# Abundant Exact Soliton Solutions of the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation Based on the Jacobi Elliptic Function Ideas

Authors: Qinghao Zhu; Jianming Qi
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The Heisenberg ferromagnetic spin chain equation (HFSCE) is very important in modern magnetism theory. HFSCE expounded the nonlinear long-range ferromagnetic ordering magnetism. Also, it depicts the characteristic of magnetism to many insulating crystals as well as interaction spins. Moreover, the ferromagnetism plays a fundamental role in modern technology and industry and it is principal for many electrical and electromechanical devices such as generators, electric motors, and electromagnets. In this article, the exact solutions of the nonlinear (2+1)-dimensional HFSCE are successfully examined by an extended modified version of the Jacobi elliptic expansion method (EMVJEEM). Consequently, much more new Jacobi elliptic traveling wave solutions are found. These new solutions have not yet been reported in the studied models. For the study models, the new solutions are singular solitons not yet observed. Additionally, certain interesting 3D and 2D figures are performed on the obtained solutions. The geometrical representation of the HFSCE provides the dynamical information to explain the physical phenomena. The results will be significant to understand and study the (2+1)-dimensional HFSCE. Therefore, further studying EMVJEEM may help researchers to seek for more soliton solutions to other nonlinear differential equations.


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## Body

## 1. Introduction and Main Results

The investigation of traveling wave solutions of nonlinear evolution equations (NLEEs) plays a key role in the study of the internal mechanisms of complex phenomena. In the last few decades, we have made imperative developments and found in the literature many powerful and skilled methods to obtain analytical traveling wave solutions, such as electromagnetism, liquid mechanics, atomic materials, complex physics, electrical engineering, optical fibers, and geochemistry [1–8]. As a result, a large number of mathematicians and physicists tried to invent various methods to obtain solutions to such equations. About describing various complex phenomena in the field of NLEEs, soliton theory plays a crucial role in a number of nonlinear models. Different scientists have studied the dynamics of solitons in different models. For instance, Belić [9] investigated analytical light bullet solutions of the generalized (3+1)-dimensional nonlinear Schrödinger equation in 2007. In recent years, Kumar et al. have made great achievements in the study of nonlinear differential equations of water wave models by using Lie symmetry analysis [10, 11].In 2011, Sabry et al. interpreted three-dimensional ion-acoustic envelope soliton excitations in electron-positron-ion magnetoplasmas through the derivation of three-dimensional nonlinear Schrödinger equation. [12] Helal and Seadawy applied function transformation methods to the D-dimensional nonlinear Schrödinger equation with damping and diffusive terms. [13] In 2012, Giulini and Groβardt [14] derived the Schrödinger-Newton equation for spherically symmetric gravitational fields in a WKB-like expansion in 1/c from the Einstein-Klein-Gordon and Einstein-Dirac system. Kumar et al. [15] obtained exact space-time periodic traveling wave solutions of the generalized (3+1)-dimensional cubic-quandt nonlinear Schrödinger equation with spatial distribution coefficients. In 2017, Seadawy and Lu [16] derived the exact bright, dark, and bright-dark solitary wave soliton solutions of the generalized higher order nonlinear NLS equation by using the amplitude ansatz method. The powerful sine-Gordon expansion method was utilized to search for the solutions to some important nonlinear mathematical models arising in nonlinear sciences by Bulut et al. [17]. Zayed et al. [18] investigated the soliton solutions to the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity.In modern magnetic theory, the (2+1)-dimension HFSCE is considered as one of the very important equations to explain the dynamics of nonlinear magnets. The (2+1)-dimensional HFSCE which is a suitable equation representing many insulating magnetic crystal properties and explaining spin-long ferromagnetic ordered interactions is of striking interest in the soliton theory. The soliton solutions for the (2+1)-dimensional HFSC equation are characterized by high quality and qualitative studies for a lot of phenomena and processes in various fields such as ferromagnetic materials, nonlinear optics, and optical fibers. In the meantime, the Heisenberg model of ferromagnetic spin chains with various magnetic interactions associated with nonlinear evolution equations exhibiting a neat and tidy behavior in the classical and semiclassical continuum limits [19]. Inhomogeneous exchange interactions are also good candidates for activating spin reversal processes in ferromagnets [20]. In 2014, Latha and Vasanthi [21] applied the modified Kudryashov and Darboux transformation method for construction of exact traveling wave solutions. In 2017, Inc et al. [22] applied the complex envelope function and the generalized tanh methods to obtain the resolution of the soliton solutions of the two-dimensional HFSC equation. In 2018, Ma et al. [23] utilized an improved F-expansion method (Exp-function method) and the Jacobi elliptic method, respectively, to explore soliton solutions of the 2D-HFSC equation. In 2018, Li and Ma [24] used the Hirota bilinear method and chose proper polynomial functions in bilinear forms, the one-order rogue waves solution and its existence condition were obtained. In 2020, the different wave structures of the 2D-HFSC equation were investigated by Osman et al. [25] via the new extended FAN subequation method. In 2020, Bashar and Islam [26] implemented the modified simple equation (MSE) and improve F-expansion method to find the exact solutions of the HFSCE. In 2021, Hosseini et al. [27] investigated the HFSCE by the Jacobi elliptic functions (JEFs) method. In 2022, Sahoo and Tripathy [28] used the modified Khater method to study the HFSCE for the new exact solitary solutions.Considering the (2+1)-dimensional HFSCE [25, 27] which is defined as
(1)i∂Vx,y,t∂t+β1∂2Vx,y,t∂x2+β2∂2Vx,y,t∂y2+β3∂2Vx,y,t∂y∂x−β4Vx,y,tVx,y,t2=0,here
(2)β1=δ4γ+γ2,β2=δ4γ1+γ2,β3=2δ4γ2,β4=2δ4B.Here the complex-valued functionVx,y,t signifies the wave propagation, x,y are the spatial variables, and t is the time variable. The lattice parameter is represented as δ with the interaction coefficients γ,γ1,γ2, while the anisotropic parameter [25, 27] is denoted as B. In this section, we present a mathematical analysis of the proposed model. The complex wave transformation which has been taken into account to seek for the solitary solutions of HFSCE is of the next form:
(3)Vx,y,t=Φξeiθ,where the amplitude is denoted as Φξ with ξ=x+y−χt and θ=−lx+ky+ωt+ε2 is the corresponding phase component. By (3), we deduce that:
(4)Vt=−χΦ′+iωϕeiθ,(5)Vxx=ϕ′′−2ilϕ′−l2ϕeiθ,(6)Vyy=ϕ′′−2ikϕ′−k2ϕeiθ,(7)Vxy=ϕ′′−il−kϕ′+lkeiθ.Inserting (3) and (4)–(7) into (1), the real part,
(8)β1+β2+β3Φ′′−ω+β1l2+β2k2−β3klΦ−β4Φ3=0,and the imaginary part,
(9)χ=2β1l−2β2k−β3k+β3l,here the β1,β2,β3,k, and l are all nonzero parameters.In 2021, based on the ideas of the Jacobi elliptic functions, Yang and Zhang [29] used the unified F-expansion method to study the Korteweg-De Vries partial differential equations. In 2021, Ünal et al. [30] also investigated the exact solutions of space-time fractional symmetric regularized long wave equation using ideas of JEFs.Twelve kinds JEFs are available in literature [31]. Basic JEFs are expressed as snξ=snξ;m or cnξ=cnξ;m or dnξ=dnξ;m and other basic JEFs such as sd,cd,nd,sc,nc,dc,ns,cs, and ds. In addition, if m=0 and m=1, then the JEFs turn into trigonometric and hyperbolic functions. Here, m is the modulus and is a complex number. If m is real, it can be arranged 0<m2<1.


## 2. The Extend Modified Version of the Jacobi Elliptic Expansion Method

Based on the ideas of [30], we consider the following form of Equation (8), and we give the details of the extend modified version of the Jacobi elliptic expansion method (EMVJEEM) and employ this method to seek for new and more general traveling wave exact solutions to Equation (8).Step 1. Regarding that the solution of Equation (8) can be expressed by a polynomial in K as follows:
(10)Φξ=∑j=−NNajKjξ,in Equation (8), K=Kξ is satisfied the following differential equations:
(11)K′2ξ=pK4ξ+qK2ξ+r,where p,q, and r are arbitrary constants. N can be determined by the uniform equilibrium term between the highest derivative and the nonlinear term appearing in (8). All the solutions Kξ of (11) are listed in Table 1.Table 1
Basic structure.
pqrK1m2−1+m21±snξ,±cdξ21−1+m2m2±nsξ,±dcξ;3−m2−1+m2−1±isnξ,±icdξ4−1−1+m2−m2±insξ,±idcξ512−m21−m2±csξ,±idnξ61−m22−m21±scξ,±indξ712−m2m2−1±icsξ,±dnξ8m2−12−m2−1±iscξ,±ndξ91−m22m2−1−m2±ncξ,±imsdξ10−m22m2−11−m2±cnξ,±imdsξ11m2−12m2−1m2±incξ,±msdξ12m22m2−1m2−1±icnξ,±1mdsξ13m4−m22m2−11±imncξ,±sdξ1412m2−1m4−m2±imcnξ,±dsξ15−m4+m22m2−1−1±1mncξ,±isdξ16−12m2−1−m4+m2±1mcnξ,±idsξ17141+m221−m224dsξ±csξ,−dsξ∓csξ,imcnξ±dnξ,−imcnξ±dnξ181−m2241+m2214snξdnξ±cnξ,−snξdnξ±cnξ,imcnξ±dnξ,−1mcnξ±dnξ19−141+m22−1−m224idsξ±csξ,−idsξ±csξ,mcnξ±dnξ,−mcnξ∓dnξ20−1−m2241+m22−14isnξdnξ±cnξ,−isnξdnξ±cnξ,1mcnξ±dnξ,−1mcnξ±dnξ211−m241+m221−m24ncξ±scξ,−ncξ∓scξ,imsdξ±ndξ,−imsdξ±ndξ22m2−141+m22m2−14incξ±scξ,−incξ±scξ,msdξ±ndξ,−msdξ∓ndξ2314m2−22m44nsξ±dsξ,−nsξ∓dsξ,dcξ±1−m2ncξ,−dcξ∓1−m2ncξ24m44m2−2214snξ1±dnξ,−snξ1±dnξ,cnξdnξ±1−m2,−cnξdnξ±1−m225−14m2−22-m4/4insξ±dsξ,−insξ∓dsξ,idcξ±1−m2ncξ,−idcξ∓1−m2ncξ26−m44m2−22−14isnξ1±dnξ,−isnξ1±dnξ,icnξdnξ±1−m2,−icnξdnξ±1−m227m24m2−22m24snξ±icnξ,−snξ∓icnξ,cdξ±i1−m2sdξ,−cdξ∓i1−m2sdξ28−m24m2−22−m24cnξ±isnξ,−cnξ∓isnξ,1−m2sdξ±icdξ,−cdξ∓i1−m2sdξ29141−m2214nsξ±csξ,−nsξ∓csξ,msnξ±idnξ,−msnξ∓idnξ,dcξ±1−m2scξ,−dcξ∓1−m2scξmcdξ±i1−m2ndξ,−mcdξ∓i1−m2ndξ30−141−m22−14insξ±csξ,−insξ∓csξ,dnξ±imsnξ,−dnξ∓imsnξ,idcξ±1−m2scξ,−idcξ±1−m2scξ1−m2ndξ±imcdξ,−1−m2ndξ∓imcdξStep 2. Taking (10) into (8) and using (11), (8) is converted into another polynomial in K. Calculating all the coefficients of the polynomial to zero produces the system algebraic equations for a1,⋯, aN, p, q, and r.Step 3. The constants a1,⋯, aN can be obtained by solving the system of algebraic equations obtained in Step 3. Since (11) may have the following many possible solutions. Thus, the exact solutions for given (8) can be derived. Here:Step 4. Putting the inverse transform T−1 into the solutions Φξξ=x+y−χt, we can get all exact solutions Vx,y,t=Φx,y,tei−lx+ky+ωt+ε2 of the original Equation (8).Remark 1.
In 2021, Hosseini et al. [27] only considered the four cases in Table 1 (such as Cases 1, 2, 9, and 10). Obviously, we consider a broader scenario in Table 1, and therefore, more new solutions will be obtained in this paper.


## 3. Employing the EMVJEEM to Equation (8)

Taking into account the homogeneous equilibrium term betweenΦ′′ and Φ2 in (8), it is easy to deduce N=1. The solution of (9) can be listed as follows:
(12)Φξ=a1K+a0+b11K,here a0,a1, and b1 are unknown constants and will be determined later.By using (12) and (11) and collecting all terms with the same power K together, we deduce
(13)Φ′ξ=a1K′ξ−b1K−2ξK′ξ,(14)Φ′′ξ=2pa1K3ξ+a1qKξ+b1qK−1ξ+2b1rK−3ξ.Substituting (14) into (9), we obtain
(15)β1+β2+β32pa1K3+a1qK+b1qK−1+2b1rK−3+β3kl−ω−β1l2−β2k2a0+a1K+b11K−β4a0+a1K+b11K3=0.Sorting out all terms with the same power ofK together, we obtain
(16)2pa1+ca13K3+β1+β2+β3a1q+a1β3kl−ω−β1l2−β2k2−3β4a02a1−3β4a12b1K+β1+β2+β3b1q+b1β3kl−ω−β1l2−β2k2−3β4a02b1−3β4a1b121K+2b1rβ1+β2+β3−β4b131K3+a0β3kl−ω−β1l2−β2k2−β4a03−6a0a1b1β4−3a0a12β4K2−3β4a0b121K2=0.For the same term of functionK, we are extracting their unknown coefficients and setting them to zero to get the next equation:
(17)a0β3kl−ω−β1l2−β2k2−β4a03−6a0a1β1β4=0,β1+β2+β3a1q+a1β3kl−ω−β1l2−β2k2−3β4a02a1−3β4a12b1=0,β1+β2+β3b1q+β3kl−ω−β1l2−β2k2b1−3β4a02b1−3β4a1b12=0,3a0a12β4=0,3β4a0b12=0,2b1rβ1+β2+β3−β4b13=0,2pa1−β4a13=0.Solving this system of equations, the unknown coefficients are found:(18)a0=0,a1=±2pβ4,b1=±2rβ1+β2+β3β4,and β1+β2+β3q+β3kl−ω−β1l2−β2k22=36prβ1+β2+β3. From (3), we know that Φξ is the real function. Hence, Kξ takes the real cases in Table 1, the forms as follows:
(19)Φξ=a1Kξ+b11Kξ=±2pβ4Kξ±2rβ1+β2+β3β41Kξ.Case 1.
Ifp=m2, q=−1+m2, and r=1, then Kξ=±snξ or ±cdξ,
(20)Φ31x+y−χt=±m2β4snx+y−χt±2β1+β2+β3β41snx+y−χt,or
(21)Φ32x+y−χt=±m2β4cdx+y−χt±2β1+β2+β3β41cdx+y−χt;V31x+y−χt=Φ31x+y−χtei−lx+ky+ωt+ε2==±m2β4snx+y−χt±2β1+β2+β3β41snx+y−χtei−lx+ky+ωt+ε2,or
(22)V32x+y−χt=Φ32x+y−χtei−lx+ky+ωt+ε2=±m2β4cdx+y−χt±2β1+β2+β3β41cdx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2−β3kl−ω−β1l2−β2k22=36m2β1+β2+β3.Case 2.
Ifp=1, q=−1+m2, and r=m2, then Kξ=±nsξ or ±dcξ,
(23)Φ33x+y−χt=±2β4nsx+y−χt±m2β1+β2+β3β41nsx+y−χt,or
(24)Φ34x+y−χt=±2β4dcx+y−χt±m2β1+β2+β3β41dcx+y−χt;V33x+y−χt=Φ33x+y−χtei−lx+ky+ωt+ε2;=±2β4nsx+y−χt±m2β1+β2+β3β41nsx+y−χtei−lx+ky+ωt+ε2,or
(25)V34x+y−χt=Φ34x+y−χtei−lx+ky+ωt+ε2=±2β4dcx+y−χt±m2β1+β2+β3β41dcx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2−β3kl−ω−β1l2−β2k22=36m2β1+β2+β3.Case 3.
Ifp=1, q=2−m2, and r=1−m2, then Kξ=±csξ,
(26)Φ35x+y−χt=±2β4csx+y−χt±21−m2β1+β2+β3β41csx+y−χt;V35x+y−χt=Φ35x+y−χtei−lx+ky+ωt+ε2=±2β4csx+y−χt±21−m2β1+β2+β3β41csx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32−m2+β3kl−ω−β1l2−β2k22=361−m2β1+β2+β3.Case 4.
Ifp=1−m2, q=2−m2, and r=1, then Kξ=±scξ,
(27)Φ36x+y−χt=±21−m2β4scx+y−χt±2β1+β2+β3β41scx+y−χt;V36x+y−χt=Φ36x+y−χtei−lx+ky+ωt+ε2=±21−m2β4scx+y−χt±2β1+β2+β3β41scx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32−m2+β3kl−ω−β1l2−β2k22=361−m2β1+β2+β3.Case 5.
Ifp=1, q=2−m2, and r=m2−1, then Kξ=±dnξ,
(28)Φ37x+y−χt=±2β4dnx+y−χt±2m2−1β1+β2+β3β41dnx+y−χt;V37x+y−χt=Φ37x+y−χtei−lx+ky+ωt+ε2=±2β4dnx+y−χt±2m2−1β1+β2+β3β41dnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32−m2+β3kl−ω−β1l2−β2k22=36m2−1β1+β2+β3.Case 6.
Ifp=m2−1, q=2−m2, and r=−1, then Kξ=±ndξ,
(29)Φ38x+y−χt=±2m2−1β4ndx+y−χt±−2β1+β2+β3β41ndx+y−χt;V38x+y−χt=Φ38x+y−χtei−lx+ky+ωt+ε2=±2m2−1β4ndx+y−χt±−2β1+β2+β3β41ndx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32−m2+β3kl−ω−β1l2−β2k22=−36m2−1β1+β2+β3.Case 7.
Ifp=1−m2, q=2m2−1, and r=−m2, then Kξ=±ncξ,
(30)Φ39x+y−χt=±21−m2β4ncx+y−χt±−2m2β1+β2+β3β41ncx+y−χt;and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=−361−m2m2β1+β2+β3.
(31)V39x+y−χt=Φ39x+y−χtei−lx+ky+ωt+ε2=±21−m2β4ncx+y−χt±−2m2β1+β2+β3β41ncx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=−361−m2m2β1+β2+β3.Case 8.
Ifp=−m2, q=2m2−1, and r=1−m2, then Kξ=±cnξ,
(32)Φ310x+y−χt=±m−2β4cnx+y−χt±21−m2β1+β2+β3β41cnx+y−χt;V310x+y−χt=Φ310x+y−χtei−lx+ky+ωt+ε2=±m−2β4cnx+y−χt±21−m2β1+β2+β3β41cnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=−361−m2m2β1+β2+β3.Case 9.
Ifp=m2−1, q=2m2−1, and r=m2, then Kξ=±msdξ,
(33)Φ311x+y−χt=±2m2−1β4msdx+y−χt±m2β1+β2+β3β41msdx+y−χt;V311x+y−χt=Φ311x+y−χtei−lx+ky+ωt+ε2=±2m2−1β4msdx+y−χt±m2β1+β2+β3β41msdx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m2−1m2β1+β2+β3.Case 10.
Ifp=m2, q=2m2−1, and r=m2−1, then Kξ=±1/mdsξ,
(34)Φ312x+y−χt=±2β4dsx+y−χt±2m2−1β1+β2+β3β41mdsx+y−χt;V312x+y−χt=Φ312x+y−χtei−lx+ky+ωt+ε2=±2β4dsx+y−χt±2m2−1β1+β2+β3β41mdsx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m2m2−1β1+β2+β3.Case 11.
Ifp=m4−m2, q=2m2−1, and r=1, then Kξ=±sdξ,
(35)Φ313x+y−χt=±2m4−m2β4sdx+y−χt±2β1+β2+β3β41sdx+y−χt;V313x+y−χt=Φ313x+y−χtei−lx+ky+ωt+ε2=±2m4−m2β4sdx+y−χt±2β1+β2+β3β41sdx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m4−m2β1+β2+β3.Case 12.
Ifp=1, q=2m2−1, and r=m4−m2, then Kξ=±dsξ,
(36)Φ314x+y−χt=±2β4dsx+y−χt±2m4−m2β1+β2+β3β41dsx+y−χt;V314x+y−χt=Φ314x+y−χtei−lx+ky+ωt+ε2=±2β4dsx+y−χt±2m4−m2β1+β2+β3β41dsx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m4−m2β1+β2+β3.Case 13.
Ifp=−m4+m2, q=2m2−1, and r=−1, then Kξ=±1/mncξ,
(37)Φ315x+y−χt=±21−m2β4ncx+y−χt±−2β1+β2+β3β4mncx+y−χt;V315x+y−χt=Φ315x+y−χtei−lx+ky+ωt+ε2=±21−m2β4ncx+y−χt±−2β1+β2+β3β4mncx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m4−m2β1+β2+β3.Case 14.
Ifp=−1, q=2m2−1, r=m2−m4, then Kξ=±mcnξ,
(38)Φ316x+y−χt=±m−2β4cnx+y−χt±21−m2β1+β2+β3β41cnx+y−χt;V316x+y−χt=Φ316x+y−χtei−lx+ky+ωt+ε2,=±m−2β4cnx+y−χt±21−m2β1+β2+β3β41cnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β32m2−1+β3kl−ω−β1l2−β2k22=36m4−m2β1+β2+β3.Case 15.
Ifp=1/4, q=1+m2/2, and r=1−m22/4, then Kξ=dsξ±csξ or −dsξ∓csξ,
(39)Φ317x+y−χt=±12β4dsx+y−χt±csx+y−χt±β1+β2+β31−m222β41dsx+y−χt±csx+y−χt,or
(40)Φ318x+y−χt=±12β4−dsx+y−χt∓csx+y−χt±β1+β2+β31−m222β41−dsx+y−χt∓csx+y−χt;(41)V317x+y−χt=Φ317x+y−χtei−lx+ky+ωt+ε2=±12β4dsx+y−χt±csx+y−χt±β1+β2+β31−m222β41dsx+y−χt±csx+y−χtei−lx+ky+ωt+ε2,or
(42)V318x+y−χt=Φ318x+y−χtei−lx+ky+ωt+ε2=±12β4−dsx+y−χt∓csx+y−χt±β1+β2+β31−m222β41−dsx+y−χt∓csx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 16.
Ifp=1−m22/4, q=1+m2/2, and r=1/4, then Kξ=snξ/dnξ±cnξ or −snξ/dnξ±cnξ,
(43)Φ319x+y−χt=±1−m222β4snx+y−χtdnx+y−χt±cnx+y−χt±β1+β2+β32β4dnx+y−χt±cnx+y−χtsnx+y−χt,or
(44)Φ320x+y−χt=±1−m222β4−snx+y−χtdnx+y−χt±cnx+y−χt±β1+β2+β32β4dnx+y−χt±cnx+y−χt−snx+y−χt;(45)V319x+y−χt=Φ319x+y−χtei−lx+ky+ωt+ε2=±1−m222β4snx+y−χtdnx+y−χt±cnx+y−χt±β1+β2+β32β4dnx+y−χt±cnx+y−χtsnx+y−χtei−lx+ky+ωt+ε2,or
(46)V320x+y−χt=Φ320x+y−χtei−lx+ky+ωt+ε2=±1−m222β4−snx+y−χtdnx+y−χt±cnx+y−χt±β1+β2+β32β4dnx+y−χt±cnx+y−χt−snx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 17.
Ifp=−1/4, q=1+m2/2, and r=−1−m22/4, then Kξ=mcnξ±dnξ or −mcnξ∓dnξ,
(47)Φ321x+y−χt=±−12β4mcnx+y−χt±dnx+y−χt±−1−m22β1+β2+β32β41mcnx+y−χt±dnx+y−χt,or
(48)Φ322x+y−χt=±−12β4−mcnx+y−χt∓dnx+y−χt±−1−m22β1+β2+β32β41−mcnx+y−χt∓dnx+y−χt;V321x+y−χt=Φ321x+y−χtei−lx+ky+ωt+ε2=±−12β4mcnx+y−χt±dnx+y−χt±−1−m22β1+β2+β32β41mcnx+y−χt±dnx+y−χtei−lx+ky+ωt+ε2,or
(49)V322x+y−χt=Φ322x+y−χtei−lx+ky+ωt+ε2=±−12β4−mcnx+y−χt∓dnx+y−χt±−1−m22β1+β2+β32β41−mcnx+y−χt∓dnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 18.
Ifp=−1/4, q=1+m2/2, and r=−1−m22/4, then Kξ=mcnξ±dnξ or −mcnξ∓dnξ,
(50)Φ323x+y−χt=±−12β4mcnx+y−χt±dnx+y−χt±−1−m22β1+β2+β32β41mcnx+y−χt±dnx+y−χt,or
(51)Φ324x+y−χt=±−12β4−mcnx+y−χt∓dnx+y−χt±−1−m22β1+β2+β32β41−mcnx+y−χt∓dnx+y−χt;(52)V323x+y−χt=Φ323x+y−χtei−lx+ky+ωt+ε2=±−12β4mcnx+y−χt±dnx+y−χt±−1−m22β1+β2+β32β41mcnx+y−χt±dnx+y−χtei−lx+ky+ωt+ε2,or
(53)V324x+y−χt=Φ324x+y−χtei−lx+ky+ωt+ε2=±−12β4−mcnx+y−χt∓dnx+y−χt±−1−m22β1+β2+β32β41−mcnx+y−χt∓dnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 19.
Ifp=−1−m22/4, q=1+m2/2, and r=−1/4, then Kξ=1/mcnξ±dnξ or −1/mcnξ±dnξ,
(54)Φ325x+y−χt=±−1−m222β41mcnx+y−χt±dnx+y−χt±−β1+β2+β32β4mcnx+y−χt±dnx+y−χt,or
(55)Φ326x+y−χt=±−1−m222β4−1mcnx+y−χt±dnx+y−χt±−β1+β2+β32β4−mcnx+y−χt±dnx+y−χt;V325x+y−χt=Φ325x+y−χtei−lx+ky+ωt+ε2=±−1−m222β41mcnx+y−χt±dnx+y−χt±−β1+β2+β32β4mcnx+y−χt±dnx+y−χtei−lx+ky+ωt+ε2,or
(56)V326x+y−χt=Φ326x+y−χtei−lx+ky+ωt+ε2=±−1−m222β4−1mcnx+y−χt±dnx+y−χt±−β1+β2+β32β4−mcnx+y−χt±dnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 20.
Ifp=1−m2/4, q=1+m2/2, and r=1−m2/4, then Kξ=ncξ±scξ or −ncξ∓scξ,
(57)Φ327x+y−χt=±1−m22β4ncx+y−χt±scx+y−χt±1−m2β1+β2+β32β41ncx+y−χt±scx+y−χt,or
(58)Φ328x+y−χt=±1−m22β4−ncx+y−χt∓scx+y−χt±1−m2β1+β2+β32β41−ncx+y−χt∓scx+y−χt;(59)V327x+y−χt=Φ327x+y−χtei−lx+ky+ωt+ε2=±1−m22β4ncx+y−χt±scx+y−χt±1−m2β1+β2+β32β41ncx+y−χt±scx+y−χtei−lx+ky+ωt+ε2,or
(60)V328x+y−χt=Φ328x+y−χtei−lx+ky+ωt+ε2=±1−m22β4−ncx+y−χt∓scx+y−χt±1−m2β1+β2+β32β41−ncx+y−χt∓scx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case 21.
Ifp=m2−1/4, q=1+m2/2, and r=m2−1/4, then Kξ=msdξ±ndξ or −msdξ∓ndξ,
(61)Φ329x+y−χt=±m2−12β4msdx+y−χt±ndx+y−χt±m2−1β1+β2+β32β41msdx+y−χt±ndx+y−χt,or
(62)Φ330x+y−χt=±m2−12β4−msdx+y−χt∓ndx+y−χt±m2−1β1+β2+β32β41−msdx+y−χt∓ndx+y−χt;(63)V329x+y−χt=Φ329x+y−χtei−lx+ky+ωt+ε2,=±m2−12β4msdx+y−χt±ndx+y−χt±m2−1β1+β2+β32β41msdx+y−χt±ndx+y−χtei−lx+ky+ωt+ε2,or
(64)V330x+y−χt=Φ330x+y−χtei−lx+ky+ωt+ε2,=±m2−12β4−msdx+y−χt∓ndx+y−χt±m2−1β1+β2+β32β41−msdx+y−χt∓ndx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=9m2−12/4β1+β2+β3.Case 22.
Ifp=1/4, q=m2−2/2, and r=m4/4, then Kξ=nsξ±dsξ or −nsξ∓dsξ, or dcξ±1−m2ncξ, or −dcξ∓1−m2ncξ(65)Φ331x+y−χt=±12β4nsx+y−χt±dsx+y−χt±m4β1+β2+β32β41nsx+y−χt±dsx+y−χt,or
(66)Φ332x+y−χt=±12β4−nsx+y−χt∓dsx+y−χt±m4β1+β2+β32β41−nsx+y−χt∓dsx+y−χt,or
(67)Φ333x+y−χt=±12β4dcx+y−χt±1−m2ncx+y−χt±m4β1+β2+β32β41dcx+y−χt±1−m2ncx+y−χt,or
(68)Φ334x+y−χt=±12β4−dcx+y−χt∓1−m2ncx+y−χt±m4β1+β2+β32β41−dcx+y−χt∓1−m2ncx+y−χt;(69)V331x+y−χt=Φ331x+y−χtei−lx+ky+ωt+ε2=±12β4nsx+y−χt±dsx+y−χt±m4β1+β2+β32β41nsx+y−χt±dsx+y−χtei−lx+ky+ωt+ε2,or
(70)V332x+y−χt=Φ332x+y−χtei−lx+ky+ωt+ε2=±12β4−nsx+y−χt∓dsx+y−χt±m4β1+β2+β32β41−nsx+y−χt∓dsx+y−χtei−lx+ky+ωt+ε2,or
(71)V333x+y−χt=Φ333x+y−χtei−lx+ky+ωt+ε2=±12β4dcx+y−χt±1−m2ncx+y−χt±m4β1+β2+β32β41dcx+y−χt±1−m2ncx+y−χtei−lx+ky+ωt+ε2,or
(72)V334x+y−χt=Φ334x+y−χtei−lx+ky+ωt+ε2=±12β4−dcx+y−χt∓1−m2ncx+y−χt±m4β1+β2+β32β41−dcx+y−χt∓1−m2ncx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β3m2−2/2+β3kl−ω−β1l2−β2k22=9m4/4β1+β2+β3.Case 23.
Ifp=m4/4, q=m2−2/2, and r=1/4, then Kξ=snξ/1±dnξ or −snξ/1±dnξ, or cnξ/dnξ±1−m2, or −cnξ/dnξ±1−m2(73)Φ335x+y−χt=±m42β4snx+y−χt1±dnx+y−χt±β1+β2+β32β41±dnx+y−χtsnx+y−χt,or
(74)Φ336x+y−χt=±m42β4−snx+y−χt1±dnx+y−χt±β1+β2+β32β41±dnx+y−χt−snx+y−χt,or
(75)Φ337x+y−χt=±m42β4cnx+y−χtdnx+y−χt±1−m2±β1+β2+β32β4dnx+y−χt±1−m2cnx+y−χt,or
(76)Φ338x+y−χt=±m42β4−cnx+y−χtdnx+y−χt±1−m2±β1+β2+β32β4dnx+y−χt±1−m2−cnx+y−χt;(77)V335x+y−χt=Φ335x+y−χtei−lx+ky+ωt+ε2=±m42β4snx+y−χt1±dnx+y−χt±β1+β2+β32β41±dnx+y−χtsnx+y−χtei−lx+ky+ωt+ε2,or
(78)V336x+y−χt=Φ336x+y−χtei−lx+ky+ωt+ε2=±m42β4−snx+y−χt1±dnx+y−χt±β1+β2+β32β41±dnx+y−χt−snx+y−χtei−lx+ky+ωt+ε2,or
(79)V337x+y−χt=Φ337x+y−χtei−lx+ky+ωt+ε2=±m42β4cnx+y−χtdnx+y−χt±1−m2±β1+β2+β32β4dnx+y−χt±1−m2cnx+y−χtei−lx+ky+ωt+ε2,or
(80)V338x+y−χt=Φ338x+y−χtei−lx+ky+ωt+ε2,=±m42β4−cnx+y−χtdnx+y−χt±1−m2±β1+β2+β32β4dnx+y−χt±1−m2−cnx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β3m2−2/2+β3kl−ω−β1l2−β2k22=9m4/4β1+β2+β3.Case 24.
Ifp=1/4, q=1−2m2/2, and r=1/4, then Kξ=nsξ±csξ or −nsξ∓csξ, or dcξ±1−m2scξ, or −dcξ∓1−m2scξ(81)Φ339x+y−χt=±12β4nsx+y−χt±csx+y−χt±β1+β2+β32β41nsx+y−χt±csx+y−χt,or
(82)Φ340x+y−χt=±12β4−nsx+y−χt∓csx+y−χt±β1+β2+β32β41−nsx+y−χt∓csx+y−χt,or
(83)Φ341x+y−χt=±12β4dcx+y−χt±1−m2scx+y−χt±β1+β2+β32β41dcx+y−χt±1−m2scx+y−χt,or
(84)Φ342x+y−χt=±12β4−dcx+y−χt±1−m2scx+y−χt±β1+β2+β32β41−dcx+y−χt±1−m2scx+y−χt;(85)V339x+y−χt=Φ339x+y−χtei−lx+ky+ωt+ε2,=±12β4nsx+y−χt±csx+y−χt±β1+β2+β32β41nsx+y−χt±csx+y−χtei−lx+ky+ωt+ε2,or
(86)V340x+y−χt=Φ340x+y−χtei−lx+ky+ωt+ε2,=±12β4−nsx+y−χt∓csx+y−χt±β1+β2+β32β41−nsx+y−χt∓csx+y−χtei−lx+ky+ωt+ε2,or
(87)V341x+y−χt=Φ341x+y−χtei−lx+ky+ωt+ε2,=±12β4dcx+y−χt±1−m2scx+y−χt±β1+β2+β32β41dcx+y−χt±1−m2scx+y−χtei−lx+ky+ωt+ε2,or
(88)V342x+y−χt=Φ342x+y−χtei−lx+ky+ωt+ε2,=±12β4−dcx+y−χt∓1−m2scx+y−χt±β1+β2+β32β41−dcx+y−χt∓1−m2scx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31−2m2/2+β3kl−ω−β1l2−β2k22=9/4β1+β2+β3.
Besides, ifm⟶0 and m⟶1, then the Jacobi elliptic functions become vestigial trigonometric and hyperbolic functions ([31]). Now we give specific examples as follows:Case3 If m⟶0 then csξ⟶cotξ, then
(89)Φ35x+y−χt=±2β4cotx+y−χt±2β1+β2+β3β4tanx+y−χt;(90)V35x+y−χt=Φ35x+y−χtei−lx+ky+ωt+ε2=±2β4cotx+y−χt±2β1+β2+β3β4tanx+y−χtei−lx+ky+ωt+ε2,and 2β1+β2+β3+β3kl−ω−β1l2−β2k22=36β1+β2+β3.Ifm⟶1 then csξ⟶cschξ, and
(91)Φ35x+y−χt=±2β4cschx+y−χt;V35x+y−χt=Φ35x+y−χtei−lx+ky+ωt+ε2=±2β4cschx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β3+β3kl−ω−β1l2−β2k22=0.Case15. If m⟶0, then dsξ±csξ⟶cscξ±cotξ or −dsξ∓csξ⟶−cscξ∓cotξ,
(92)Φ317x+y−χt=±12β4cscx+y−χt±cotx+y−χt±β1+β2+β32β41cscx+y−χt±cotx+y−χt,or
(93)Φ318x+y−χt=±12β4−cscx+y−χt∓cotx+y−χt±β1+β2+β32β41−cscx+y−χt∓cotx+y−χt;(94)V317x+y−χt=Φ317x+y−χtei−lx+ky+ωt+ε2=±12β4cscx+y−χt±cotx+y−χt±β1+β2+β31−m222β41cscx+y−χt±cotx+y−χtei−lx+ky+ωt+ε2,or
(95)V318x+y−χt=Φ318x+y−χtei−lx+ky+ωt+ε2=±12β4−cscx+y−χt∓cotx+y−χt±β1+β2+β31−m222β41−cscx+y−χt∓cotx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β31+m2/2+β3kl−ω−β1l2−β2k22=91−m22/4β1+β2+β3.Case23. If m⟶1, then snξ/1±dnξ⟶tanhξ/1±sechξ and
(96)Φ335x+y−χt=±12β4tanhx+y−χt1±sechx+y−χt±β1+β2+β32β41±sechx+y−χttanhx+y−χt;V335x+y−χt=Φ335x+y−χtei−lx+ky+ωt+ε2=±12β4tanhx+y−χt1±sechx+y−χt±β1+β2+β32β41±sechx+y−χttanhx+y−χtei−lx+ky+ωt+ε2,and β1+β2+β3−1/2+β3kl−ω−β1l2−β2k22=9/4β1+β2+β3.


## 4. Comparison

In this paper, employing the EMVJEEM to the HFSCE, we found that various forms exact solutions for HFSCE. In 2021, Ünal et al. [30] investigated the exact solutions of space-time fractional symmetric regularized long wave equation using ideas of the Jacobi elliptic functions. In our paper, Equation (10) is Φξ=∑j=−NNajKjξ. Here, j starts from the negative number to the corresponding positive number. However, in [30], j only takes 0,1,⋯,N, this is the biggest difference of [30]. From there, j can take the negative number; our form is more extensive than [30].Comparing with [29, 32], our article still varies greatly. In this paper, Equation (11) is K′2ξ=r+qK2ξ+pK4ξ. However, in [29, 32], what is similar to Equation (11) is K′2ξ=a0+a1K+a2K2+a3K3+a4K4. Due to the importance of Formula (11), this is obviously a big difference between our paper and [29].From the perspective of the complex calculation process, our paper is more complex than [27, 30]. At the same time, the results in our paper are more concise and clearer than [29, 32].Recently, the complex method [33, 34] is used to study the exact solutions of nonlinear evolution equations and found the Weierstrass elliptic functions (WEFs) solution, hyperbolic function or trigonometric meromorphic solutions, and rational solutions. In our paper, we also can find more solutions by the EMVJEEM, including hyperbolic function solutions or trigonometric solutions. The WEFs solutions ℘ξ satisfy the equation K′2ξ=4K3ξ−g2Kξ−g3. In this paper, Equation (11) is K′2ξ=pK4ξ+qK2ξ+r. In the above two equations, K of 3 power varies greatly from K of 4 power. At the same time, the relationship between the JEFs and WEFs is
(97)℘ξ,g2,g3=k2−k2−k3cn2k1−k3;m,here m2=k2−k3/k1−k3 is the modulus number of JEFs and kii=1,2,3,k1≥k2≥k3 are the roots of equation 4ξ2−g2ξ−g3=0.It is well know that ifm⟶1 and e2⟶e1, then cnξ;m⟶sechξ, or if m⟶0 then cnξ;m⟶cosξ. Equation (97) is the bridge linking among the WEFs, hyperbolic function, trigonometric function, and JEFs.


## 5. Computer Simulations

In this section, we are trying to explain the results through computer simulation images and further analyze the nature of theΦ35ξ, Φ317ξ, Φ318ξ, and Φ335ξ in the equations.Figure1 The 3D, 2D, and contour images of Φ35ξ, when m⟶0, we take the values of β1=1, β2=4, β3=3, β4=4, y=1, and q=1; the graphs demonstrate the localized interactions of diverse solitons of Φ35ξ on the domain. We also find the waves train in Figure 1(a).Figure 1
The 3D, 2D, and contour images ofΦ35ξ..
(a)(b)(c)Figure2: The 3D, 2D, and contour images of Φ35ξ, when m⟶1, we take the values of β4=2, y=1, and q=1; the graphs depict the single kink solitary wave solution Φ35ξ on the domain.Figure 2
The 3D, 2D, and contour images ofΦ35ξ.
(a)(b)(c)Figure3: The 3D, 2D, and contour images of Φ317ξ, we take the values of β1=1, β2=4, β3=3, β4=2, y=1, and q=1; the graphs clearly demonstrate the multiperiodic characteristic of Φ317ξ on the domain.Figure 3
The 3D, 2D, and contour images ofΦ317ξ.
(a)(b)(c)Figure4: The 3D, 2D, and contour images of Φ318ξ, we take the values of β1=1, β2=4, β3=3, β4=2, y=1, and q=1; the graphs show the periodic curved parabolic structure of Φ318ξ on the domain.Figure 4
The 3D, 2D, and contour images ofΦ318ξ.
(a)(b)(c)Figure5: The 3D, 2D, and contour images of Φ335ξ by considering the values β1=1, β2=4, β3=3, β4=2, y=1, and q=1; the graphs demonstrate the interactions between one soliton with kink waves of Φ335ξ on the domain.Figure 5
The 3D, 2D, and contour images ofΦ335ξ.
(a)(b)(c)The all above wave profiles solutions of HFSCE keep their velocities, shapes, and amplitudes invariant during the appropriate value of some specific parameters. In Figures1, 3,and 4(a), we can see M-shaped waves train. In Figures 2 and 5(a), we also see the multiple bright and dark peak solitons and also seen their behavior in contour (Figures 1–5(c)) and 2D (Figures 1–5(b)) structures, respectively.


## 6. Conclusion and Future Study

The EMVJEEM was implemented perfectly for the first time in the framework of these techniques to design these new types of soliton solutions for this model. The achieved soliton solutions show the potentially traveling wave solutions to the (2+1)-dimensional HFSCE. The resulting sample of the achieved solutions offers a rich podium to study the nonlinear spin dynamics in magnetic materials. The derived solutions having abundant applications to handle spin dynamics in magnetic materials and transmission of high-frequency waves in tranquil medium. We have successfully used EMVJEEM to construct a rich variety of exact solutions of HFSCE under various family cases in this paper. By choosing suitable best values for the constant parameters, these new complex soliton solutions established the dynamical behaviors through 3D and 2D wave profiles via simulation. When compared with [27, 35], our paper has the largest number solutions. Because there are 42 types of linear independent solutions for 23 different cases in the presented method. Hence, these new exact solutions will play an important role in understanding and investigating the HFSCE.In summary, the results of the full text eloquently prove that the above EMVJEEM is very efficient and powerful in solving the exact solutions of nonlinear evolution equations now and in the future. We can apply EMVJEEM of this research to other nonlinear evolution equations.In the meanwhile, we notice if there is additional noise, the equation becomes a stochastic differential equation (SDE),(98)β1+β2+β3Φ′′−ω+β1l2+β2k2−β3klΦ−β4Φ3−ξ=0.This model, compared to the previous model (8), is more pervasive in scientific computing experiments, for the reason that such noise exists in both complicated nature phenomenon and artificial model errors. Usually, the noise is Gaussian white noise. According to the theory of existence and uniqueness of solution to SDE, one can find solution to the SDE problem (98) under certain conditions [36].Numerical solutions can also be found by various methods, for example, the Runge-Kutta method [37], the Euler method [38, 39], and the Milstein method [40, 41]. Since the Runge-Kutta method is far more complicated than the Euler method and the Milstein method, one usually performs the Euler method and the Milstein method in real applications. One can perform numerical experiments to the SDE and compare the strong and weak convergence, numerical errors, and stability of the numerical methods [42].For the noise perturbed system (8), one can observe the property of energy landscape and the phenomenon of exit from basin of attraction under the framework of the large deviation theory [43]. We will discuss about that in later work. From the data assimilation point of view, when Equation (8) has unobservable state Φ and noisy observations of Φ, one can perform filtering strategy to recover the state Φ, based on a the Bayesian framework [44]. We will also explore this issue in future work.



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