic magnetic particle may be constructed using the geometrical scalars associated with the embedding of the particle worldline in Heisenberg spacetime as building blocks for the action. In this investigation, we consider a new theory of optical magnetic spherical antiferromagnetic spin of timelike spherical magnetic flows of the t-magnetic particle by the spherical Sitter frame in de Sitter space. Also, the extended direct algebraic method is used to find new soliton solutions of the fractional (4.1) equations in de Sitter space. Many partial differential equations are proceeding in mathematical physics and engineering modeling applications [31-46]. We express that the suggested approach is applicable to investigate the plenty of illustrations based on engineering and science.

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Received: March 09, 2021; Accepted: April 02, 2021

e-mail: minc@firat.edu.tr

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