# Solitary Wave Solutions to the Multidimensional Landau–Lifshitz Equation

Authors: Ahmad Neirameh
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

In this paper, we study the different types of new soliton solutions to the Landau–Lifshitz equation with the aid of the auxiliary equation method. Then, we get some special soliton solutions for this equation. Without the Gilbert damping term, we present a travelling wave solution with a finite energy in the initial time. The parameters of the soliton envelope are obtained as a function of the dependent model coefficients.


---

## Body

## 1. Introduction

Nonlinear partial differential equations have different types of equations; one of them is the Landau–Lifshitz equation that is relevant to the classical and quantum mechanics. Nonlinear evolution equations (NEEs) which describe many physical phenomena are often illustrated by nonlinear partial differential equations. So, the exact solutions of NLPDE are explored in detail in order to understand the physical structure of natural phenomena that are described by such equations. A variety of powerful methods have been used to study the nonlinear evolution equations, for the analytic and numerical solutions. Some of these methods, the Riccati Equation method [1], Hirota’s bilinear operators [2], exponential rational function method [3], the Jacobi elliptic function expansion [4], the homogeneous balance method [5], the tanh-function expansion [6], first integral method [7, 8], the subequation method [9], the exp-function method [10], the Backlund transformation, and similarity reduction [11–29], are used to obtain the exact solutions of NLPDE.In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the processional motion of magnetizationM in a solid. It is a modification by Gilbert of the original equation of LLG equation (see [30]) can be written down as
(1)∂∂tS=αSΛΔS−βSΛSΛΔS,here, S=S1t,x→,S2t,x→,S3t,x→∈S2↪R3,α≥0,α2+β2=1, Λ denotes the cross product. The term multiplying with α represents the exchange interaction, while the β term denotes the Gilbert damping term. Especially, two extreme cases of (1) (β=0 and α=0, respectively) include as special cases the well-known Schrödinger map equation and harmonic map heat flow, respectively. The well-posedness problem of the LL(G) equation are intensively studied in mathematics, to list a few, in 1986, of the weak solution of the LL(G) equation. Under the small initial value, the global existence of the solution in different spaces [31–33] was proved. The first progress on the existence of partially regular solutions to the LLG equation was found [30, 33–36]. Even for the small initial data, the exact form of the solution is still unknown. On the other hand, whether the LLG equation admits a global solution will develop a finite time singularity from the large initial data is an open equation.


## 2. Auxiliary Equation Method

Let us consider a typical nonlinear PDE forq=qx,t, given by
(2)Uqx,qt,⋯.=0.Under the wave transformations ofqx,t=Qξ and ξ=σx−lt, Equation (2) becomes an ordinary differential equation given by
(3)UQ,σQ′,−lQ′,⋯.=0.We assume that the solution ϕξ of the nonlinear Equation (18) can be presented as
(4)Qξ=∑i=0MAiϒiξ,in which Aii=0,⋯,n are all real constants to be determined, the balancing number M is a positive integer which can be determined by balancing the highest order derivative terms with the highest power nonlinear terms in Equation (2) and ϒiξ expresses the solutions of the following auxiliary ordinary differential equation:
(5)dϒdξ2=s1ϒ2ξ+s2ϒ3ξ+s3ϒ4ξ,where s1,s2,ands3 are real parameters. Equations (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) with Δ=s22−4s1s3 give the following solutions:Case 1.
Fors1>0,(6)−s1s2sech2s1/2ξs22−s1s31+ξtanhs1/2ξ.Case 2.
Fors1>0,(7)−s1s2csch2s1/2ξs22−s1s31+ξtanhs1/2ξ.Case 3.
Fors1>0,Δ>0,
(8)2s1sechs1ξεΔ−s2sechs1ξ.Case 4.
Fors1<0,Δ>0,
(9)2s1sec−s1ξεΔ−s2sec−s1ξ.Case 5.
Fors1>0,Δ<0,
(10)2s1cschs1ξε−Δ−s2cschs1ξ.Case 6.
Fors1<0,Δ>0,
(11)2s1csc−s1ξεΔ−s2csc−s1ξ.Case 7.
Fors1>0,s3>0(12)−s1sech2s1/2ξs2+2ξs1s3tanhs1/2ξ.Case 8.
Fors1<0,s3>0,
(13)−s1sec2−s1/2ξs2+2ξ−s1s3tan−s1/2ξ.Case 9.
Fors1>0,s3>0(14)s1csch2s1/2ξs2+2ξs1s3coths1/2ξ.Case 10.
Fors1<0,s3>0,
(15)−s1csc2−s1/2ξs2+2ξ−s1s3cot−s1/2ξ.Case 11.
Fors1>0,Δ=0,
(16)−s1s21+ξtanhs12ξ.Case 12.
Fors1>0,Δ=0,
(17)−s1s21+ξcoths12ξ.Case 13.
Fors1>0,(18)4s1eεs1ξeεs1ξ−s22−4s1s2.Case 14.
Fors1>0,s2=0,
(19)±4s1εeεs1ξ1−4s1s3e2εs1ξ.Stage 2: substituting Equations (4) and (5) into Equation (3) and collecting all terms with the same order of ϒj together, we convert the left-hand side of Equation (3) into a polynomial in ϒj. Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for A0,A1,andA2. By solving these algebraic equations, we obtain several cases of variables solutions [15, 37]Stage 3: by substituting the obtained solutions in stage 2 into Equation (4) along with general solutions of Equations (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), it finally generates new exact solutions for the nonlinear PDE (1)


## 3. Travelling Wave Solutions of the LL Equation via Aem

In this section, we construct a travelling wave solution without the Gilbert term [30]. Under the condition of the auxiliary equation method, we consider the following wave transformations:
(20)Wc,wt,r¯=e−iwtϕr¯−cteiψr¯−ct,where c and w are constants undetermined.Here, we assume−c2+4w>0. Substitute (5) into (1) [30], the separate real and the virtual parts, respectively, as
(21)ϕξw−2ξ2+cψ′ξ−ψ′ξ2+ϕξ3w+cψ′ξ+ψ′ξ2+ϕξ2ϕ′′ξ=0,(22)ϕ′ξ−c+2ψ′ξ−ϕξ2ϕ′ξc+2ψ′ξ+ϕξψ′′ξ+ϕξ3ψ′′ξ=0,where ξ=r¯−ct. (21) and (22) are the nonlinear constant coefficients of ordinary differential equation system with the variable ξ. According to (22), we can obtain a relationship between ψ and ϕ:
(23)ψ′ξ=1+ϕξ2−c+2C1+2C1ξ22ϕξ2,where C1 is the arbitrary constant. If we set C1=0, we have
(24)ψ′ξ=−c1+ϕξ22ϕξ2.Substituting (24) into (21), we get
(25)c2+3c2ϕξ2+c2−4wϕξ6+ϕξ23c2−4w+8ϕ′ξ2−4ϕξ31+ϕξ2ϕ′′ξ=0.


## 4. Results

By the auxiliary equation method, the solution of (25) is assumed as
(26)ϕξ=A1ϒξ+A0.From (5), we have
(27)ϒ′2=s1ϒ2ξ+s2ϒ3ξ+s3ϒ4ξ,ϒ″=12s1ϒξ+3s2ϒ2ξ+4s3ϒ3ξ,where A1 and A0 are constants. Substituting (26) into (25) along with (27) and comparing the coefficients of alike powers of ϒξ provides an algebraic system of equations, and solving this set of algebraic equations by used of the Maple, we obtain several case solutions. For example is as follows.Set 1:(28)A1=−203s33s1−12s2240s3−4w,A0=3s1−12s2240s3−4w,s2=0,c=1301025s33s12+−3120w−10800s3s33s1+144w+90s32+150s1−10800s3+480w.From set 1 and Eq. (26), we have
(29)ϕξ=−203s33s1240s3−4wϒξ+3s1240s3−4w.So, solutions of Equation (5) are obtained in (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), we have final solutions of Equation (1) and Equation (25) as follows:
(30)ϕ1r¯,t=−203s33s1240s3−4w×−s1csc2−s1/2r¯−ct2ε−s1s3cot−s1/2r¯−ct+3s1−12s2240s3−4w.In Figure1, the graphical behavior of solutions ϕ1 for s1=−1,s3=1,andw=1 in different values of r¯ and t is illustrated.
(31)ϕ2r¯,t=−203s33s1240s3−4w×2s1sec−s1r¯−ctε−4s1s3+3s1−12s2240s3−4w.Figure 1
Graphical behavior forϕ1r¯,t, for r¯,t=−π..π,s1=−1,s3=1,andw=1.In Figure2, the graphical behavior of solutions ϕ2 for s1=−1,s3=1,andw=1 in different values of r¯ and t is illustrated.
(32)ϕ3r¯,t=−203s33s1240s3−4w×s1csch2s1/2r¯−ct2εs1s3coths1/2r¯−ct+3s1240s3−4w.Figure 2
Graphical behavior forϕ2r¯,t, for r¯,t=−π..π,s1=−1,s3=1,andw=1.In Figure3, the graphical behavior of solutions ϕ3 for s1=1,s3=1,andw=1 in different values of r¯ and t is illustrated.
(33)ϕ4r¯,t=−203s33s1240s3−4w×4s1eεs1r¯−cteεs1r¯−ct2+3s1240s3−4w,ϕ5r¯,t=−203s33s1240s3−4w×±4s1εeεs1r¯−ct1−4s1s3e2εs1r¯−ct+3s1240s3−4w,(34)Wc,wt,r¯=e−iwtϕr¯−cteiψr¯−ct,ψ′ξ=−c1+ϕξ22ϕξ2soψξ=−∫c1+ϕξ22ϕξ2dξ,c=1301025s33s12+−3120w−10800s3s33s1+144w+90s32+150s1−10800s3+480w.Figure 3
Graphical behavior forϕ3r¯,t, for r¯,t=−π..π,s1=1,s3=1,andw=1.Set 2:(35)A1=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24,A0=−115s1s2,s3=0,c=415s1s2+4w/6−115s1s2.From set 2 and Eq. (26), we have
(36)ϕξ=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24ϒξ−115s1s2.So, solutions of Equation (5) are obtained in (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), we have final solutions of Equation (1) and Equation (25) as follows:
(37)ϕ6r¯,t=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24×−s1s2sech2s1/2r¯−cts22−115s1s2.In Figure4, the graphical behavior of solutions ϕ6 for s1=1,s2=1 in different values of r¯ and t is illustrated.
(38)ϕ7r¯,t=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24×−s1s2csch2s1/2r¯−cts22−115s1s2.Figure 4
Graphical behavior forϕ6r¯,t, for r¯,t=−π..π,s1=1,ands2=1.In Figure5, the graphical behavior of solutions ϕ7 for s1=1,s3=1 in different values of r¯ and t is illustrated.
(39)ϕ8r¯,t=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24×2s1sechs1r¯−ctεs2−s2sechs1r¯−ct−115s1s2,ϕ9r¯,t=−s12s12+225s22−90s2+s112w2s14−75s13s2+118125s24×4s1eεs1r¯−cteεs1r¯−ct−s22−4s1s2−115s1s2,(40)Wc,wt,r¯=e−iwtϕr¯−cteiψr¯−ct,ψ′ξ=−c1+ϕξ22ϕξ2soψξ=−∫c1+ϕξ22ϕξ2dξ,c=415s1s2+4w/6−115s1s2.Figure 5
Graphical behavior forϕ7r¯,t, for r¯,t=−π..π,s1=1,ands2=1For more convenience and understanding, the graphical behavior of the answers is considered (see Figures1–5).


## 5. Conclusions

This paper derived new optical soliton solutions of the Landau–Lifshitz equation, which describe the propagation of ultrashort pulses in nonlinear optical fibers by using the auxiliary equation method. We boldly say that the work here is valuable and may be beneficial for studying in other nonlinear science. The exact solutions obtained from the model equations provide important insight into the dynamics of solitary waves. The solutions obtained in this paper have not been reported in the old research.



---

*Source: 5538516-2021-03-31.xml*