Solitons in the stripe domain structure of an easy-axis ferromagnet

Chaos, Solitons and Fractals 127 (2019) 302–311

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Solitons in the stripe domain structure of an easy-axis ferromagnet V.V. Kiselev a,b,∗, A.A. Raskovalov a,b a b

Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences, Sofia Kovalevskaya str., 18, Ekaterinburg, 620108, Russia Institute of Physics and Technology, Ural Federal University, Mira str., 19, Ekaterinburg 620002, Russia

a r t i c l e

i n f o

Article history: Received 12 February 2019 Revised 18 June 2019 Accepted 19 June 2019

Keywords: Solitary domains Domain boundaries Solitons Landau–Lifshitz equation Riemann problem

a b s t r a c t New solutions of the Landau–Lifshitz model have been found and investigated by the “dressing” technique on a torus. They describe solitons strongly associated with the domain structure of an easy-axis ferromagnet. Solitons serve as elementary carriers of macroscopic shifts of the structure and are, under certain conditions, nuclei of the magnetic reversal of a material. It is shown, that the inhomogeneous elliptic precession of magnetization in a soliton core leads to oscillations of the neighboring domain walls of the structure. The connection of the mobility of solitons in the domain structure with the construction of their cores is investigated.

1. Introduction The presence of the periodic domain structures complicates a theoretical description of the solitons and spin waves in magnets. In the bulk ferromagnets with a large constant anisotropy of an easy-axis type a stripe domain structure is energetically more favorable, than a structure with closed domains [1]. However, even in this (the most simple) case an equilibrium state of the medium represents essentially nonlinear configuration - a one-dimensional lattice of domain boundaries, in which the deviations of magnetization from alternative homogeneous states in the neighboring domains are not small (but about the magnetization of saturation) and are localized within a narrow space regions. The basic model describing equilibrium domain structures and their collective excitations is the nonlinear Landau–Lifshitz equation. In the framework of the static Landau–Lifshitz model, the main types of the domain structures in the bulk samples and films were studied (see [2,3]). Exact solutions of the linearized dynamical Landau–Lifshitz equations, describing spin waves in the stripe domain structure of an easy-axis ferromagnet were found [4,5]. The magnetization in the spin waves makes only small oscillations around the equilibrium positions of the domain structure which are not accompanied by shifting the center of the domain walls. To analyze the oscillations of the domain boundaries of the stripe structure, the complete Landau–Lifshitz equations are usually replaced by the more simple ∗

Corresponding author. E-mail address: [email protected] (V.V. Kiselev).

https://doi.org/10.1016/j.chaos.2019.06.026 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

ones [1,6,7]. The domain walls are considered to be infinitely thin and plane. Interactions of the discrete domain walls at small deviations from equilibrium states are described by an effective potential. In a such approach self-localized waves of the longitudinal deformations of the domain structure were studied [8], taking into account a nonlinear interaction between the neighboring domain boundaries. The obtained solitons can be only moving: their velocities exceed the limiting Walker velocity of the single domain wall. The domain structure seems to get rid of such long-periodic modulations. We suppose, that at a theoretical description of quasi-onedimensional solitons in the stripe domain structure of an “easyaxis” ferromagnet in the first appoximation we can neglect the magnetostatic forces in the Landau–Lifshitz equation, since their basic contribution is accounted for by fixing the period of the domain structure. This approximation is better to describe the structure with Bloch domain boundaries, which do not induce the magnetostatic fields. At such approximation the Landau–Lifshitz equation has the form:

  ∂t S = S × ∂x2 S + (n · S )[S × n], S2 = 1,

(1)

where S is the magnetization, n = (0, 0, 1 ) sets an easy-axis anisotropy; x, t are spatial coordinate and time. Here and below, to simplify formulas we use dimensionless variables. In the parametrization S = (sin  cos , sin  sin , cos ) the stripe domain structure corresponds to the following solution of Eq. (1) [4,9]:

 = ϕ0 = const,

(χ ) = θ0 = π /2 − am(χ , k );

χ = x/k;

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

sin θ0 = cn(χ , k ),

cos θ0 = sn(χ , k ),

k ∂x θ0 = ∂χ θ0 = −dn(χ , k ).

(2)

Here am(χ , k) is the Jacobi elliptic amplitude, sn(χ , k), cn(χ , k), dn(χ , k) are the Jacobi elliptic functions with the modulus 0 ≤ k ≤ 1 [10–12]. The value k defines the period 4 Kk of the domain structure; K = K (k ) is the complete elliptic integral of the first kind. The distribution (2) describes domains of width L0 = 2 Kk, within which the magnetization distribution S(0) is almost homogeneous: ϕ0 = const, θ 0 ≈ π n, where n is an integer. Domains are divided by the transition layers – the domain boundaries of thickness l0 = 2K  k/π (K  = K (k ), k = 1 − k2 is the complementary modulus) [13]. In these layers the vector S(0) rotates in such a way, that the neighboring domains have opposite equilibrium directions of magnetization (0, 0, ± 1):



S (0 ) ≈



cos ϕ0 sin ϕ0 , , th ξ , ch ξ ch ξ

ξ = l0−1 (x − L0 p).

The angle ϕ 0 provides the geometry of the reversal of magnetization. In walls of the Bloch (ϕ0 = ±π /2) and Neel (ϕ0 = 0, π ) types the vector S(0) rotates respectively in the plane Oyz or Oxy. Domain walls of the Bloch type do not create magnetic charges and demagnetizing fields (within their divS(0 ) = 0). The model (1) is integrable. As a result, nonlinear waves and solitons against the homogeneous ground state of an easy-axis ferromagnet were studied in detail [14–16]. At the same time, solitons in the stripe domain structure are still difficult to describe because of the essentially nonlinear and inhomogeneous ground state (2) of the medium. The method of finite-gap integration does not give an effective solution of such problems because it leads to little-known multidimensional theta functions and complicated transcendental equations for the parameters of soliton-like states [17–19]. In the work [20] we have proposed a special version of the inverse scattering method for constructing the required solutions of model (1) using the Riemann problem on a torus. We emphasize the principial difference of model (1) from the models that were first integrated using the Riemann problem on a torus [21–23]. In [21–23] solitons were studied on the background of the homogeneous ground state of the medium in models with the Lax representations initially involving a dependence on the spectral parameter through doubly periodic functions. This specific feature leads to the Riemann problems on a torus. In our case, the appearance of the Riemann problem on a torus has a different origin: it is related to the presence of the periodic ground state of the medium. Under shifts on the period in terms of the spectral parameter and/or the space coordinate the Riemann problem functions acquire additional multipliers. As a result, the procedure for solitons constructing becomes more complicated. The solutions of the Riemann problem with zeros give explicit formulas for the solitons in the stripe domain structure in terms of well-studied elliptic functions, permitting detailed analysis. In the works [24,25] we have used this techinque to study motionless solitons in the domain structure. The approach [20] allows also to find integrals of motion, providing the dynamical stability of solitons against the background of the stripe structure. Constructing the integrals of motion for solitons in the domain structure by any other techniques is very embarrassing. In this work we have shown, that the solitons can be both mobile and immobile. We give a complete analysis of changes in the form of the soliton cores, depending on the velocity of their motion and their interactions with the neighboring domain walls of the structure. As well as dislocations in crystals [26], such solitons can serve as elementary carriers of the macroscopic shifts of the domain structure. However, unlike the case of dislocation in crystals, an elementary shift of the domain structure, accompaning the formation and motion of the soliton, does not depend on the pe-

303

riod of the stripe structure, but is determined solely by the construction of the soliton core. The pulsations of the soliton core (due to inhomogeneous precession of magnetization within the core) induce the reciprocal oscillations of the domain walls in the stripe structure, shifted by the core. The properties of solitons in the domain structure of an easy-axis ferromagnet are impossible to analyze without an explaining in sufficient detail the method to integrate the model (1), using the Riemann problem on a torus. Other methods are few effective to study the nontrivial dynamics of the soliton cores and the features of their strong coupling with the domain structure. The article has the following structure. In the Section 2 we present the main relations of the Riemann problem method on a torus which are necessary for constructing and studying new exact solution of the Landau–Lifshitz Eq. (1), describing solitons in the stripe domain structure. The Appendix contain several lengthy formulas excluded from the basic text. In the Sections 3–5 we show, that the dynamical properties of solitons in the domain structure depend on the construction and size of their cores. If the size of the soliton core does not exceed the width of the domain, than the solitons in the domain structure inherit key peculiarities of solitons against the homogeneous ground state. In the case, when the size of the soliton core approaches the width of the domain, the soliton is slowing down. An additional energy of the soliton is localized in the oscillations of the domains neighboring to the core. At last, if the size of the core tends to the period of the domain structure, than the solitons transform into strongly nonlinear long-periodic modulations of the stripe structure, that move with large velocities and represent analogues of the shear solitons [8] in the anharmonic chain of the domain walls. Near the boundaries of the region of existence of a soliton, it is transformed either into low-amplitude spin-wave perturbations, or into surges of the strong extended modulations of the stripe domain structure (see Section 3). The last case is specially studied in the Section 5. We have shown, that instability develops only on the initial time interval, after which the system returns to its original state. In this regard, the wide solitons incorporated into the domain structure are reminiscent of Kuznetsov-Ma, Akhmediev, and Peregrine solitons over unstable states, which are reproduced with excellent precision in laboratory experiments to study extreme modulations in an optical fiber and rogue waves in a water basin [27].

2. The Riemann problem and soliton excitations We have shown, that the formation of solitons is always accompanied by the macroscopic translations of the stripe domains [20]. Therefore, the soliton solutions of the model (1) in the domain structure correspond to the boundary conditions:

S(x, t ) → S2(0) = (sin θ2 cos ϕ0 , sin θ2 sin ϕ0 , cos θ2 ), (0 )

S(x, t ) → S1 = (sin θ1 cos ϕ0 , sin θ1 sin ϕ0 , cos θ1 ),

χ → +∞ , χ → −∞, (3)

where θ j = π /2 − am(χ + j , k ); j = 1, 2; 1 = , 2 = 0. In the further analysis, the structure shift is correlated with zeros of the Riemann problem, which define the form and velocity of solitons. An integration technique for the model (1) is based on the auxiliary linear system (U-V–pair) [28]:

304

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

i

∂x  = − [w1 (S1 σ1 + S2 σ2 ) + w3 S3 σ3 ] ≡ U  , 2 i ∂t  = − w1 ([S × ∂x S]1 σ1 + [S × ∂x S]2 σ2 ) + w3 [S × ∂x S]3 σ3 − 2  − w21 S3 σ3 − w1 w3 (S1 σ1 + S2 σ2 )  ≡ V  . (4) Its compatibility condition is equivalent to Eq. (1). Here σ i are the Pauli matrices (i = 1, 2, 3). The coefficients w1, 3 are interrelated as w21 − w23 = 1, what agrees with the properties of the periodic structure (2) in uniformization: w1 = cn(u, k ), w3 = i sn(u, k ). Here and below snu, cnu, dnu are the Jacobi elliptic functions with the modulus k. The solution S(0) (2) of the nonlinear Landau–Lifshitz Eq. (1) corresponds to the following solution of the linear system (4):



0 (u, χ , t ) = exp −

i ϕ0 2



σ3 M˜ (u, χ ) exp (A(u, χ , t )σ3 );



it cn(u, k ) dn(u, k ), 2k

1 η ζ ( u + iK  ) + ζ ( u − iK  − 2 K ) − 1 ( u − K ) p( u ) = 2i K Z (u, k ) = ; 2i ˜ (u, χ ) = M (u, χ ) diag([σ (−u + K + iK  )]−1 , M

 = {u : Im p(u ) = 0}

= {Reu = 0, K; Imu ≤ 2K  },



1 ( u ) =  2 ( u ) T ( u ) ,

u ∈ .

(8)

In the work [20] it is shown, that the matrix T(u) has the form:



a (u ) b( u )



−b¯ (u ) , a¯ (u )

where a¯ (u ) = a∗ (−u∗ ), b¯ (u ) = b∗ (−u∗ ), a(u )a¯ (u ) + b(u )b¯ (u ) = 1. Let us introduce the matrix functions + ( u ) ≡ (2(1) (u ), 1(2) (u )), − (u ) ≡ (1(1) (u ), 2(2) (u )). Here the no-

tation 1(i,)2 means ith column of the matrix  1,2 (u); i = 1, 2. The matrix function + (u ) can be analytically continued from the contour  to the regions

D+ = {u : Im p(u ) > 0} = {−K < Reu < 0,

[σ (u + K + iK  ]−1 );



Imu ≤ 2 K  },

D− = {u : Im p(u ) < 0}



  σ (χ + u − K + iK  ) eη1 (χ +iK +u) σ ( χ − u + K + iK  ) =  η ( χ +iK  −u )  1 σ ( χ − u − K + iK ) e σ ( χ + u + K + iK )  1/2 η uχ

σ ( iK  ) σ ( iK  + 2 K ) 1 × m(χ ) exp σ3 ; m ( χ ) = . 2K 2 σ ( χ + iK  ) σ ( χ + iK  + 2 K ) (5)

Here σ (u) and ζ (u) are the Weierstrass sigma and zeta functions with the periods [4 K, 2iK  ]; Z(u, k) is the Jacobi zeta function; the transformation properties of the Weierstrass functions are determined by the real parameter η1 = ζ (2 K ) and the imaginary parameter η3 = ζ (iK  ) [10–12]. The word “diag” denotes a diagonal matrix. The branch of the square root m(χ ) is fixed by the condition: m(χ ± 4 K ) = −m(χ ) exp[∓2η1 (χ + iK  + K ± 2 K )]. The function M(u, χ ) is periodic in terms of χ with the period 4K, whereas the function  0 (u, χ ) is quasiperiodic: 0 (u, χ ± 4K, t ) = 0 (u, χ , t ) exp [±4K i p(u )σ3 ]. This implies, that the value p(u) for  0 (u) represents “the Bloch quasi-momentum”. The parametrization in terms of the elliptic functions means that we formulate the Riemann problem not on the complex plane of the spectral parameter, as it was on the homogeneous ground state of the medium, but on the Riemann surface, which is topologically equivalent to a torus. Boundary conditions (3) are associated with the fundamental Jost solutions of the auxiliary linear system (4) with asymptotic conditions:

2 → 2(0) = 0 where



at

χ → −∞;

at

χ → +∞,

 (0) (u, χ , t ) = exp − −

i ϕ0 2

η1 u

2K

(6)

 σ3 M˜ (u, χ + ) exp (A(u, χ , t )σ3 

σ3 .

mod(2K, 4iK  ),

whereas the function − (u ) is analytic in the regions

M (u, χ )

1 → 1(0) =  (0)

mod(2K, 4iK  )

the solutions  1,2 (u) have oscillating behavior. The set  corresponds to the continuous spectrum of the problem (4), (6). For u ∈  the Jost functions  1,2 are interrelated by the transition matrix:

T (u ) =

det 0 = 1; A(u, χ , t ) = i p(u )χ +

On the contour

(7)

= {0 < Reu < K,



Imu ≤ 2 K  },

mod(2K, 4iK  ).

Let us extract from the functions  ± (u) the multipliers (0 ) 

(u ), that contain all their essential peculiarities on the con/2 tour  :

± (u ) = W± (u )  (0/)2 (u ).

(9)

Then, we can reduce the problem of integrating the auxiliary linear system (4) to the Riemann problem, which is formulated as follows: we should construct two functions W+ (u ) and W− (u ), analytical in the respective regions D+ and D− , that satisfy the conjugacy condition on the contour  :

W− (u ) = W+ (u ) G(u ), G (u ) =

  (0/)2 (u ) 1 b( u ) a¯ (u )



b¯ (u ) (0 ) [ 

(u )]−1 , /2 1

u ∈ ,

(10)

the reductions

W+ (u ± 2iK  ) =

σ3W+ (u )σ3 σ3W− (u )σ3 W+∗ [(u ± 2 K )∗ ] = σ1W+ (u )σ1 W−∗ [(u ± 2 K )∗ ] = σ1W− (u )σ1 W− (u ± 2iK  ) =

exp[± η3 ], exp[∓ η3 ], exp[± η1 /2], exp[∓ η1 /2],

(11)

and the restriction on  :

W+∗ (−u∗ ) = σ2W− (u )σ2 .

(12)

The restriction (12) and reductions (11) follow from the symmetry properties of the linear system (4) and asymptotic conditions (6). The conjugacy condition (10) represents another form of writing the relation (8) between fundamental solutions on  . The Riemann problem (10)–(12) has no normalization condition in terms of the spectral parameter. We will redefine its solutions by the asymptotics:

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

 + → 2(0) 1 0  a − → 2(0) b  a¯ + → 1(0) −b  (0 ) 1 − → 1 0



−b¯ a¯

at

χ → +∞;

0 1

at

χ → +∞;

at

χ → −∞;

σ (u − μ ) σ (u − μ∗ + 2 K ) η1 (μ+μ∗ ) , g2 ( u ) = e , σ ( u + μ∗ ) σ (u + μ + 2 K )  

σ ( u + K )σ ( u − K ) μ − μ∗ g3 ( u ) = exp η1 u + , (19) ∗ σ ( u + μ )σ ( u + μ + 2 K ) 2 η

σ ( μ + μ∗ ) σ ( μ − μ∗ + 2 K ) 1 κ (μ ) = exp ( 3 μ∗ − μ ) . σ ( μ + K )σ ( μ − K ) 2 g1 ( u ) =



0 1





b¯ a

at

χ → −∞,

(13)

following from the relation (8). The relation (8) also gives the expressions

a(u ) = det W− (u ),

a¯ (u ) = det W+ (u ),

(14)

which show, that the elements a(u) and a¯ (u ) of the transition matrix can be analytically continued from the contour  into the respective regions D− and D+ . If the functions a(u) and a¯ (u ) have zeros in their analyticity regions, then the set of zeros is the discrete spectrum of the problem (4), (6). We assume that all zeros are simple. In this case, according to reductions (11), zeros of the function a¯ (u ) in the regions D+ have the form:

u = μs ,

μ∗s − 2 K,

mod(4K, 2iK  ).

s = 1, 2, . . . , N,

(15) Zeros of the coefficient a(u) are related with (15) by the property a¯ (u ) = a∗ (−u∗ ) and can be written as:

u = −μ∗s , −μs + 2 K,

mod(4K, 2iK  ).

s = 1, 2, . . . , N,

(16) Solitons correspond to the solutions of the Riemann problem (10)–(13), when coefficients a(u) and a¯ (u ) in their analyticity regions have zeros and b(u ) = b¯ (u ) ≡ 0. The quasi-periodic meromorphic function a¯ sol (u ) for solitons is recovered from its zeros, poles and reductions [10,16]. For N-soliton state it has the form:

a¯ sol (u ) =

σ (u − μs )σ (u − μ∗s + 2 K ) η1 (μs +μ∗s ) e . σ (u + μ∗s )σ (u + μs + 2 K ) s=1

(μs + μ∗s ) = − /2,

mod(4 K ).

s=1

For definiteness, below we choose −K < Reμs < 0, |Imμs | < K . Since the reductions (11) contain the multipliers exp [ ± η3 ], exp [ ± η1 /2], the matrix W+ (u ) is not doubly periodic, and its construction is nontrivial (see Appendix). One-soliton matrix W+ (u, χ , t ) has the form:

W+ (χ , t, u μ, m )

 2



r m 2 g 2 ( u ) κ ∗ + m 1 2 g 1 ( u ) κ

κ −|κ|2 g3 (u ) m2 m∗1

=

 −|κ|2 g3 (u ) m1 m∗2

2

2 .

m 1 κ ∗ g 2 ( u ) + m 2 g 1 ( u ) κ (17)

Here

|r |

−2

= |m1 | + |m2 | + 2|m1 m2 | (1 + |snμ| )|cnμ| 4

4

2

2

The complex parameter μ (−K < Reμ < 0, |Imμ| < 2K ) and the vector m, depending on x, t, with the components m1,2 specify the matrix W+ . One can directly check, that the matrix (17) does satisfy all the algebraic relations of the Riemann problem (10)–(13). Using the representation (9), from the linear system (4), we express the matrices U(u) and V(u) through W+ (u ). Since W+−1 (u ) = †

W+ (−u∗ ), we have:

U (u ) = − W+ (u )[∂x − U /2 (u )]W+† (−u∗ ), V (u ) = − W+ (u )[∂t − V /2 (u )]W+† (−u∗ ),

−2

,

(18)

the coefficients g1,2,3 (u) and κ can be written in terms of the Weierstrass sigma-functions with periods [4K, 2iK ]:

(20)

where the matrices U /2 and V /2 are obtained from U, V by re(0 ) placing:  → ϕ0 = const, θ → θ

= π /2 − am(χ + /2, k ). The /2 left-hand side of (20) contains extra poles, coming from the soliton † matrices W+ (u ) and W+ (−u∗ ). The requirement, that the residues in these poles are zero, gives the dependence of the vector m on x, t [16,29]:



  

σ3 M˜ μ, χ + 2  η1 μ

× exp A(μ, χ , t )σ3 − σ3 c ,

iϕ (0 ) m = 

(μ, χ , t )c = exp − 0 /2 2 4K

(21)

where c is an arbitrary constant complex vector. Equating the residues from the right- and the left-hand side of the first of the “dressing” relations (20) at the point u = −iK  , we obtain an explicit solution of the Landau–Lifshitz model (1), corresponding to one-soltion excitation strongly connected with the stripe domain structure: (0 ) Sk σk = W+ (u )(S

) σ W † (−u∗ )|u=−iK  , /2 k k +

N 

Here μs ∈ D+ , and the Weierstrass sigma-functions have periods [4K, 2iK ]. The transformation properties of the function a¯ sol (u ) are valid only under the condition, that its zeros are related with the shift of the domain structure [20]: N 

305

(22)

(0 ) (0 ) (0 ) (0 ) where S

= (sin θ

cos ϕ0 , sin θ

sin ϕ0 , cos θ

), θ (0/)2 = /2 /2 /2 /2 π /2 − am(χ + /2, k ). We note, that the solution + (u ) = (0 ) W+ (u )

(u ) of the system (4) satisfies the conditions (13). /2 Therefore, the proposed procedure gives the soliton solution (22) with the required boundary conditions at infinity (3). The results obtained can be generalized. We have expressed any multisoliton excitation of the domain structure of an easy-axis ferromagnet in terms of the one-soliton matrices of the Riemann problem [20]. The obtained results allow us to find out the important and interesting distincitons of the properties of solitons in the domain structure from the properties of solitons against the homogeneous ground state of the medium. Elastic pairwise collisions between solitons in the domain structure are accompanied not only by changes of coordinates of the “centers of mass” of solitons and the phases of oscillations of their cores, but also by the local macroscopic shifts of the stripe structure due to the motion of each soliton [20].

3. Precessing solitons The solution (22) for the solitons in the domain structure is quite complicated. However, some specific features of the intradomain solitons are the same as for the solitons against the homogeneous background [14,16]. It would be useful to remind some of them.

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1. Solitons against the homogeneous distribution of magnetization We obtain the soliton solution on the homogeneous background from the solution (22) in the limit k → 1, when the period of the domain structure infinitely grows. It has the form:

S3 (x, t ) = 1 −

2s2ρ cρ2

,

(s2ρ + cθ2 )(cρ2 + s2y ) 2 i sρ cρ (cρ cθ cy − isρ sθ sy ) −iϕ S1 (x, t ) − iS2 (x, t ) = − e , (s2ρ + cθ2 )(cρ2 + s2y )

(23)

−2

y + i ϕ = +x thμ + i t ch

μ + i ϕ0 + y0 ≡ −l (x − V t ) + i (−η x + ω t + ϕ0 ) + y0 . −1

The structure and properties of the soliton is determined by the complex spectral parameter μ = −ρ + iθ , where ρ > 0, −π /2 ≤ θ ≤ π /2. Here we use denotations: sρ = shρ , cρ = chρ , sy = shy, cy = chy, sθ = sin θ , cθ = cos θ ; y0 , ϕ 0 are real constants of integration. −2 The soliton (23) has the velocity V = −l Im ch μ. The magnetization in it makes precession around the Oz-axis with the fre−2 quency ω = Re ch μ. The wave number of oscillations in the soliton is: η = −Im thμ. The amplitude of oscillations exponentially decays in the regions of the width l = −(Re thμ )−1 > 0. In the vicinities of the points μ = ±π i/2 the soliton solution (23) describes narrow regions of remagnetized material, which size is less than the width of two domain walls. Such solutions violate the condition of applicability of continuous approximation. Therefore, we will not consider small vicinities of the points μ = ±i π /2. Let us estimate these vicinities. Near the points μ = −ρ ± i (π /2 − δ ), 0 < ρ , δ < 1 at finite δ the value l (δ, ρ ) ≈ ρ + δ 2 /ρ has minimum 2δ at ρ = δ . We assume, that l (ρ , ρ = δ ) = 2δ is equal to the width of the single domain wall (it is equal to 1 in dimensionless variables). Then, we can estimate the boundary values of the parameters δ , ρ : δm = ρm ≈ 1/2. At ρ < < 1, −π /2 + δm < θ < π /2 − δm the soliton (23) represents “cutted” spin wave:

S3 = 1 + O ( ρ 2 ),

S1 − i S2 ≈ −

2 i ρ iϕ e cθ cy

with the length l  ρ −1 , the dispersion law: ω ≈ η2 + 1 and the finite velocity V = −2 tgθ . The region of remagnetization within the soliton we will call the soliton core. Projection S3 within the core changes from its edges, where S3 = Sa ≈ ±1, to its center, where S3 = Sc (|Sc | < 1). We will call the amplitude of the soliton the value A = |Sa − Sc | (0 < A ≤ 2). Then the width of the core would be determined by the value d, that is, by the distance between the points, where S3 = ( Sa + Sc )/2. At the values of parameters ρ , θ from the region ρ m < ρ ≤ Arcsh[cos θ ] ≤ 0.88 (arc on the Fig. 1), S3 -projection of magnetization within the soliton is nonnegative (A ≤ 1). In this case the soliton is localized in the region of the width d = q l, measured on the half of its height (see Fig. 1). The multiplier q ≡ 2 Arcsh[chρ ] weakly depends on ρ : 1.76 ≤ q ≤ 2.3. Hence, we obtain d ≈ 2 l. At sufficiently large ρ ≥ 1.8 the soliton represents the magnetization reversal center with the width d ≈ 2ρ l ≥ 3.6, separated from the another part of the sample by the domain walls of the width l ≈ 1 < d. In the reference frame associated with the moving soliton, the precession frequency of the vector S around the Oz-axis is constant. However, the initial phase of precession acquires an additional term:



0 (x − V t ) = arctg



sρ sθ sy , cρ cθ cy

Fig. 1. S3 -component of magnetization for the soliton (23) on the homogeneous background for different values of ρ , θ .

inhomogeneous within the soliton core. The direction of the velocity V of the soliton is given by the sign of derivative ∂ x 0 , which is opposite to the sign of the parameter θ : signV = sign(∂x 0 ) = −sign θ . The immobile solitons (23) are described by the more simple expressions at θ = 0 and θ = ±π /2. For θ = 0 we have [14,16]:

S3 = 1 −

2 s2ρ s2ρ + cy2

,

S1 − i S2 = −

2 i sρ cy −i ϕ e ; s2ρ + cy2

(24)

where y = −sρ (x − x0 )/cρ , ϕ = ϕ0 + (t − t0 )/cρ2 , x0 = const, t0 = const. In the soliton localization region the vector S makes inphase circle precession. The case θ = ±π /2:

S3 = 1 −

2cρ2 cρ2 + s2y

,

S1 − i S2 = ∓

2 cρ sy −i ϕ e , cρ2 + s2y

(25)

where y = −cρ (x − x0 )/sρ , ϕ = ϕ0 − (t − t0 )s−2 ρ is peculiar. The material in its center is completely remagnetized: S(y = 0 ) = (0, 0, −1 ). Such soliton is difficult to generate, since the rotation phases of magnetization to the left and to the right from its center has a jump (equal to π ). 2. Solitons in the stripe structure The solution of the Landau–Lifshitz model (1), describing a soliton in the domain structure, is written as (see (5), (17), (21), (22)):

(|α|2 − |β|2 )s˜ + (β ∗ α + α ∗ β )c˜ , |α|2 + |β|2 (α 2 − β 2 )c˜ − 2α β s˜ −iϕ0 S1 − iS2 = e , |α|2 + |β|2 S3 =

(26)

∗ |m |2 , β = −i m m∗ (s + s∗ ), where α = cμ |m2 |2 + cμ 1 1 2 μ μ

m1 = a− ey+iγ + b+ e−y−iγ , m2 = b− ey+iγ + a+ e−y−iγ ; η μχ˜

σ (χ˜ + K + iK  ± μ ) 1 exp ∓ , a± = m(χ˜ ) σ ( K + iK  ± μ ) 2K

σ (χ˜ − K + iK  ± μ ) σ ( K + iK  ± μ ) η μχ˜

1 × exp ∓ + η1 (χ˜ + iK  ± μ ) , 2K  1/2 σ ( iK  ) σ ( iK  + 2 K ) m (χ ) = ; 2 σ ( χ + iK  ) σ ( χ + iK  + 2 K ) η1 μ

+ y0 , y = Re A(μ, χ , t ) − 4K η μ

γ = Im A(μ, χ , t ) − 1 + γ0 . b± = m(χ˜ )

4K

Here we use denotations: cμ = cnμ, sμ = snμ, s˜ = snχ˜ , c˜ = cnχ˜ ; χ˜ = χ + /2, = −4Reμ; y0 , γ 0 are real integration constants.

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

307

Here a = k cμ /dμ , ω (μ ) = k−1 cμ dμ ; cμ = cnμ, dμ = dnμ; s4 = sn(χ + 4 ρ ), c4 = cn(χ + 4 ρ ), sχ = snχ , cχ = cnχ ; s0 = sin ϕ0 , c0 = cos ϕ0 . At k = 0.9994 the coefficient a ≈ 1. Hence, the end of the vector S on the Oe1 e2 -plane moves in an ellipse, close to a circle of small radius |X ± |. The function

(u, χ )  N (u, χ ) ei p (u) x , p (u ) = −i k−1 Z (u, k );  σ (χ − u ) σ (χ − u + 2 K ) η1 u χ  exp 2 η χ + N (u, χ ) = 3 σ (χ + i K  ) σ (χ + i K  + 2 K ) K is the solution of the Lame equation:

[k−2 ∂χ2 − 2 sn2 (χ , k )]  = −[k−2 + cn2 (u, k )] .

Fig. 2. The plane of parameters ρ , θ for the soliton (26) in the domain structure.

The dynamics of the soliton is characterized by the velocity V of the motion of its core and the frequency ω of the precession of magnetization in the core:

A(μ, χ , t ) = [−l −1 (x − V t ) − i(η x − ω t )]/2, l = −k [ReZ (μ )]−1 > 0, V = −l k

−1

Im(cnμ dnμ ),

η = −k−1 ImZ (μ ), ω = k−1 Re(cnμ dnμ ).

(27)

Here η is the wave number of the precession of magnetization around Oz-axis in the soliton core. The value l represents the size of the regions, within which the sharp changes in soliton magnetization occur. The structure and properties of the soliton (26) are determined by the complex parameter μ. We note, that upon the replacement μ → μ ± 2 iK the values V, ω, l do not change, the wave number η acquires an additional shift η ± 2π /L0 , where L0 = 2 Kk is the domain size, whereas the matrix M(μ, χ ) is transformed as:

M (μ ± 2 iK  , χ ) = σ3 M (μ, χ )σ3 exp[±π i x/L0 ]. From this it is follows, that the soliton solutions (26) with the parameters μ and μ ± 2 iK are distinguished only by unsufficient redefining the integration constants. For the following analysis we will write the parameter μ in the form: μ = −ρ + i θ and restrict ourself by considering the region from the rectangle 0 < ρ < K, −K  ≤ θ ≤ K  (see Fig. 2). In the vicinities of points μ = ±i (K  − δ ) − ρ , 0 < δ < δ m ≈ 1/2, 0 < ρ < ρ m ≈ 1/2 in the domain structure, as well as against the homogeneous background, there are narrow solitons, violating the continuous approximation. Below we will not consider such regions. Far from the center of soliton (at |y| > > 1) the solution (26) describes small-amplitude precession of magnetization around its equilibrium positions in the domain structure:





S = SR e1 (χ ) + SI e2 + 1 + O(e−4 |y| ) e3 (χ ).

(28)

The vectors ei (i = 1, 2, 3) are orthonormalized. At x << −1 we obtain:

e1 = c0 s4 i + s0 s4 j − c4 k ,

e2 = −s0 i + c0 j,

e3 = S2(0) ;

SR ≈ −Re[a X− ], SI ≈ −Im X− ,

∗ ∗ X−  ∗ (χ + , −μ∗ ) e−i ω (−μ ) t ;

(29)

and at x > > 1:

e1 = c0 sχ i + s0 sχ j − cχ k , SR ≈ −Re[a X+ ],

e2 = −s0 i + c0 j,

SI ≈ −Im X+ ,

e3 = S1(0) ;

X+  (χ , μ ) ei ω (μ) t . (30)

The asymptotics (29), (30) show, that the soliton (26) moves the stripe domains apart on the value k = −4 k Reμ. The macroscopic shift is localized within the region of the width k . We will treat this region as the core of the soliton in the domain structure. Like dislocations in crystals [26], the soliton (26) serves as an elementary carrier of the macroscopic shift of the domain structure. However, unlike the case of dislocations, the shift k = 4kρ (0 < ρ < K) depends only on the shape of the soliton core, but not on the period 4K(k)k of the domain structure (the parameters ρ and k are independent). The regions of sharp changes in magnetization to the left and to the right of the soliton core have the length l. For the soliton (23) on the homogeneous background the parameter l defined the size of the regions of transition from the soliton core to the homogeneous distribution of magnetization. From (29), (30) it follows, that for the soliton in the domain structure the value l characterizes the length of intervals near the soliton core, where the core pulsations cause the oscillations of the domain structure. So, we estimate the resulting size of the soliton in the domain structure as k + 2l. Here k = 4kρ is the size of the core, 2l is the length of magnetization oscillations in the domain structure, shifted by the core. The values l can vary in a wide range: from the width of one domain wall to the length of several domains. The maximal values of l are obtained at ρ → 0 and ρ → K. Let us denote nontrivial connection of extended solitons with linear modes of the domain structure. The values of the parameter μ = i θ , −K  < θ < K  correspond to the intra-domain spin waves [4] with the real frequencies:

1 dn(θ , k ) ≤ ω1 = < ∞. k k cn2 (θ , k )

(31)

Inside narrow hatched strip to the left from the line ad on the Fig. 2 the soliton (26) transforms into the “cutted” small-amplitude wave of the length  ρ −1 . The values μ = −K + i θ , −K  < θ < K  correspond to the imaginary frequencies of the second branch of the linear modes:

ω2 =

i k2 sn(θ , k )cn(θ , k ) . 2 k dn (θ , k )

In the bulk samples: L0 /l0 > > 1 [4]. It is possible only at k < < 1. We choose L0 /l0 ≈ 9.5 (k = 0.9994, K ≈ 4.75, K ≈ π /2). Than, instable mode has the negligibly small increment of increase over time: |ω2 |max = (1 − k )/k3/2 ≈ 6 · 10−4 << 1. In such a situation, the growth of longitudinal deformations of a domain structure is restrained by the law of conservation of the Oz-projection of magnetization:



R 0

S3 (x, t )dx = const,

(where R is the size of the sample) and by the nonlinear interactions that are not considered in the linear analysis of stability of

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study the scenarios of changing the soliton limiting velocity, due to rearrangements of its core. 4. Dynamics of the solitons with the cores less than the domain size

Fig. 3. The value Vmax (ρ ) (32).

the domain structure [24]. In this work we show, that the nonlinear interactions lead to localization of perturbations and appearance of the particle-like solitons, causing the macroscopic translations of the domain structure. In the hatched strip to the right from the line bc on the Fig. 2 soliton (26) is analogue of the wide shear soliton within the anharmonic chain of the domain walls [8]. The real parameters ρ and θ specify all the characteristics of the soliton: its frequency, wave number, velocity, amplitude and localization region. The soliton velocity V and the wave number η of its pulsations are odd, whereas the values ω and l are even functions of the parameter θ . In the detalied form the velocity (27) of the soliton (26) is:

V =−

i sρ sθ (dρ2 dθ2 + k2 cρ2 cθ2 )

(1 − k2 s2ρ s2θ )(k2 sρ cρ dρ s2θ − (1 − k2 s2ρ s2θ )Z (ρ , k ))

,

(32)

where sρ = snρ , cρ = cnρ , dρ = dnρ , sθ = sn(i θ ), cθ = cn(i θ ), dθ = dn(i θ ). The soliton is immobile at θ = 0 and θ = ±iK (μ ∈ a b, μ ∈ c d on the Fig. 2). The corresponding wave numbers of pulsations are: η = 0 and η = ±π /L0 . In the other cases the direction of the soliton velocity in the domain structure, as well as for the solitons against the homogeneous background, is set by the sign of the parameter θ : V = −sign θ . When the parameter |θ | grows from zero to K , the absolute value of the velocity begin to increase, approaching the limiting value Vmax (ρ ), and then decreases up to zero. The values of ρ and θ , corresponding to the maximal soliton velocity, are shown by two hatching-dotted lines on the Fig. 2. The values Vmax (ρ ) essentially depend on ρ , that is, on the size of the soliton core (see Fig. 3). In the following sections we will

We should distinguish two cases of elementary translation of the domain structure caused by soliton. At first of them (ρ m < ρ < K/2, |θ | ≤ K ) the soliton core is less than the size of the domain: k < 2 Kk. When the center of a soliton is within one of the domains of the structure, its core causes remagnetization of the material and moves apart the neighboring domain walls of the parent domain. As a result of it, all the domain structure shifts on the distance k . To illustrate it, let us discuss the immobile solitons at 0.4 K < ρ < 0.6K with the centers in one of the domain of the structure. Such solitons were studied in detail in the works [24,25]. They are qualitatively the same as the solitons (24) and (25) against the homogeneous background. The immobile soliton with the parameter μ = −ρ , θ = 0, 0.4 K < ρ < 0.6K is shown on the Fig. 4 and represents the nucleus of remagnetizing in the center in the one of the domain of the stripe structure. Its internal precession frequency: 0 < ω = k−1 cρ dρ < k−1 lies within the energy gap of the spectrum of spin waves (31). Therefore, such solitons are well-generated and diagnosized in the numerical experiments [25]. To excite this soliton we should enlarge and perturb one or several domains of the structure so that they would become a resonator for the core of the soliton. In the numerical experiments the immobile solitons and the pairs of the mobile intra-domain solitons moving in the opposite directions were excited by remagnetizing the part of the domain structure under the action of the step pulse of an external magnetic field. The solid and hatching lines on the Fig. 4 correspond to the angle  = arccos S3 at the moments of time t = 0 and t = T /2, when S3 (χ˜ = K ) approaches a maximum (T - is the precession period). The noncircular precession in the soliton localization region induces periodic synphase shifts of the domains along the sturucture on the value χ ≤ 0.4K and the changes of the angle  on the value 0.05π . On the Fig. 4 the mutual directions of translations are depicted by the arrows. Outside of the soliton core they are exponentially decreasing. The numbers in circles enumerate peculiar points, in which the shifts of the domains are absent. The position of these points in the structure changes, depending on the values ρ . For the immobile soltion with μ = −ρ ± iK  there are no such points (see Fig. 5). On the Fig. 5 the soliton with the parameter μ = −ρ + iK  (0.4 K < ρ < 0.6K) is shown, which also represents the magnetization reversal center inside one of the domains of the stripe structure. An internal precession frequency of the soliton lies within the wide interval: 0 < |ω| = k−1 cρ dρ s−2 ρ . Within the soliton core there is always a complete remagnetization of the parent

Fig. 4. Precession of the domain structure nearby the core of the soliton (26) at θ = 0.

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

309

Fig. 5. The soliton (26) at θ = K  . The denotations on the Fig. 5 are the same as on the Fig. 4.

domain: the magnetization in the center of the soliton reaches saturation and does not depend on time. The phases of precession around Oz-axis to the right and to the left from the center differ by π . In makes difficult to excite the immobile solitons with μ = −ρ + iK  in the numerical experiments. The core of the mobile soliton (26) resembles the soltion on the homogeneous background, but only on the time intervals, when it is situated inside one of the domains of the stripe structure (see Figs. 4 and 5). When the core passes through the domain wall of the stripe structure, the magnetization in the neighboring domains rotates so, that the magnetization reversal center disappears in the one domain and arises in the another domain with an opposite S3 -projection of magnetization in its center. The soliton core as if “overturns” without changing its form. Thus, the motion of the soliton in the domain structure is accompanied by the periodically repeating local rotations of magnetization and intricate oscillations at the transitions of the soliton core through the domain walls of the stripe structure. On the interval ρ m < ρ < K/2 the limiting velocity Vmax (ρ ) of the soliton has a following behavior. At ρ = ρm ≈ 1/2 the value Vmax (ρ ) tends to the minimal phase velocity of the linear activational modes in the domain structure. In the limit of wide domains (at k → 1) in the dimensionless variables: Vmax (ρ m ) ≈ 2 (see Fig. 3). The values Vmax decrease with the growing ρ . At ρ ≈ K/2 the size of the soliton core tends to the length 2 Kk of the domain. Even if the center of soliton is in the middle of the domain, the soliton core strongly interacts with its both domain walls and causes their longitudinal oscillations. During the motion of the core along the structure, a symmetry of its environment and a character of soliton oscillations are changing. It makes difficult a forward motion of the soliton through the stripe structure. As the result of it, at ρ ≈ K/2 the maximal velocity of the soliton decreases up to the value Vmax (K/2) ≈ 2k .

The behavior of solutions in the regions of instability of the structure in the linear approximation is of a special interest. On the small interval ρ = K − ε (0 < ε ≤ ε 0 ) at any θ by exception of small vicinities of the points θ = 0, ±K  the weakly localized wide core solitons move with the large velocities: Vmax ≈ k 2 K/(ε E), where E = E (k ) is the complete elliptic integral of the second kind. Their complete length is: 2l + k ≈ 2k/[ε (EK −1 − k2 dθ−2 )] + 4 Kk. The system as if tends to move them away. The cases μ → −K + 0 and μ → −K + 0 + i K  , when the modulations of the structure are immobile, are the most appropriate to observe. At such μ the soliton shifts the domain structure on the period. Therefore, the solution (26) has one and the same asymptotics at χ → +∞ and χ → −∞: S → S2(0 ) (see (3)). In the limit μ = −K + iK  + 0 from (26) we obtain the aperiodic weakly localized elliptic-polynomial excitation of the domain structure:

5. The nonlinear dynamics of solitons with the large cores

 π  ˜ (S1 − i S2 ) exp i ϕ0 − ≡ S1 − i S˜2 .

In the second case, - at K/2 ≤ ρ < K − ε0 (ε 0 0.1K), |θ | ≤ K – the soliton core qualitatively changes. The size of the core is more than the length 2 Kk of the domain, but less than the period 4 Kk of the structure. The core shifts the domain structure because of two processes, that is due to the translation of the nearby domain walls and the enlargement of the domains as the result of the rotation of magnetization. When ρ increases, the complete length 2l + k of the soliton incorporates several nearby domains. As a result, the maximal velocity of the soliton on the whole interval K/2 ≤ ρ < K − ε0 is almost unchanged and is comparatively small: Vmax ≈ 2k . At large ρ the oscillations of the domains near the soliton core accumulate an additional energy. That impedes an exciting the solitons with the wide cores in the numerical experiments [25]: remagnetization of an extended part of the domain structure leads to the formation of the several intra-domain solitons, which are energetically more favorable, than one soliton with large core.

S˜1 = −

2 sϕ κ , r2 + 1

S˜2 = −cnχ +

2 cϕ κ , r2 + 1

S3 = snχ −

2rκ , r2 + 1 (33)

where κ ≡ r snχ + cϕ cnχ , sϕ ≡ sin ϕ , cϕ ≡ cos ϕ ,

ϕ = arg[(2g2 + i t¯ )/(2g1 + i t¯ )] + arg[snχ ], k r (χ , t ) = 2

 m m 

2 1

m − m , 1

2

η1 χ

g1 ( χ ) =

ζ (χ ) −

g2 ( χ ) =

ζ (χ + 2 K ) −

2K

+

t¯ = t k2 /k2 ,

m1 k snχ = m2 dnχ + cnχ





2g1 + i t¯ , 2g2 + i t¯

E (χ − χ0 ), 2K

η1

2K

(χ + 2 K ) +

E (χ − χ0 ). 2K

Here we have extracted from (26) a rotation on the angle ϕ 0 in the Oxy-plane:

2

The character of the excitation (33) significantly depends on the position of its center, which is set by the real constant χ 0 . If the center of excitation is in the middle of the domain wall (χ0 = 0), then the solution (33) describes intricate process of the magnetic reversal of the domain structure, occurring in two stages (see Fig. 6). At first, the domain walls with the centers at the points χ = ±2K move, as t increases, towards a motionless domain wall at the point χ = 0. The domains adjacent to this domain wall (domains 1) are shortened, and the ones following them (domains 2) are extended. On Fig. 6, the solid line represents the projection S3 corresponding to the background structure; the directions of motion of the domain walls are indicated by arrows. Over time −∞ < t < −t0 (t0 = k2 /k = 27.4) the moving domain walls reach the points χ = ±K, which occupy the positions marked on Fig. 6 by the hatching-dotted lines. On the second stage (for

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V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

Fig. 6. S3 -component of magnetization for the excitation (33).

−t0 < t < 0) the magnetization is rotated in the narrow domains 1, taking on the values corresponding to homogeneous states in the following wider domains 2. As a result, for t = 0, the narrow domains 1 completely disappear, and the two wide domains 2 occupy the region |χ | < 4K, divided by the domain wall with the center at the point χ = 0 (hatching line on Fig. 6). Other parts of the structure are insignificantly shifted in the direction toward the central domain wall. For 0 < t < +∞, the above process of the magnetization reversal occurs in the reverse sequence. In the small vicinities of the points μ = −K + 0, μ = −K + i K  + 0 the aperiodic modulations are replaced by the periodic weakly localized excitations with longitudinal oscillations, translations and remagnetization within the corresponding part of the domain structure [24,25]. Thus, these excitations may be referred to as analogues of the “Peregrine soliton” [27] in the domain structure.

Acknowledgment This work was performed within the state assignment of the Ministry of Education and Science (the theme “Quantum”, number AAAA-A18-118020190095-4) and supported by the project of the Russian Foundation of Fundamental Investigations for young scientists “My first grant” 18-32-00143. Appendix Let us costruct the matrix function W+ (u, χ , t ) for one-soliton excitation in the stripe domain structure. For soliton states the conjugacy condition of the Riemann problem acquires the simple form:

W− (u ) = W+ (u )/a¯ sol (u ),

u ∈ .

(A1)

The restrictions (12), (A1) are equivalent to the equality: 6. Conclusion

W+ (u ) = σ2W+∗ (−u )σ2 a¯ sol (u ),

In this work the new exact solutions of the Landau–Lifshitz equation (1) are obtained and analyzed in detail, which describe solitons in the domain structure of an easy-axis ferromagnet. These solitons are structurally stable and serve as elementary carries of macroscopic shifts of the structure. Noncircular precession of magnetization in the soliton core causes the pulsations of the core and longitudinal oscillations of the neighboring domain boundaries of the structure. At low precession frequencies solitons represent nucleation sites for remagnetizing the material. When the frequencies of solitons approach the boundaries of the spin-wave spectrum, they transform into small-amplitude spin waves or periodic “splashes” of strong modulations of the domain structure. These modulations are accompanied by the reciprocating motion of a group of the domain boundaries along the structure, as well as the processes of rotation of magnetization within the several neighboring domains. We have studied the connection of the velocity of solitons in the domain structure with the internal dynamics and the length of their cores. We have found the intra-domain soliton excitations, that can be generated experimentally applying the external magnetic field. A strong interaction of solitons with the domain structure should be taken into account, when analyzing the processes of magnetization reversal of the material. It also can be important for the development of new devices for storing and processing the information.

which can be analytically continued from the contour  into the parallelogram of periods [4 K, 4iK  ], since for the soliton states the functions W+ (u ), a¯ sol (u ) are meromorphic. Taking into account, that any nondegenerate 2 × 2 matrix satisfies the identitiy

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A−1 =

σ2 A T σ2 det A

u ∈ ,

(A2)

,

where “T” means transposition, from (A2), (14) we obtain the restriction on the matrix function W+ (u ) in the parallelogram of periods [4 K, 4iK  ]:

W+† (−u∗ )W+ (u ) = I.

(A3)

The reductions (11) contain multipliers exp [ ± η3 ], exp [ ± η1 /2]. That is mean, that the matrix W+ (u ) is not doubly periodic. To find W+ (u ), at first we will construct the meromorphic function f† (u∗ ) with periods [4K, 4iK ], satisfying the restrictions:

f ( u ± 2 iK  ) =

σ3 f (u )σ3 , f (−u ± 2 K ) = σ3 f (u )σ3 , f (−u ) f † (u∗ ) = I,

(A4)

where extra multipliers are absent. The second relation (A4) replaces the reductions (11), (12) with complex conjugation. According to (A4), one-soliton function f † (u∗ ) in the parallelogram of periods has four zeros:

u = μ∗ ,

μ∗ − 2iK  , −μ∗ + 2 K, −μ∗ − 2K + 2iK  ,

(A5)

and four poles u = μi (i = 1, 2, 3, 4):

μ1 = −μ, μ2 = −μ + 2iK  , μ3 = μ + 2 K, μ4 = μ − 2K − 2iK  .

(A6)

V.V. Kiselev and A.A. Raskovalov / Chaos, Solitons and Fractals 127 (2019) 302–311

The sum of zeros (A5) is commensurable with the sum of poles (A6) by the modulus of periods [4K, 4iK ]. Hence, the function f† (u∗ ), unlike W+ (u ), may be doubly periodic [10] and we will seek it as the expansion in terms of quasi-periodic Weierstrass zetafunctions with periods [4K, 4iK ]:

f (u ) = A0 + A ζ1 + B ζ2 + C ζ3 + D ζ4 , †

ζi ≡ ζ (u − μi ). (A7)

∗

Here A0 , A, B, C, D are matrix coefficients independent of u; zetafunctions have properties [10]:

ζ ( u ± 4 i K  ) = ζ ( u ) ± 2 η3 , ζ ( u ± 4 K ) = ζ ( u ) ± 2 η1 , ζ (−u ) = −ζ (u ); η1 + η2 + η3 = 0, η2 = −ζ (2 K + 2 iK  ), η1 = ζ (2 K ), η3 = ζ (2 iK  ). Substituting (A7) into the first two relations (A4), we find:





f † ( u∗ ) = I +

B˜11 ϕ (u ) B˜21 γ (u )

 

B˜12 γ (u ) B˜22 ϕ (u )

r1 0



0 , r2

ϕ (u ) = ζ1 + ζ2 − ζ3 − ζ4 , γ (u ) = ζ1 + ζ3 − ζ2 − ζ4 + 2η1 ,

(A8)

where B˜i j (i, j = 1, 2) and r1 , r2 are unknown scalar functions of x, t. It can be readily verified, that the sum of residues of the function f† (u∗ ) (A8) in its poles within the parallelogram of periods [4K, 4iK ] is equal to zero, and therefore this function is indeed doubly periodic [10]. We express ϕ (u) and γ (u) in terms of more popular and tabulated Jacobi elliptic functions snu, cnu, dnu [10–12]:

cnu dnμ , snu + snμ  snu − snμ∗  cnμ dnμ ϕ ( u ) − ϕ ( μ∗ ) = − . (snμ + snμ∗ ) snu + snμ

γ (u ) =

Here all the Jacobi functions have the modulus k. The requirement, that there are no poles in the product f (−u ) f † (u∗ ) = I is equivalent to one independent matrix equation B˜† f † (u∗ )|u=μ∗ = 0. It has nontrivial solution only if the matrix B˜ is degenerate: B˜i j = mi X j (i, j = 1, 2) [16,29]. From this it follows, that:

m∗1 , | m 1 | ϕ ( μ ) + | m 2 | 2 γ ( μ∗ ) ∗ m2 X2 = − , | m 2 | 2 ϕ ( μ∗ ) + | m 1 | 2 γ ( μ∗ ) X1 = −

2

∗

(A9)

where m1,2 are still unknown functions of x, t. The denominators in (A9) can be eliminated by redefining the elements r1,2 . Then, f† (u∗ ) takes the form: f † ( u∗ ) =

 |m2 |2 − |m1 |2 β ( u ) −m2 m∗1 α (u )



−m1 m∗2 α (u ) |m1 |2 − |m2 |2 β ( u )

r1 0



0 , r2

where α (u) ≡ γ (u)/γ (μ∗ ), β (u ) ≡ (ϕ (u ) − ϕ (μ∗ ))/γ (μ∗ ). In the work [20] it is shown, that the solution of the original Riemann problem (10)–(13) is:

W+ (u ) = f (u )g2 (u ),

(A10)

where arg r1 = arg r2 = 0. The multiplier g2 (u) is determined in (19). The modules of the functions r1,2 are found from the

311

condition (A3) and are equal to the value r (see (18)). The expression for W+ (u ) (A10) is equivalent to the one-soliton matrix (17) in the Section 2. References [1] Gurevich LE, Liverts EV. Quasiacoustical and quasioptical oscillations of the domain structure of a uniaxial ferromagnet. JETP 1982;55(1):132–4. [2] Hubert A. Theory of domain walls in ordered media. Berlin-Heidelberg-New York: Springer; 1974. [3] Brown WF. Micromagnetics. New York: Interscience Publishers; 1963. [4] Filippov BN, Tankeev AP. The dynamical effects in ferromagnets with a domain structure. Moscow: Nauka; 1987. [in Russian]. [5] Baryachtar VG, Ivanov BA. About the high-frequency properties of a ferromagnet with the domain structure. Fiz Met Metalloved 1973;36(4):690–7. [in Russian]. [6] Baryachtar VG, Gorobetz YI, Denisov SI. Stability and spectrum of natural oscillations of the plane-parallel domain structure. Ukr Fiz Zh 1983;28(3):436–40. [in Russian]. [7] Philippov BN, Soloviev MM. The nonlinear dynamics of interacting domain boundaries in a thin ferromagnet film under an external periodic magnetic field. I The frequencies of natural oscillations of the domain boundary. Fiz Met Metalloved 1995;80(2):20–3. [in Russian]. [8] Shamsutdinov MA, Rakhimov SE, Kharisov AT. Nonlinear waves in a chain of plane-parallel domain walls in ferromagnets. Phys Sol St 2001;43(4):718–21. [9] Shirobokov MK. To the theory of ferromagnetic magnetization mechanism. Zh Eksp Teor Fiz 1945;15(1–2):57–76. [in Russian]. [10] Akhiezer AI. Elements of the theory of elliptic functions. Am Math Soc Providence 1990. [11] Bateman H, Erdelyi A. Higher transcendental functions. Vol. 3: Elliptic and automorphic functions, Lame and Mathieu functions. New York: McGraw-Hill; 1954. [12] Byrd PF, Friedman MD. Handbook of elliptic integrals for engineers and scientists. Springer Verlag; 1971. [13] Borisov AB, Kiselev VV. Topological defects in incommensurable magnetic and crystal structures and quasi-periodic solutions of the elliptic sine–gordon equation. Physica D 1988;31:49–64. [14] Kosevich AM, Ivanov BA, Kovalev AC. The nonlinear magnetization waves. Dynamical and topological solitons. Kiev: Naukova Dumka; 1983. [in Russian]. [15] Borovik AE, Klama S, Kulinich SI. Integration of the Landau–Lifshitz equation with preferred–axis anisotropy by the method of the inverse scattering problem. Physica D 1988;32(1):107–34. [16] Borisov AB, Kiselev VV. Quasi-one-dimensional magnetic solitons. Moscow: Fizmatlit; 2014. [in Russian]. [17] Bikbaev R.F., Bobenko A.I., Its A.R.. The Landau–Lifshitz equations. The theory of exact solutions II – Preprint Don FTI – 84 - 7, Donetsk, 1984;(82). [in Russian]. [18] Bikbaev RF, Bobenko AI, Its AR. Finite-zone integration of the Landau–Lifshitz equation. Dokl Akad Nauk SSSR 1983;272(6):1293–8. [19] Mitropolskii YF, Bogolubov NN, Prikarpatskii AK, Samoilenko VG. Integrable dynamical systems: spectral and differential-geometrical aspects. Kiev: Naukova Dumka; 1987. [in Russian]. [20] Kiselev VV, Raskovalov AA. Solitons in the domain structure of the ferromagnet. Theor Math Phys 2018;196(3):1317–34. [21] Mikhailov AV. The Landau–Lifshitz equation and the Riemann boundary problem on a torus. Phys Lett A 1982;92(2):51–5. [22] Borisov AB, Kiselev VV. Many-soliton solutions of asymmetric chiral SU(2), SL(2,R)-theories (d = 1 ). Theor Math Phys 1983;54(2):246–57. [23] Borisov AB. The Hilbert problem for matrices and a new class of integrable equation. Lett Math Phys 1983;7:195–9. [24] Borisov AB, Kiselev VV, Raskovalov AA. Precessing solitons in the stripe domain structure. Low Temp Phys 2018;44(8):765–74. [25] Kiselev VV, Raskovalov AA, Batalov SV. Localized nonlinear excitations of the domain structure of a ferromagnet. Phys Met Metallogr 2019;120(2):107–20. [26] Hirth J, Lothe J. Theory of dislocations. John Wiley and Sons, Inc.; 1982. ˝ [27] Peregrin DH. Water waves, nonlinear Schrodinger equations and their solutions. J Austr Math Soc Ser-B 1983;25:16–43. [28] Sklyanin E.K. On complete integrability of the Landau–Lifshitz equation. – preprint LOMI 79-E-3. – Leningrad. 1979. [29] Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE. Theory of solitons. The inverse scattering method. New–York: Plenum; 1984.