# New Traveling Wave Solutions for the (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation

Authors: Tianyong Han; Jiajin Wen; Zhao Li; Jun Yuan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The principal objective of this article is to construct new exact soliton solutions of the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation which investigates the nonlinear dynamics of magnets. Through using the complete discrimination system method, the traveling wave solutions are obtained. As a result, we get the traveling wave solutions of the Heisenberg ferromagnetic spin chain equation, which include rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, inverse trigonometric function, logarithmic function. Some graphical representations of the problems and that of comparison are also provided.


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## Body

## 1. Introduction

In this paper, we are devoted to studying the traveling wave solutions for (2 + 1)-dimensional nonlinear equation describing nonlinear wave propagation in Heisenberg ferromagnetic spin chain equation(HFSC) which was derived by Latha and Christal Vasanthi [1],(1)iut+α1uxx+α2uyy+α3uxy−α4u2u=0,where α1=γ4J+J2, α3=2γ4J2, and α4=2γ4Ω. In (1), ux,y,t is the complex function of the normalized spatial variables x and y and temporal variable t and represents the appropriate continuum approximation of the coherent magnetism amplitude to the bosonic operators at spin-lattice sites; γ is lattice parameter; J and J1 correspond to the coefficients of bilinear exchange interactions along the x and y directions, respectively; J2 refer to the neighboring interaction along the diagonal; and Ω is the uniaxial crystal field anisotropy parameter.Traveling wave solution is an important research content of nonlinear partial differential equations. It is difficult to find the traveling wave solution of nonlinear partial differential equations (PDE), but in recent years, many methods of spherical traveling wave solution have been studied. A large number of reports show that researchers have obtained traveling wave solutions of partial differential equations involving many fields, such as nuclear physics, chemical reaction, signal processing, optical fiber, hydrodynamics, plasma, nonlinear optics, and ecology. Traveling wave solutions play an important role in revealing the properties of these nonlinear partial differential equations and predicting the trend of these phenomena [2–21].Heisenberg ferromagnetic spin chain equation is used to describe the nonlinear wave propagation in a ferromagnetic spin chain system. It is also a generalization of (2 + 1)-dimensional nonlinear Schrödinger equation. With the development of technology, the transistor size is reduced to the nanometer scale. How to develop electronic components with higher density and faster storage speed is the practical reason why the HFSC equation is valued [22–25].Since the traveling wave solution is helpful to explain many physical properties of magnetic materials, many scholars are attracted to the study of the traveling wave solution of the HSFC equation. In reference [1], equation (1) was derived by using the coherent state ansatz connected to a Holstein–Primakoff bosonic representation of spin operators. Then, the multisoliton solution is constructed by the Darboux transform, and the modulation instability is discussed. Three kinds of traveling wave solutions of equation (1) are obtained by using traveling wave transformation [26]. The bilinear form and dark soliton solution of equation (1) are derived by the auxiliary function method, and the soliton interaction is studied. There are two types of elastic and inelastic collisions between these solitons [27]. In references [28, 29], dark multiple solitons are derived, and then, the propagation, interaction, and linear stability analysis of solitons are discussed, respectively. Some complex solutions were obtained by the auxiliary ordinary differential equation method in [30]. In [31], a series of new solutions are constructed by the improved F-expansion method combined with the Jacobian ellipse method. They are exported to understand the constraints that exist. These solutions include periodic wave solutions, double periodic wave solutions, dark soliton solutions, and bright soliton solutions. Lax pair and generalized Darboux transform are used in equation (1) to construct a class of n-order strange waves [32]. In reference [33], the bifurcation of the solution of equation (1) and some traveling wave solutions are obtained by using the dynamic system method. In [34], the authors find the soliton solutions for equation (1) by considering the Bäcklund transformation. By using the Hirota bilinear method, the one-order rogue waves solutions are obtained in [35], and the interaction behaviors between breather and rogue wave are studied in [36]. By applying the bilinear method, the lump wave solution is constructed in [37]. Recently, Li studied a (2 + 1)-dimensional nonlinear ferromagnetic spin chain system with variable coefficients in [38] and obtained the breather and rogue wave by using the algebraic iteration method.The scholars mentioned above have obtained different traveling wave solutions by different transformations, so they have obtained the properties of HFSC from different directions. Although many traveling wave solutions of the HFSC equation have been obtained in the above literature, there are few studies on the classification of all possible traveling wave solutions. Driven by the above reasons, we will use the complete discrimination system method to find more traveling wave solutions of HFSC to enrich the research results of HFSC.The rest of this continuing article is methodized as follows: in Section2, we propound the formation of the polynomial complete discrimination system. In Section 3, we implement this technique to find traveling wave solutions to the HFSC equation. In Section 4, with the help of Maple, the graphical illustration of the modulus of the traveling wave solutions is described by using 2-dimensional and 3-dimensional plots. Finally, some conclusions are given in Section 5.


## 2. Overview of the Complete Discrimination System

To show the basic idea of our method, consider the following nonlinear differential equation:(2)Pu,ux,uy,ut,uxx,uxt,uyt,⋯=0,where u is an unknown function and P is a polynomial of u and its partial fractional derivatives. Using the traveling wave transformation,(3)ux,y,t=Uξ,ξ=η1x+η2y−ct,where c is a nonzero velocity of the traveling wave in (3). We get an ordinary differential equation of the polynomial form(4)Qu,u′,u″,⋯=0,where Q is a polynomial in u and its derivatives and notation ′ is the derivative with respect to ξ. Equation (4) can be written as(5)u′ξ2=Gu,θ1,θ2,…,θm,where θ1,θ2,…,θm are parameters. Then, integrating the above formula once, we have(6)±ξ−ξ0=∫1Gu,θ1,θ2,…,θmdu,where Gu is a polynomial function.According to the complete discrimination system forGu, the roots of Gu can be classified, and the detailed classification will be given in Section 3.


## 3. Traveling Wave Solution of the (2 + 1)-Dimensional HFSC

To find the traveling wave solutions of equation (1), we assume that(7)ux,y,t=φξeiρx,y,t,where(8)ξ=η1x+η2y−ct,ρx,y,t=k1x+k2y−μt,where ξ is the traveling coordinate, φξ is the real amplitude function to be determined, and ρx,y,t is the phase of the envelope. The parameters η1andη2 represent the wavenumbers in the x and y directions, respectively, c is the wave velocity, and μ is the frequency of the pulse.Utilizing the wave transformation (8) into equation (1), separating the real part and the imaginary part, respectively, we obtain that(9)μ−α1k12−α2k22−α3k1k2φ+α1η12+α2η22+α3η1η2φξξ−α4φ3=0,and(10)2α1k1η1+2α2k2η2+α3k1η2+k2η1−cφξ=0.Letting(11)A=μ−α1k12−α2k22−α3k1k2,B=α1η12+α2η22+α3η1η2,C=2α1k1η1+2α2k2η2+α3k1η2+k2η1−c.It follows(12)Cφξ=0,(13)Aφ+Bφξξ−α4φ3=0.We can see that systems (12) and (13) will be compatible in the case C=0. We always assume that B≠0, otherwise equation (13) is not a differential equation. Multiply both sides of equation (13) by φ′ξ and integrate once, we get(14)φξ2=α42Bφ4−ABφ2+C0,where C0 is the integration constant.Leta4=α4/2B,a2=−A/B,a0=C0. Equation (14) can be written as(15)φξ2=a4φ4−a2φ2+a0.Taking(16)φξ=±4a4−1/3wξ1,b1=4a24a4−2/3,b0=4a04a4−1/3,ξ1=4a41/3ξ.Equation (15) can be written as follows:(17)wξ12=ww2+b1w+b0.Equation (17) becomes the following integration form:(18)±ξ1−ξ0=∫dwwFw,where Fw=w2+b1w+b0, and noting Δ=b12−4b0.Case 1.
Δ=0. Since w>0, we obtain that(19)±ξ1−ξ0=∫dww+b1/2w.
Ifb1<0, the solution of equation (19) is(20)±ξ1−ξ0=2−b1ln2w−−b12w+−b1.
The traveling wave solutions of equation (17) are(21)w1ξ1=−b12tanh212−b12ξ1−ξ0,(22)w2ξ1=−b12coth212−b12ξ1−ξ0.
Ifb1>0, the traveling wave solution of (17) is(23)w3ξ1=b12tan212b12ξ1−ξ0.
Ifb1=0, the traveling wave solution of equation (17) is(24)w4ξ1=1ξ1−ξ02.
That is to say, whenΔ=0, we get the following solutions of equation (17): solution (21) is the solitary wave solution, solution (22) is the hyperbolic function solution, solution (23) is the trigonometric function solution, and solution (23) is the rational function solution of the equation.Case 2.
Δ>0,b0=0. Since w>−b1, equation (19) can be written as(25)±ξ1−ξ0=∫dwww+b1.
Ifb1>0, then, equation (25) is(26)±ξ1−ξ0=2b1ln2w+b1−p12w+b1+p1.
The solutions of equation (17) are(27)w5ξ1=b12tanh212b12ξ1−ξ0−b1,(28)w6ξ1=b12coth212b12ξ1−ξ0−b1.
Ifb1<0, then, the solution of equation (25) is(29)±ξ1−ξ0=−2−2b1arc tan2w+b1−b1.
So, the solution of (17) is(30)w7ξ1=−b12tan212−b12ξ1−ξ0−b1.Case 3.
Δ>0,b0≠0. Suppose α1<α2<α3, and one of them is zero, and the other two are the roots of Fw.
Ifα1<w<α2, take w=α1α2−α1sin2θ , then, equation (18) can be rewritten as(31)±ξ1−ξ0=2α3−α1∫dθ1−m2sin2θ.
Here,m2=α2−α1/α3−α1. According to the definition of Jacobian elliptic function sn, we obtain the solution of equation (17) in the following form:(32)w8ξ1=α1+α2−α1sn212α3−α1ξ1−ξ0,m.
Ifw>α3, take w=α3−α2sin2θ/cos2θ, the solution of (17) can be constructed as follows:(33)w9ξ1=−α2sn1/2α3−α1ξ1−ξ0,m+α3cn1/2α3−α1ξ1−ξ0,m.
Because sn and cn are periodic functions, we get two periodic solutions. Note thatlimk⟶1snx,k=tanhx, limk⟶1cnx,k=sechx. When α2⟶α3, m⟶1. Then,(34)w8ξ1=α1+α2−α1tanh212α3−α1ξ1−ξ0,w9ξ1=−α2tanh1/2α3−α1ξ1−ξ0+α3sech1/2α3−α1ξ1−ξ0.Case 4.
Δ<0. Since w>0, take the transformation(35)w=b0tan2θ2.
By (18) and (32),(36)±ξ1−ξ0=b0−1/4∫dθ1−k2sin2θ,where k2=1/21−b1/2b0. From the definition of Jacobian elliptic function cn, we obtain(37)cnb0−1/4ξ1−ξ0,k=cosθ.
From equation (34), we get(38)cosθ=2b0w+b0−1.
By using equations (36) and (37), the traveling wave solution of equation (17) can be derived in the following form:(39)w10ξ1=2b01+cnb0−1/4ξ1−ξ0,k−b0.
Now, we get another periodic solution of (17).
With the help of (7) and (16), we get the classification of all single traveling wave solutions of (1) as follows:(40)u1x,y,t=±Aa4tanh2−5/3A21/3a4/B1/3η1x+η2y−ct−ξ0B1/3a42/3×expik1x+k2y−μt,u2x,y,t=±Aa4coth2−5/3A21/3a4/B1/3η1x+η2y−ct−ξ0B1/3a42/3×expik1x+k2y−μt,u3x,y,t=±Aa4tanA325Ba421/62aB1/3η1x+η2y−ct−ξ0×expik1x+k2y−μt,u4x,y,t=±B2a41/6expik1x+k2y−μt2a4/B1/3η1x+η2y−ct−ξ0,u5x,y,t=±Aa4−2+tan22−5/6A1/2Ba421/32a4B1/3ct−η1x−η2y+ξ0×expik1x+k2y−μt,u6x,y,t=±Aa4−2+coth22−5/6A1/2Ba421/32a4B1/3ct−η1x−η2y+ξ0×expik1x+k2y−μt,u7x,y,t=±−Aa42+tan2122a42B1/32a4B1/3ct−η1x−η2y+ξ0×expik1x+k2y−μt,u8x,y,t=±a42B1/6a1+a2−a1sn2a3−a122a4B1/3η1x+η2y−ct+ξ0a2−a1a3−a1×expik1x+k2y−μt,u9x,y,t=2a4B−1/6−a2sn1/2a3−a12a4/B1/3η1x+η2y−ct+ξ0a2−a1/a3−a1+a3cn1/2a3−a12a4/B1/3η1x+η2y−ct−ξ0a2−a1/a3−a1×expik1x+k2y−μt,u10x,y,t=32BC03a41/121−cn2a4/B1/3ct−η1x−η2y+ξ0/21/3C01/4B/a41/12,2/21−2−2/3A/C0Ba41+cn2a4/B1/3ct−η1x−η2y+ξ0/21/3C01/4B/a41/12,2/21−2−2/3A/C0Ba41/2×expik1x+k2y−μt.


## 4. Numerical Simulation

This section contains the 2-dimensional and 3-dimensional solution graph of some of the obtained traveling wave solutions of the HFSC equation. Here, the numerical simulation has been performed in Figures1–5 for showing the nature of the obtained solution. The plots of the modulus of u1x,y,t,u3x,y,t,u4x,y,t,u8x,y,t,andu9x,y,t, by choosing suitable parameters are shown in Figures 1–4 and Figure 5.Figure 1
The plots of the modulus ofu1x,y,t with A=25/6,B=−2,α4=1/2,C0=−1,η1=1,η2=2,c=1,andξ0=0. (a) The modulus of u1 when t = 1. (b) The modulus of u1 at y = 0.
(a)(b)Figure 2
The plots of the modulus ofu3x,y,t by choosing suitable parameters: A=25/6,B=−2,α4=1/2,C0=1,η1=1,η2=2;c=1,andξ0=−3. (a) The modulus of u3 when t = 3. (b) The modulus of u3 at y = t = 0.
(a)(b)Figure 3
The plots of the modulus ofu4x,y,t with A=0,B=2,α4=1/2,C0=0,η1=1,η2=2,c=1,andξ0=0. (a) The modulus of u4 when t = 3. (b) The modulus of u4 at y = t = 0.
(a)(b)Figure 4
The plots of the modulus ofu8x,y,t with A=−3,B=1,α4=2,C0=1,η1=1,η2=2,c=1,andξ0=0. (a) The modulus of u8 when t = 3. (b) The modulus of u8 at y = 0, t = 2.
(a)(b)Figure 5
The plots of the modulus ofu9x,y,t with A=−3,B=1,α4=2,γ=3,C0=1,η1=1,η2=2,c=1,andξ0=0. (a) The modulus of u9 when t = 3. (b) The modulus of u9 at y = 0, t = 2.
(a)(b)


## 5. Concluding Remarks

In this paper, the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation has been investigated via the complete discriminant system method. A range of new traveling wave solutions is obtained, such as periodic solutions, rational wave solutions, Jacobi elliptic solutions, triangular functions solutions, and hyperbolic function solutions. By selecting appropriate parameters, some representative solutions are drawn. These solutions may help us to explore new phenomena which appear in equation (1). This paper gives a new idea to study the dispersive traveling wave solutions of the Heisenberg ferromagnetic spin chain equation. In addition, these results are also helpful to understand the dynamics of nonlinear waves in optics, hydrodynamics and magnetic materials. This method can solve the single wave solutions of more types of PDE, such as fractional or random terms.



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