Gauge equivalent structure and solitary wave solution for a modified Landau–Lifshitz equation

Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50

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Gauge equivalent structure and solitary wave solution for a modified Landau–Lifshitz equation LiYuan Ma, ZuoNong Zhu∗ Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, PR China

a r t i c l e

i n f o

Article history: Received 2 December 2015 Accepted 8 April 2016 Available online 19 April 2016 Keywords: Gauge equivalence Given curvature condition Modified Landau–Lifshitz equation Perturbed defocusing nonlinear Schrödinger equation

a b s t r a c t In this paper, the gauge equivalence between the modified Landau–Lifshitz equation and the perturbed defocusing nonlinear Schrödinger equation is proved from the perspective of geometry of given curvature condition. By using the gauge equivalence and the soliton perturbation theory of defocusing nonlinear Schrödinger equation, the first-order approximate 1-soliton solution to the modified Landau–Lifshitz equation is given. © 2016 Elsevier B.V. All rights reserved.

1. Introduction It is well known that nonlinear dynamics of classical and modified Heisenberg spin chains presents a crucial class of problems in condensed matter physics. As generalizations of the classical and modified Heisenberg ferromagnet model, the Landau–Lifshitz (LL) equation and modified Landau–Lifshitz (mLL) equation are discussed in a series of papers [1–9]. Those models describe nonlinear spin waves in the localized ferromagnet. By modifying the effective magnetic field of Landau– Lifshitz equation, Visintin represented in [4] a modified Landau–Lifshitz equation

∂M = λ1 M × HE − λ2 M × (M × HE ), ∂t

(1.1)

where the modified effective magnetic field HE is defined as HE = He − ηα ∂∂M . It can be used to construct a new vectorial t Preisach-type model of ferromagnetic hysteresis. The dynamical behaviors including soliton solution, spin-wave excitations and domain-wall resonance of different mLL equations have been discussed [4,9]. In this paper, we investigate the following modified Landau–Lifshitz equation

˙ xx + Ju×u ˙ , ut = u×u

(1.2)

with magnetization vector u = (u1 , u2 , u3 ) ∈ H 2 ⊂ R2,1 satisfying u21 + u22 − u23 = −1 and the diagonal matrix J = ˙ = ( a2 b3 − diag(J1 , J2 , J3 ), J1  J2  J3 . ×˙ denotes the pseudo cross product in R2,1 defined by the Minkowski metric a×b a3 b2 , a3 b1 − a1 b3 , −(a1 b2 − a2 b1 )). We will address its gauge equivalent structure. As an application, we will construct its first-order approximate 1-soliton solution. The work presented in this paper is motivated by the following facts:

∗

Corresponding author. Tel.: +86 13611688394. E-mail addresses: [email protected] (L. Ma), [email protected] (Z. Zhu).

http://dx.doi.org/10.1016/j.cnsns.2016.04.019 1007-5704/© 2016 Elsevier B.V. All rights reserved.

L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50

45

The concept of gauge equivalence between completely integrable equations plays an important role in the study of soliton [10]. The papers [10,11] showed the gauge equivalence between the continuous isotropic Heisenberg ferromagnet model(HF): ut = u × uxx , where u = (u1 , u2 , u3 ) ∈ S2 ⊂ R3 , and the focusing nonlinear Schrödinger equation (NLS+ ): iqt + qxx + 2|q|2 q = ˙ xx is gauge equivalent to the defocusing 0. It is also proved that the modified Heisenberg ferromagnet model(mHF): ut = u×u nonlinear Schrödinger equation (NLS− ): iqt + qxx − 2|q|2 q = 0. Although the dynamical properties of NLS+ and NLS− are different, it has been shown in [12] that there is a unified geometric interpretation of NLS: iqt + qxx + 2k|q|2 q = 0 for k = −1, 0, 1, i.e., they are exactly the Schrödinger flow of maps from R1 to H2 , C and S2 respectively. As Ref.[12] shown, the definition of Schrödinger flow of maps from R1 to a symplectic manifold (N, ω) is given by the Hamiltonian system of the energy functional E(u) on X: ut = J (u )∇ E (u ) that in a local chart can be written as ut = J (u )τ (u ), where τ (u) is tension field of map u and J is a complex structure on N. For NLS− , J = u×˙ : Tu H 2 → Tu H 2 is a complex structure on H2 compatible with the standard symplectic structure ω = du1 ∧ du2 /u3 on H2 and the tension field of a map u : R1 → H 2 ⊂ R2,1 is given by τ (u ) = uxx + |ux |2 u. Therefore the Schrödinger flow becomes mHF. In addition, the concept of gauge equivalence is also applied to the discrete nonlinear integrable systems and nonintegrable systems [13–18]. Gauge equivalence between (2+1)dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrödinger-type equations was studied in [19]. By introducing the conception of given curvature formulation and the corresponding gauge transformation, Ding and Zhu in [20] have proved that LL equation

ut = u × uxx + Ju × u,

(1.3)

with magnetization vector u = (u1 , u2 , u3 ) ∈ S2 ⊂ R3 and the diagonal matrix J = diag(J1 , J2 , J3 ), J1  J2  J3 characterizing the interaction anisotropy, is gauge equivalent to a perturbed nonlinear Schrödinger+ equation

iqt + qxx + 2|q|2 q = iH[q],

(1.4)

where H[q] is a given function of q. As an application, they also derive the explicit first-order approximate 1-soliton solutions through soliton perturbation theory of the nonlinear Schrödinger+ equation. Their work gives out an answer to the question of Bordag and Yanovski in [21] that “... it is natural, then, to ask whether the Landau–Lifshitz equation is gauge equivalent to some (nonlinear) Schrödinger-like equation”. Thus, it is natural to ask whether the modified LL equation (1.2) is gauge equivalent to some nonlinear Schrödinger-like equation. In this paper, following the ideas in [20], we will prove that mLL equation (1.2) is gauge equivalent to the following perturbed defocusing nonlinear Schrödinger (pNLS− ) equation:

iqt − qxx + 2|q|2 q = iR[q].

(1.5)

Then, if the interaction anisotropy J1 , J2 and J3 in Eq. (1.2) are suitable small, the first-order approximate 1-soliton solution to the mLL equation is constructed on the basis of the soliton perturbation of the defocusing nonlinear Schrödinger equation. The paper is organized as follows. In Section 2, it is proved that the gauge equivalence of the modified Landau–Lifshitz equation and the perturbed defocusing nonlinear Schrödinger equation via the geometric concepts of given curvature formulation. In Section 3, the first-order approximate 1-soliton solution to the modified Landau–Lifshitz equation is constructed by the gauge transformation and the soliton perturbation of the defocusing nonlinear Schrödinger equation. We give conclusions in the last section. 2. Gauge equivalence between mLL equation and pNLS− equation In this section, our main purpose is to prove the gauge equivalence between the modified Landau–Lifshitz equation and the perturbed defocusing nonlinear Schrödinger equation. Set S = iu3 σ1 + u2 σ2 + u1 σ3 , where S2 = −I and σi (i = 1, 2, 3 ) are Pauli’s matrices



σ1 =

0 1



1 , 0



σ2 =

0 i



−i , 0



σ3 =

1 0



0 . −1

Then mLL equation (1.2) can be rewritten as the following matrix form:

St = with

 1 1 SJ , S , [S, Sxx ] + 2 2

 S=

u1 i ( u3 + u2 )

(2.1)



i ( u3 − u2 ) , −u1

 SJ =

J1 u1 i(J3 u3 + J2 u2 )



i(J3 u3 − J2 u2 ) . −J1 u1

A direct computation shows that Eq. (2.1) is equivalent to holding given curvature condition as follows:

FA = dA − A ∧ A = Kdt ∧ dx,

(2.2)

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L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50

where the connection A and the curvature K satisfy the following conditions respectively:

A = iλSdx + (2λ2 S + iλSSx )dt, K = iλt S + Set

 U=

0 q¯

(2.3a)

 iλ  SJ , S . 2

(2.3b)



q 0

and Q is an su(1, 1)-matrix satisfying the relation:



−Qx + [U, Q ] = −R,

R=

0 R[q]



R[q] , 0

(2.4)

where R[q] is a given function of q, one can check that the following given curvature condition

FA˜ = dA˜ − A˜ ∧ A˜ = K˜ dt ∧ dx,

(2.5)

with

A˜ = (λσ3 + U )dx + (−2iλ2 σ3 − 2iλU + i(U 2 + Ux )σ3 + Q )dt,

(2.6a)

K˜ = λt σ3 + λ[σ3 , Q ],

(2.6b)

yields a

pNLS− :

iqt − qxx + 2|q|2 q = iR[q].

(2.7)

Consider the linear system

t = − (i(U 2 + Ux )σ3 + Q )  − V,

x = − U,

(2.8)

with solution (x, t) ∈ SU(1, 1) being form



=

−C¯ −B¯



−C , B

(2.9)

in which B and C are some functions of q and det = −BC¯ − B¯C = 1. We can see that its zero curvature representation Ut − Vx + [U, V ] = 0 is just the pNLS− (2.7) with any given R[q]. Note that d + A can be considered to a connection on an SU(1, 1) principle bundle over R2 , so for ∀ ∈ C ∞ (R2 , SU (1, 1 )) there exists gauge transformation

A = A˜ −1 + d −1 .

(2.10)

Under this gauge transformation, we have the curvature relation between FA and FA˜

FA = FA˜ −1 .

(2.11)

S  −i σ3 −1 .

(2.12)

Set

Using (2.8), the right of (2.10) is reduced to iλSdx + (2λ2 S + iλSSx )dt, that is, A in the left of (2.10) is the connection of (2.1) given in (2.3a) and substituting (2.9) into (2.12), we obtain

u1 = 2Im(B¯C ),

u2 = |B|2 − |C |2 ,

Besides, from (2.11), we have K = restriction on Q:

u3 = |B|2 + |C |2 .

K˜ −1 ,

(2.13)

where K and K˜ are given by (2.3b) and (2.6b) respectively. This results in a

[SJ , S] = 2[S, Q −1 ].

(2.14)

One can check that the matrix S defined by (2.12) satisfies Eq. (2.1) through the given curvature condition for A. On the other hands, from (2.4) we have

[SJ , S] = 2[S, ∂x−1 ( R −1 )]

(2.15)

that only involves R and hence further get ∂x−1 ( R −1 ) = − 12 SJ + φ S with undetermined function φ . Next, in order to fix accurate expressions for φ and R[q], we take a derivative with respect to (2.12) Sx = i (U σ3 − σ3U ) −1 , i.e.

u1x = 2i(qB¯C¯ − q¯ BC ),

u2x = 2Re(qB¯ 2 + q¯C 2 ),

u3x = 2Re(qB¯ 2 − q¯C 2 ).

(2.16)

L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50

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Using (2.13) and (2.16), through a careful and tedious calculation, we finally get

1 4

φ = − (J1 u21 + J2 u22 − J3 u33 ),

(2.17)

and

iq 2 2 2 [J1 (2Im(B¯C )) + J2 (|B|2 − |C |2 ) − J3 (|B|2 + |C |2 ) ] 2 + i[2J1 B¯C¯ (qB¯C¯ − q¯ BC ) − J2 (B¯ 2 + C¯ 2 )Re(qB¯ 2 + q¯C 2 ) + J3 (B¯ 2 − C¯ 2 )Re(qB¯ 2 − q¯C 2 )].

R[q] = −

(2.18)

In summary, we have the following conclusion: The modified Landau–Lifshitz equation (1.2) is gauge equivalent to the perturbed defocusing nonlinear Schrödinger equation (2.7) with R[q] being given by (2.18) under the action of the gauge transformation (2.10). 3. The approximation soliton solutions to mLL equation In this section, by using the soliton perturbation of the defocusing nonlinear Schrödinger equation, we construct the first-order approximation 1-soliton solution to the mLL equation. Firstly, we suppose that |J1 | < |J3 | and let ε = |J3 |  0. As 2 pointed out in [12] and [22], let q = ve2iρ t with ρ being a positive real constant, the defocusing nonlinear Schrödinger 2 equation ivt − vxx + 2|v| v = 0 becomes iqt − qxx + 2(|q|2 − ρ 2 )q = 0, which has the single-soliton solution

q(x, t ) = e−iβ1 [ρ cos β1 + iρ sin β1 · tanh(ρ sin β1 (x − x1 − 2ρ cos β1 · t ))]  e−iβ1 (λ1 + ik1 tanh θ1 ),

(3.1)

where the parameters β 1 , λ1 and k1 satisfy ζ1 = λ1 + ik1 = ρ eiβ1 , θ1 = k1 (x − x1 − 2λ1 t ), under the non-vanishing boundary conditions



q→

ρ, x → +∞, ρ e−2iβ1 , x → −∞.

With the help of the conclusions in [22], we can obtain the perturbed solution to pNLS− equation:

iqt − qxx + 2(|q|2 − ρ 2 )q = iR[q].

(3.2)

Following the idea of soliton perturbation theory in [22–26], we let tn = expansions:

ε n t, n

= 0, 1, 2, . . . , and introduce the asymptotic

∂t = ∂t0 + ε∂t1 + · · ·,

(3.3)

q = q(0) + ε q(1) + ε 2 q(2) + · · ·,

(3.4)

R[q] = −iε (R(0) [q(0) ] + ε R(1) [q(0) , q(1) ] + · · · ).

(3.5)

with the initial condition q(x, 0 ) = q(0 ) (x, 0 ), q( j ) (x, 0 ) = 0( j  1 ). One can get the zeroth-order approximate solution to the pNLS− equation (3.2)

q(0) (z, t0 ) = e−iβ1 (λ1 + ik1 tanh θ1 ),

(3.6)

where θ1 = k1 (z − zc ) with z = x − x1 − 2λ1 t0 and the parameter zc |t0 =0 = 0 describing the shift of the soliton centre. As Ref. [22] shown, since there exists the perturbation, the soliton parameters depend on the slow-variant time ti , i = 1, 2, . . .. If we consider only the first-order perturbation, then the slow-variant time-dependent relations of soliton parameter are given by

λ1,t1 =

1 2



∞

−∞

dθ1 sech

2

θ1 Im(eiβ1 ε R(0) [q(0) ] ), 

∞

(−2L + 1 )ρερt1 + λ1 k1 εβ1,t1 + 2k31 ε zc,t1 = dθ1 k1 sech θ1 eθ1 Re(eiβ1 ε R(0) [q(0) ] ) −∞    ∞ λ1 2 2 − dθ1 λ1 θ1 sech θ1 + sech θ1 e2θ1 Im(eiβ1 ε R(0) [q(0) ] ), 2

−∞

(3.7)

(3.8)

where 2L → ∞ is the length of the system, and the first-order correction is

q(1) (z, t0 ) = where



C

dζ q(1) (ζ , t0 )φ12 (z, ζ ) +



C

dζ q˜(1) (ζ , t0 )ψ12 (z, ζ ),

(3.9)

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L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50



   −2iβ −2iκ z 1e (ζ − ζ1 + ik1 sech θ1 e−θ1 )2 φ12 (z, ζ ) e φ (z, ζ ) = = , 2 −1 i β − θ 2 1 1 φ2 (z, ζ ) (iρζ (ζ − ζ1 ) − e k1 sech θ1 e ) (ζ − ζ¯1 )2  2    2 iκ z (iρζ −1 (ζ − ζ¯1 ) + e−iβ1 k1 sech θ1 e−θ1 )2 ψ1 (z, ζ ) e ψ (z, ζ ) = = , ψ22 (z, ζ ) (ζ − ζ¯1 )2 (ζ − ζ¯1 − ik1 sech θ1 e−θ1 )2  2 A  T (ζ − ζ¯1 − ik1 sech θ1 e−θ1 )2 φ1 (z, ζ ) −e2iκ z φ (z, ζ )A = = , 2 φ2 (z, ζ ) (ρζ −1 (ζ − ζ¯1 ) − ik1 e−iβ1 sech θ1 e−θ1 )2 (ζ − ζ¯1 )2  2 A  T (ρζ −1 (ζ − ζ1 ) + ik1 eiβ1 sech θ1 e−θ1 )2 e−2iβ1 e−2iκ z ψ1 (z, ζ ) ψ (z, ζ )A = = , ψ22 (z, ζ ) (ζ − ζ1 + ik1 sech θ1 e−θ1 )2 (ζ − ζ¯1 )2 ∞ A −∞ dzφ (z, ζ )R (1 ) q ( ζ , t0 ) = − (1 − exp(4iκ (λ − λ1 )t0 )), 2 8π κ a(ζ ) (1 − ρ 4 ζ −4 ) ∞ A −∞ dzψ (z, ζ )R q˜(1) (ζ , t0 ) = − (1 − exp(4iκ (λ − λ1 )t0 )), 2 8π κ a(ζ ) (1 − ρ 4 ζ −4 )

N N  R(0) [q(0) ] − iqt(10) ζ − ζn iθ R= , a (ζ ) = e 2 , θ = −2 βn , ( 0 ) ¯ −(R¯ (0) [q(0) ] + iq¯t1 ) n=1 ζ − ζn n=1 ζn = ρ eiβn , 0 < βn < π , n = 1, 2, . . ., where φ (z, ζ )A and ψ (z, ζ )A are defined as φ (z, ζ )A = ψ (z, ζ )T (iσ2 ) and ψ (z, ζ )A = φ (z, ζ )T (iσ2 ), and λ = 12 (ζ + ρ 2 ζ −1 ) and κ = 12 (ζ − ρ 2 ζ −1 ) ( The formulae (3.7), (3.8), and (3.9) can be found in Ref. [22]. See [22] for details). Next, we turn to seek first order approximation 1-soliton solutions to the mLL equation. We first note that for the pNLS− equation (3.2), the corresponding linear system (2.8) should be changed to

t = − (i(U 2 + Ux − ρ 2 )σ3 + Q )  − V.

x = − U,

One can deduce that  =

x = U  ,

−1

(3.10)

is a solution to conjugate linear problem:

t = (i(U + Ux − ρ 2 )σ3 + Q ) . 2

(3.11)

Expanding  , Q and S as follows:

 = 0 ( I + ε  1 + · · · ) ,

(3.12)

Q = Q (0) + ε Q (1) + · · ·,

(3.13)

S = − −1 σ3 i = S0 + ε S1 + · · ·.

(3.14)

From Eqs. (3.11)–(3.14), we have

0 x = U ( 0 ) 0 ,

 0t0 = i (U (0) )2 + Ux(0) − ρ 2 σ3 0 ,

(3.15)

   1t0 = 0−1 i U (0)U (1) + U (1)U (0) + Ux(1) σ3 + Q (1) 0 − 0−1 0t1 ,

1x = 0−1U (1) 0 ,

(3.16)

where we have taken Q (0 ) = 0, and





q (i ) , i = 0, 1 0

0 q¯ (i )

U (i ) =

Then, it follows that

S0 = −0−1 σ3 i0 ,

S 1 = S 0  1 − 1 S 0 .

(3.17)

For given q(0) (z, t0 ), solving Eq. (3.15) gives (also see Ref. [12])

⎛

1 1+ 2ρ

1 ⎜  0 = √ ⎜ 2⎝

1+

 

B0 B¯ 0

1 2ρ





1 + sin β1 1+i cos β1



1−i

1 + sin β1 cos β1







(q

(0 )

− ρ)

eρ x



(q¯ (0) − ρ ) eρ x



1 − 1+ 2ρ



1+

1 2ρ





1 − sin β1 1−i cos β1

1+i

1 − sin β1 cos β1

C0 . −C¯0

Therefore the zeroth-order approximate 1-soliton solution to mLL equation is determined by





 (q

(0 )

− ρ ) e −ρ x



(q¯ (0) − ρ ) e−ρ x

⎞ ⎟ ⎟ ⎠

L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50

49

Fig. 1. The graph of zeroth approximate 1-soliton solution u1(0) , u2(0) and u3(0) with parameters chosen by ρ = 4, β1 = 0.3, x1 = 0.4, zc = 0.

 S0 = −0−1 σ3 i0 =



u1(0)

i(u3(0) − u2(0) )

i(u3(0) + u2(0) )

−u1(0)

 =

2Im(B¯ 0C0 ) 2 i|B 0 |2



2i|C0 |2 , −2Im(B¯ 0C0 )

(3.18)

where

u1(0) = u2(0) =

sin β1 (tanh2 θ1 − 1 ), 2 cos β1 2

1 + sin β1 (1 + cos2 β1 − 2 sin β1 tanh θ1 + sin2 β1 tanh2 θ1 )e2ρ x 4 cos2 β1 −

u3(0) =

1 − sin β1 (1 + cos2 β1 + 2 sin β1 tanh θ1 + sin2 β1 tanh2 θ1 )e−2ρ x , 4 cos2 β1

1 + sin β1 (1 + cos2 β1 − 2 sin β1 tanh θ1 + sin2 β1 tanh2 θ1 )e2ρ x 4 cos2 β1 +

1 − sin β1 (1 + cos2 β1 + 2 sin β1 tanh θ1 + sin2 β1 tanh2 θ1 )e−2ρ x . 4 cos2 β1

We remark here that solving matrix equation (3.15) is quite difficult, which is distinguished from the case for LL equation (see Ref. [20]). Fig. 1 gives the shape of the zeroth-order approximate 1-soliton solution u1(0 ) , u2(0 ) and u3(0 ) with the parameters ρ = 4, β1 = 0.3, x1 = 0.4, zc = 0. Moreover, we can get specific expression of R(0) [q(0) ]:

R ( 0 ) [q ( 0 ) ] =

q (0 ) 2 2 2 [c1 (2Im(B¯ 0C0 )) + c2 (|B0 |2 − |C0 |2 ) − (|B0 |2 + |C0 |2 ) ] 2 − [2c1 B¯ 0C¯0 (q(0) B¯ 0C¯0 − q¯ (0) B0C0 ) − c2 (B¯ 20 + C¯02 )Re(q(0) B¯ 20 + q¯ (0)C02 ) + (B¯ 20 − C¯02 )Re(q(0) B¯ 20 − q¯ (0)C02 )]. (3.20)

50

L. Ma, Z. Zhu / Commun Nonlinear Sci Numer Simulat 40 (2016) 44–50 J

J

where c1 = J1 , c2 = J2 and 0 ≤ |c1 |, |c2 | ≤ 1. Substituting R(0) [q(0) ] into (3.9) yields accurate expression of q(1) (z, t0 ). Further 3 3 we obtain S1 = 1 S0 − S0 1 , where  1 is given by



1 = ∂x−1

−q(1) B¯ 0C¯0 − q¯ (1) B0C0 q¯ (1) B2 − q(1) B¯ 2 0

0



−q¯ (1)C02 + q(1)C¯02 . ( q¯ 1) B0C0 + q(1) B¯ 0C¯0 )

(3.21)

We thus have constructed first-order approximate 1-soliton solution S = S0 + ε S1 to the mLL equation. 4. Conclusions In conclusion, we have shown the gauge equivalence between the modified Landau–Lifshitz equation and the perturbed defocusing nonlinear Schrödinger equation from the view of given curvature. Then, based on the soliton perturbation theory for defocusing nonlinear Schrödinger equation, the first order approximation 1-soliton solution to the mLL equation is constructed through gauge transformation between mLL equation and pNLS− equation. Acknowledgments The work of ZNZ is supported by the National Natural Science Foundation of China under grants 11271254 and 11428102, and that of ZNZ in part by the Ministry of Economy and Competitiveness of Spain under contract MTM2012-37070. References [1] Takhtajan LA. Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys Lett A 1977;64(2):235–7. [2] Mikhailov AV. The Landau–Lifschitz equation and the Riemann boundary problem on a torus. Phys Lett A 1982;92(2):51–5. [3] Rodin Yu L. The Riemann boundary problem on a torus and the inverse scattering problem for the Landau–Lifschitz equation. Lett Math Phys 1983;7(1):3–8. [4] Visintin A. Modified Landau–Lifshitz equation for ferromagnetism. Physica B 1997;233:365–9. [5] Li ZD, Liang JQ, Li L. Soliton solution of continuum magnetization equation in a conducting ferromagnet with a spin-polarized current. Phys Rev E 2004;69:066611. [6] Li Z, Zhang S. Domain-wall dynamics and spin-wave excitations with spin-transfer torques. Phys Rev Lett 2004;92:207203. [7] He PB, Xie XC, Liu WM. Domain-wall resonance induced by spin-polarized current in metal thin films with stripe structure. Phys Rev B 2005;72:172411. [8] Li ZD, Liang JQ, Li L, et al. Soliton solution for the spin current in a ferromagnetic nanowire. Phys Rev E 2007;76:026605. [9] Bazaliy Ya B, Jones BA, Zhang SC. Modification of the Landau–Lifshitz equation in the presence of a spin-polarized current in colossal- and giant-magnetoresistive materials. Phys Rev B 1998;57(6):R3213. [10] Zakharov VE, Takhtadzhyan LA. Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theor Math Phys 1979;38:17–23. [11] Lakshmanan M. Continuum spin system as an exactly solvable dynamical system. Phys Lett A 1977;61(1):53–4. [12] Ding Q. A note on the NLS and the Schrödinger flow of maps. Phys Lett A 1998;248:49–56. [13] Ishimori Y. An integrable spin chain. J Phys Soc Jpn 1982;52:3417–18. [14] Porsezian K, Lakshmanan M. Discretised Hirota equation, equivalent spin chain and backlund transformations. Inverse Probl 1989;5:L15–19. [15] Ding Q. On the Gauge equivalent structure of the discrete nonlinear Schrödinger equation. Phys Lett A 20 0 0;266:146–54. [16] Hennig D, Sun NG, Gabriel H, et al. Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations. Phys Rev E 1995;52:255. [17] Ding Q. Chaotic properties between the nonintegrable discrete nonlinear Schrödinger equation and a nonintegrable discrete Heisenberg model. J Phys A: Math Theor 2007;40:1991–2011. [18] Ma LY, Zhu ZN. On nonintegrable semidiscrete hirota equation: gauge equivalent structures and dynamical properties. Phys Rev E 2014;90:033202. [19] Myrzakulov R, Nugmanova GN, Syzdykova RN. Gauge equivalence between (2+1)-dimensional continuous heisenberg ferromagnetic models and nonlinear schrödinger-type equations. J Phys A: Math Gen 1998;31:9535–45. [20] Ding Q, Zhu ZN. The gauge equivalent structure of the Landau–Lifshitz equation and its application. J Phys Soc Jpn 2003;72(1):49–53. [21] Bordag LA, Yanovski AB. Polynomial lax pairs for the chiral o(3)-field equations and the Landau–Lifshitz equation. J Phys A: Math Gen 1995;28:4007–13. [22] Ao SM, Yan JR. A perturbation method for dark solitons based on a complete set of the squared jost solutions. J Phys A: Math Gen 2005;38:2399–413. [23] Herman RL. A direct approach to studying of soliton perturbations. J Phys A: Math Gen 1990;23:2327–62. [24] Yan J, Tang Y, Zhou G. Direct approach to studying of soliton perturbations of nonlinear Schrödinger equation and the sine-Gordon equation. Phys Rev E 1998;58(1):1064–73. [25] Kaup DJ. A perturbation expansion for the Zakharov–Shabat inverse scattering transform. SIAM J Appl Math 1976;31:121–33. [26] Yu HY, Yan JR. Direct approach to study of soliton perturbations of defocusing nonlinear Schrödinger equation. Commun Theor Phys 2004;42:895–8.