Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions

Author’s Accepted Manuscript Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions Victor K. Kuetche

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To appear in: Journal of Magnetism and Magnetic Materials Received date: 6 June 2015 Revised date: 28 August 2015 Accepted date: 30 August 2015 Cite this article as: Victor K. Kuetche, Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.08.120 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Inhomogeneous exchange within ferrites: Magnetic solitons and their interactions Victor K. Kuetche The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera, 11-I34151 Trieste, Italy National Advanced School of Engineering, University of Yaounde I, P.O. Box 8390, Cameroon Centre d’Excellence en Technologies de l’Information et de la Communication (CETIC), University of Yaounde I, P.O. Box 812, Cameroon Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon

Abstract In the wake of the recent investigation of inhomogeneous exchange effects within ferrites [Kuetche et al., J. Magn. Magn. Mater. 3374 (2015) 1], we pay a particular interest to the magnetic solitons and their dynamics. We study extensively the interactions between these waves while depicting the two-soliton features and the three-soliton scattering scenarios of the waveguides. As a result, we find that these typical head-on collisions are actually elastic. Discussing deeply the results, we determine the individual shifts of the interacting waves and we find that they actually comprise two parts: the first one relates to the nonlinear character of the interactions and the second one characterizes the motion of the smaller soliton along the larger one. Additionally, we depict the energy density of the interacting waves and we address the physical implications of the previous results. Highlights: -Inhomogeneous exchange within ferrites induces some soliton structure in the dynamics. -The head-on collisions between the magnetic waveguide excitations depict elastic features. -The individual shifts of the interacting waves comprise two parts: the first part is due to nonlinearity and the second one characterizes the relative motion of the individual wave excitations. Keywords: inhomogeneous exchange, magnetic solitons, head-on collisions, shifts, energy density 1

1. Introduction In relation with the increasing interests in advanced magnetic information storage and data process element, a proper and detailed understanding of the micromagnetic structure in microsized and nanosized magnets become more crucial. In the wake of these interests, Kuetche et al [1] regarded recently in a leading attempt a ferromagnetic slab of 0.5mm thickness while investigating deeply the effects of inhomogeneous exchange within the material. As a result, they derived the governing coupled system given by [1] B xt = BC x + B xx − sB x , C xt = −BB x ,

(1a) (1b)

arising from the nonlinear dynamics of magnetic polaritons in the medium. The variables x and t are generic spacelike and timelike coordinates, respectively, while the physical meanings of the observables B and C will be reviewed briefly in the next section. The constant s is the first-order dimensionless Gilbertdamping parameter and the quantity  stands for an arbitrary parameter expressed in term of the second-order dimensionless inhomogeneous exchange parameter within the magnet. Already at the heart of the information technologies developed in the second half of the twentieth century, the tailoring of magnetic materials [2–4] is arguably undergoing a second revolution with the development of spintronics [5, 6]. Below the Curie transition temperature, the spin degrees of freedom carried by localized electrons in ferromagnetic materials tend to spontaneously long-range order. Micromagnetic description actually consists in studying the local order parameter -the magnetization averaged over a few lattice cells. In solid physics, the understanding of complex magnetic structures can be achieved by the Heisenberg model of spin-spin interactions. Such a model has successfully explained the existence of ferromagnetism below the Curie temperature, and also attracted considerable attention in nonlinear science and condensed matter physics [7]. The dynamics and kinetics of a ferromagnet is dictated by the variations in its magnetization. When a ferromagnet is used to store information, bits are encoded in the orientation of the local magnetization. Controlling Email address: [email protected] (Victor K. Kuetche) Preprint submitted to J. Mag. Mag. Mater.

August 31, 2015

the state of a ferromagnet crystal unambiguously described by the magnetization vector stands to be fundamental in the understanding of the magnetic storage process of the data elements. Different aspects of magnetization dynamics are involved in magnetic storage technologies in extended layers and nanostructures [8–12]. Investigating the magnetic nanoelements is not only in focus in view of their technological potential but they are often as test systems for the analysis of debated microscopic mechanisms involved in magnetization dynamics such as coupling to spin-currents [13–15], heat gradients [16] and the role of spinorbit coupling in domain wall dynamics [17, 18], just to name a few. Dissipation -relaxation of excited magnetic textures towards an equilibrium state -is faced usually when magnetic textures are dynamically manipulated. Actually, magnetization dissipation, expressed in terms of the Gilbert-damping parameter is a key factor determining the performance of magnetic material in a host of applications. As a matter of fact, we mention the enhancement of the damping of ferromagnetic materials in contact with nonmagnetic metals [19, 20] in magnetic memory devices based upon ultrathin magnetic layer [21, 22], which paves the way to tailoring the Gilbert-parameter for particular materials and applications. Nonlinear behavior in magnetic systems actually constitutes a great topic of continuing interest [23–25]. There has been a number of experiments [26–28] which have shown an interesting nonlinear mixing of two signals in magnetic materials. The tradeoff between the nonlinearity -tendency to increase the wave slope and the dispersion -tendency to flatten the wave, generates some typical self-confined structures coined as solitons. In ferromagnetism, these structures known as magnetic solitons are sometimes referred to magnons. Owing to attraction, a cluster of magnons in ferromagnetism tends to be self-localization. In one-dimensional ferromagnets, the previous attraction becomes critical due to the fact that it produces a bound state of quasi particles. Because of the self-localization of the magnon cluster, a spin wave which can be regarded as a cluster of microscopic number of coherent magnons becomes unstable. The topological soliton and the dynamic soliton [29] are the result of the magnetization localization induced by the developing instability. The topological soliton which refers to the magnetic domain wall is regarded as a potential hill separating two degenerated magnetic states. This type of soliton actually forms a spatially localized configuration of magnetization where the magnetic moments inverse gradually. The second type describes the localized states of magnetization which is likely to reduce to a uniform magnetization by continuous deformation where the excited ferromagnet transits to the ground state. These typi3

cal solitons which result from the tradeoff between the Maxwell equations and the Landau-Lifshitz-Gilbert equation constitute a family of waves known as autonomous solitons which are similar to the classical soliton concept introduced firstly by Zabusky and Kruskal [30] for autonomous nonlinear and dispersive dynamic systems [31, 32]. They can completely preserve their localized form and speed during propagation and even after suffering some complex head-on collision with one another. Since bits are encoded in the orientation of the local magnetization, our main motivation in this work is to better understand the magnetic information storage and data process elements in advanced magnetic devices. Thus, our leading objective is to investigate properly the dynamics of a ferromagnet slab of zero conductivity where inhomogeneous exchange become crucial. We are then resorted to study properly the governing system above given by Eq. (1), from the physical viewpoint consisting of the propagation and nonlinear interactions between the magnetic polaritons within the slab. Accordingly, we organize our work as follows. In Section 2, we review in a concise presentation the physical ground of the system given by Eq. (1). Next, in Section 3, we study the soliton structure of the previous system while expressing in detail the two-soliton and the three-soliton solutions alongside their energy functionals -kinetic, potential, and total energy densities. Then, in Section 4 we discuss the above results while computing the shifts of the individual solitons. Finally, we end the present work with a brief summary. 2. Physical ground: Inhomogeneous exchange within ferrites We consider a quasi one-dimensional ferrite slab lying in the x-axis, the transverse dimension being negligible. This slab is magnetized to saturation by an inplane external field H∞ 0 directed along the transverse y-axis perpendicular to the propagation x-direction as presented in Fig. 1. Due to the absence of eddy currents, electromagnetic waves are likely to propagate. We consider a thick enough film in view of ensuring an homogeneous magnetization over the ferrite. We assume that the crystalline and surface anisotropy of the sample is negligible. The use of the Maxwell’s equations combined to the Landau-Lifshitz-Gilbert (LLG) equation [33, 34] for a ferrite yields the following dimensionless system −∇ · (∇ · H) + ΔH = ∂2t (H+M), ∂t M = −M ∧ Heff + σM ∧ ∂t M/m,

(2a) (2b)

where vectors H and M stand for the dimensionless magnetic induction and magnetization density, respectively. From a practical viewpoint, the above coupled 4

6

z

(a)

4

x

ω, V, Vg

y H∞ 0

V

ω V Vg

(b)

2 0 −2 −4 −6 −6

−4

−2

0

κ

2

4

6

Figure 1: Ferrite slab and dispersion relation. In panel (a), vectors V and H 0 stand for the velocity of the wave propagation and the in-plane external magnetic field, respectively. In panel (b), the variations of the wave-frequency ω, the phase velocity V of the plane wave, and the group velocity Vg of the bulk wave are plotted against the wave-number κ of the wave.

equations are actually fundamental for investigation of the data loading processes in reversal magnetic memory devices in the ferrites. The constants m and σ refer to the dimensionless saturation magnetization and Gilbert-damping parameter [33, 34], respectively. The expression of the effective field Heff is given by [33, 34] Heff = H − βn(n · M) + αΔM,

(3)

where α and β are the constants of the inhomogeneous exchange and the magnet anisotropy, respectively, and n is the unit vector directed along the anisotropy axis. For a simple tractability, we assume n ≡ ez directed along the z-axis. Now, we use the following expansion series of the magnetization density and the magnetic induction expressed as M=

∞ 

M j , j

H=

j=0

∞ 

H j j,

(4)

j=0

where the parameter represents the small perturbation related to the wavelength of the short-wave perturbation along the ferrite. Replacing Eq. (4) into (3) and following a blend of transformations where two independent spacetime coordinates X and T express [1] X = − −1 m(x − t)/2, 5

T = mt,

(5)

Eq. (2) transforms to BXT = BC X + BXX − sBX , C XT = −BBX ,

(6a) (6b)

where observables B and C, and constants  and s are defined by [1]  X y C = −X − (H0 /m)dX, B = Mx1 /2m,  = α2 m2 /4, s = −σ1 /2.

(7)

The observables B and C expressed above represent the ”relative magnetization” of the ferrite [1] with respect to the saturation magnetization and the ”integral effective magnetic strength” of the ferrite, respectively. Parameter σ1 stands for the first-order expansion cœfficient of the dimensionless Gilbert-damping against the short-wavelength perturbation. Constant α 2 refers to second-order expansion cœfficient of the inhomogeneous exchange parameter. The quantities H 0 and M1 instead, refer to the zeroth and first-order expansion cœfficients of the external magnetic field and the magnetization, respectively. According to Eq. (6), these observables evolve dynamically within the spacetime-like manifold described by variables X and T , respectively, with the boundary conditions lim X→∞ H0 = (0, μm, 0) and limX→∞ M1 = 0, parameter μ (μ > 0) being the strength of the internal magnetic field of the ferrite [1]. For the sake of simplicity, these two independent variables will be written into their lower cases x and t, respectively. 3. Soliton structure and nonlinear interactions In account of the asymptotical boundary conditions lim X→∞ B = 0 and limX→∞ C X = −(1 + μ) to Eq. (6) above, we consider the scale transformations given by B → (1 + μ)B,

C → −(1 + μ)C,

T → −T/(1 + μ),

X → X.

(8)

Under the above transformations, the system given by Eq. (6) is invariant provided the parameters  and s now read  = −α2 m2 /4(1 + μ),

s = σ1 /2(1 + μ).

(9)

In this section, the dependent variables B and C, and the independent variables T and X expressed below are actually derived from Eq. (8) to which we refer when writing down the genuine expressions of the original variables. Then, the system under our investigation takes the form given by Eq. (1) above where conventionally spacetime variables X and Y have been written into their lowercase forms, respectively. 6

Thus, Eq. (1) possesses Lagrangian and Hamiltonian formulations as follows. With observables B and C regarded as two degrees of freedom of the system above, the Euler-Lagrange equation of motion of the system reads δL δL = sB x , = 0, δB δC where the Lagrangian L of the system is expressed as   L = B x Bt + C x Ct − B2x + B2C x /2.

(10)

(11)

From this Lagrangian formulation, the Hamiltonian of the system is given by   (12) H = B x Bt + C xCt − B2x /2. The expression of the Hamiltonian above shows that it does not depend explicitly on time t. Consequently, the system under investigation is regarded as natural in such a way that its kinetic-like T and potential-like energy V densities can be derived seemingly. These quantities are then expressed as   T = B x Bt + C x Ct − B2x + B2C x /2 /2, V = −B2C x /4. (13) The energy-like functional of the system refers to the sum of its kinetic-like T and potential-like energy V densities. Since the system is natural, we refer this sum to its total energy-like density represented by the Hamiltonian H above. In order to tread into the scattering properties of the localized solutions, we pay particular interests to weak damping and we transform the system (1) into its bilinear form by means of the Hirota method [35–37]. As a result, we find (Dx Dt − D2x + sD x )G · F = GF,

(D2t − D x Dt )F · F = G2 /2,

(14)

provided B = G/F,

C = x + 2 (ln F) x − 2 (ln F)t + C0 ,

(15)

where the quantity C0 is an arbitrary parameter. The symbol D x , Dt refer to the Hirota’s operators [35–37] with respect to the variable x, t, respectively. According to the usual procedure, the dependent functions G and F are expanded into suitable power series of a perturbation parameter ε. In this paper, in view of investigating the dynamics of localized excitations with vanishing tails, we arguably expand the functions G and F as follows G = εG1 + ε3G3 + · · · ,

F = 1 + ε 2 F 2 + ε4 F 4 + · · · ,

(16)

where the functions G i , F i , (i = 1, 2, 3, · · ·) are expansion coefficients of the above series. Substituting this expansion into Eq. (14) and collecting the terms of each order of ε, we obtain the results presented below. 7

(a)

3 L

2.5

B

2 1.5 H 1 h

0.5 −6

−4

−2

0 C

2

4

6

Figure 2: Multivalued single waveguide channel solution to Eq. (1). The snapshot is taken at initial time t = 0 and parameters are chosen arbitrarily as A = 1, κ = 0.5 and  = 1. Panel (a) presents the profile of the wave and its details. The panel (b) presents the density plot of such a traveling structure while showing a constant phase velocity of the wave propagation.

3.1. The one-soliton solution The one-soliton solution to Eq. (1) is obtained from the following truncation G = A exp(η),

F = 1 + (A2 κ/16ω) exp(2η),

(17)

where the phase η is defined by η = κx + ωt + η0 ,

(18)

with constants κ and ω being wave number and frequency of the traveling wave, respectively, and constant η0 being an arbitrary parameter. According to Eq. (14), the dispersion equation reads ωκ − κ2 = 1,

(19)

and the damping vanishes. The representation of the dispersion equation is expressed in Fig. 1 in terms of the variations of the wave-frequency ω, the phase velocity V of the plane wave, and the group velocity V g of the bulk wave against the wave-number κ of the wave. From this picture, it appears that there exit two family of waves propagating within the (x, t)-spacetime manifold, namely, the right-moving waves-waves moving towards positive-definite x-axis and the left-moving waves-waves moving towards negative-definite x-axis. The plane 8

20

(a)

3

H h L Energy density

15 H, h, L

(b)

2.5

10

2 1.5 1 0.5

5

0

0 0

0.2

0.4

κ

0.6

0.8

0.5

1

1

1.5

2

2.5

3

B

Figure 3: Profile details of the loop-like wave solution given by Eq. (20) and its energy functional. Panel (a) shows the variations of H, h and L with the wave-number. Accordingly, they are monotone decreasing functions of the wave-number. Panel (b) depicts the variations of the total energy, potential and kinetic energy densities of the wave against the amplitude of the wave. In this panel, the black, blue and magenta colors represent the total energy, the potential energy and the kinetic energy densities of the wave, respectively. Accordingly, the most stable part of the wave is the region 0 ≤ B ≤ H.

phase velocity of these waves is monotone from one family to another and hence renders the bulk system absolutely left-moving. Nonetheless, we will focus our interest to the dynamics of the first wave-family where we arbitrarily select the branch κ > 0. Thus, the one-soliton solution expressed by Eq. (15) reads  B = 2 ω/κsech(η + δ0 ), C = x − (2/κ) tanh(η + δ0 ), (20) provided = ±1, C0 = 2/κ and δ0 = (1/2) ln(A2 κ/16ω). Without lost of generality, we pay attention only to waves having positive-valued amplitudes. Figure 2 depicts the variations of the observable B with respect to variable C at t = 0. As it is observed, the shape of the profile is a loop propagating at a constant velocity towards the negative-definite C-axis. This velocity increases with the amplitude of the waves. Looking forward into the shape of this looplike solitary wave, we define the quantities L, H and h standing for the maximum width of the loop, the height at which this occurs, and the height at which the crossover point occurs. This is all summarized in the above figure. We note that requiring symmetry in (C − t)-space, for the one-soliton solution above, we have chosen η0 = −δ0 . Therefore, the crossover point will occur at C = 0 9

B

(a) 3 2 1

20 10

50 0

t

0 −10

C −20

−50

Figure 4: The interaction process (panel (a)) for two “similar” multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.51) and (A 2 = 1, κ2 = 0.6), and the amplitude ratio r = 0.85. The panel (b) presents the density plot of the previous interaction process. From this depiction, the small soliton is shifted forwards whereas the larger one is shifted backwards by the interaction. Also, the interaction area is estimated as t ∈ [−10, 12] and C ∈ [−30, 20] yielding a magnitude of about tC = 1110.

corresponding to η = η1 so that tanh η1 = κx1 /2.

(21)

Since the calculations are made at initial time, when η = η1 , the height h is expressed as  h = 2 (ω/κ)(1 − κ2 x21 )/4. (22) We solve Eq. (21) numerically and we obtain κx1 = 1.9150. In order to derive the width L and height H, we have to solve the equation 1 − (κ/ω)B2 /2 = 0. This leads to √  √ √ H = 2 ω/κ, L = (2/κ)( 2 − ln( 2 + 1)). (23) The plot of the previous details of the traveling loop-wave above is shown in Fig. 3. As the wave-number κ increases, the amplitude of the traveling waves decreases. Since the width L and the heights H and h above are proportional to the amplitude of the wave, they also decrease with the wave number. 10

(a)

(b) 4 Energy density

Energy density

2

1.5

1

3

2

1

0.5

0

0 0.2

0.4

0.6 B

0.8

1

1.2

0.2

0.4

0.6

0.8 B

1

1.2

1.4

1.6

Figure 5: Panel (a) presents the energy density from the interaction process for two multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.51) and (A 2 = 1, κ2 = 0.6). Panel (b) presents the energy density from the interaction process for two multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.35) and (A 2 = 1, κ2 = 0.6). In the figure, the black, blue and magenta colors represent the total energy, the potential energy and the kinetic energy densities of the system, respectively.

The evaluation of the energy density functional of this one-loop solitary wave is depicted in Fig. 3. As observed, the energy density H increases with the amplitude of the wave. The kinetic-like energy density of the wave increases until it reaches its maximum and then decreases further to zero. The potentiallike energy density instead vanishes for B ∈ {0, H} such that the part of the wave 0 ≤ B ≤ H is more stable than the upper part. 3.2. The two-soliton solution We truncate the expansion of Eq. (16) up to a fourth-order of parameter ε. Therefore, we obtain G = A1 exp(η1 ) + A2 exp(η2 ) +C12 exp(η1 + 2η2 ) + C21 exp(η2 + 2η1 ), F = 1 + B11 exp(2η1 ) + B22 exp(2η2 ) +B12 exp(η1 + η2 ) + E12 exp 2(η1 + η2 ),

(24a) (24b)

where B11

A21 κ1 = , 16ω1

A22 κ2 B22 = , 16ω2 11

+ A1 A2 κ12 B12 = , 2ω+12

(25a)

B

(a) 54 231

4 2

20

t

0

0 −2

−20

C

−4 −40

Figure 6: The interaction process (Panel (a)) for two “dissimilar” multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.35) and (A 2 = 1, κ2 = 0.6) and the amplitude ratio r = 0.5834. The panel (b) presents the density plot of the previous interaction process. From this depiction, the small soliton is shifted forwards whereas the larger one is shifted backwards by the interaction. Also, the interaction area is estimated as t ∈ [−3, 3] and C ∈ [−17, 5] yielding a magnitude of about tC = 132.

+ + ω−12 κ12 ω−12 κ12 , C = A B 21 2 11 + − , − ω+12 κ12 ω12 κ12  − + 2 ω κ12 = B11 B22 12 , − ω+12 κ12

C12 = A1 B22

(25b)

E 12

(25c)

with ηi = κi x + ωi t + η0i and ωi κi − κi2 = 1, κi > 0, (i = 1, 2). The quantities ω±12 ± and κ12 are defined by ω±12 = ω1 ± ω2 ,

± 1/κ12 = 1/κ1 ± 1/κ2 .

(26)

A two-soliton solution feature is presented in Fig. 4 which describes the interaction between two similar amplitudes with the ratio r = 0.85. As we can see in this figure, the two single solitary waves seem to attract elastically each other while moving to the lhs of the C-axis. However, at the interaction area, it happens as the two waves head-on and partly coalesce together while exchanging amplitudes, but they do not overlap. Actually, the small soliton travels along the large one before being shifted. The density plot presented by Fig. 4 clearly represents such a phenomenon. Besides, the computation of the energy densities 12

B

(a) 86 42

2 1

40 20

0

t

0 −20

−1

−40

C

−2 −60

Figure 7: The interaction process (Panel (a)) for two “dissimilar” multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.2) and (A 2 = 1, κ2 = 0.6), and the amplitude ratio r = 0.3334. The panel (b) presents the density plot of the previous interaction process. From this depiction, the small soliton is shifted forwards whereas the larger one is shifted backwards by the interaction. Also, the interaction area is estimated as t ∈ [−2, 2] and C ∈ [−20, 5] yielding a magnitude of about tC = 100.

of these waves is also depicted in Fig. 5 (panel (a)). The relative stable part of the two-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −0.04405 < V < 0. As depicted in Fig. 5, this stable part is given by 0 ≤ B ≤ 1.055. Another typical two-soliton solution feature is presented in Fig. 6 which describes the interaction between two dissimilar amplitudes with the ratio r = 0.5834. As we can see in this figure, the two single solitary waves attract elastically each other while moving to the lhs of the C-axis. However, at the interaction area, it happens as the two waves head-on and do not coalesce together, but exchange amplitudes. Actually, the small soliton travels along the large one before being shifted. In this typical feature, the interaction area is less than the previous one. The density plot presented by Fig. 6 clearly represents such a phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 5 (panel (b)). The relative stable part of the two-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −0.1328 < V < 0. As depicted in Fig. 5, this stable part is given by 0 ≤ B ≤ 1.337. It appears from the two previous features that as the amplitude ratio decreases, the interacting area also decreases. Let us reduce the amplitude ratio 13

16

(a)

14

(b)

20

Energy density

Energy density

12 10 8 6 4

15 10 5

2 0

0 0.5

1

1.5

2 B

2.5

3

3.5

1

2

3

4

5 B

6

7

8

9

Figure 8: Panel (a) presents the energy density from the interaction process for two multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.2) and (A 2 = 1, κ2 = 0.6). Panel (b) presents the energy density from the interaction process for three multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.2), (A2 = 1, κ2 = 0.51) and (A 3 = 1, κ3 = 0.6). In the figure, the black, blue and magenta colors represent the total energy, the potential energy and the kinetic energy densities of the system, respectively.

to 1/3 and observe what happens. In this typical case, we consider a two-soliton solution feature presented in Fig. 7 with details in the figure caption. This figure describes the interaction between two dissimilar amplitudes with the ratio r=1/3. As we can see in this figure, the two single solitary waves attract elastically each other while moving to the lhs of the C-axis. However, at the interaction area, it happens as the two waves head-on and do not coalesce together, but exchange amplitudes. Actually, the small soliton travels along the large one before being shifted. In this typical feature, the interaction area is reduced significantly. The density plot presented by Fig. 7 clearly represents such a phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 8 (panel (a)). The relative stable part of the two-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −1.037 < V < 0. As depicted in Fig. 8, this stable part is given by 0 ≤ B ≤ 3.47. As a result of analysis, the interaction between two loop-like waves is attractive. The waves interact elastically while retrieving their initial properties. However, as the features move from similar amplitudes to dissimilar amplitudes, the scattering area reduces significantly. Let us analyze further whether this observation also prevails in the case of multisoliton interaction. 14

B

(a) 86 42

4 2

t

50

0

0 −2

−50 −4

C

−100

Figure 9: Interaction process of three multivalued waveguide channels with (A 1 = 1, κ1 = 0.2), (A2 = 1, κ2 = 0.51) and (A 3 = 1, κ3 = 0.6) so that the amplitude ratios r 21 = 0.3922, r31 = 0.333 and r 32 = 0.85 (panel (a)). The density plot (panel (b)) presents the shifts in space of the individual excitations in interactions. We can see that only the larger soliton is shifted backwards by the interaction. Also, the interaction area is estimated as t ∈ [−10, 11] and C ∈ [−48, 25] yielding a magnitude of about tC = 1533.

3.3. The three-soliton solution The one-soliton solution to Eq. (1) is obtained from the following truncation G = A1 exp(η1 ) + A2 exp(η2 ) + A3 exp(η3 ) + C12 exp(η1 + 2η2 ) +C13 exp(η1 + 2η3 ) + C21 exp(η2 + 2η1 ) + C23 exp(η2 + 2η3 ) +C31 exp(η3 + 2η1 ) + C32 exp(η3 + 2η2 ) + C123 exp(η1 + η2 + η3 ) +D123 exp(η1 + 2η2 + 2η3 ) + D231 exp(η2 + 2η3 + 2η1 ) +D312 exp(η3 + 2η1 + 2η2 ), (27a) F = 1 + B11 exp(2η1 ) + B22 exp(2η2 ) + B33 exp(2η3 ) +B12 exp(η1 + η2 ) + B13 exp(η1 + η3 ) + B23 exp(η2 + η3 ) +E 12 exp 2(η1 + η2 ) + E13 exp 2(η1 + η3 ) + E23 exp 2(η2 + η3 ) +F 123 exp(2η1 + η2 + η3 ) + F231 exp(2η2 + η3 + η1 ) (27b) +F 312 exp(2η3 + η1 + η2 ) + Q123 exp 2(η1 + η2 + η3 ), where Bii =

A2i κi , (i = 1, 2, 3), 16ωi 15

(28a)

B

(a) 54 231

10 5

50

t

0

0 −5

−50

C

−10

Figure 10: Interaction process of three multivalued waveguide channels with (A 1 = 1, κ1 = 0.35), (A2 = 1, κ2 = 0.4) and (A 3 = 1, κ3 = 0.6) so that the amplitude ratios r 21 = 0.8750, r31 = 0.5833 and r32 = 0.667 (panel (a)). The density plot (panel (b)) presents the shifts in space of the individual excitations in interactions. We can see that only the smaller soliton is shifted forwards by the interaction. Also, the interaction area is estimated as t ∈ [−8, 8] and C ∈ [−55, 38] yielding a magnitude of about tC = 1488.

Bi j = Ci j

Ai A j κi+j

, (i  j = 1, 2, 3), 2ω+i ω−i j κi+j = Ai B j j + − , (i  j = 1, 2, 3), ωi j κi j

C123 =

3 

Ai B jk

i jki=1

ω−i j ω−ik

, ω+i j ω+ik

Q123 =

(28b) (28c) E 12 E 13 E 23 , B11 B22 B33

Ci j C ji , (i  j = 1, 2, 3), Ai A j Ci j Cik E jk = , (i  j, i  k, j  k = 1, 2, 3), Ai B j j Bkk C ji Cki B jk = , (i  j, i  k, j  k = 1, 2, 3), Bii A j Ak

Ei j = Di jk F i jk

(28d) (28e) (28f) (28g)

with ηi = κi x + ωi t + η0i and ωi κi − κi2 = 1, κi > 0, (i = 1, 2, 3). The quantities ω±i j and κi±j are defined by ω±i j = ωi ± ω j ,

1/κi±j = 1/κi ± 1/κ j . 16

(29)

(a)

6

(b)

20

Energy density

Energy density

5 4 3 2

15

10

5

1 0

0 0.5

1

1.5

2 B

2.5

3

3.5

1

2

3

4

5 B

6

7

8

9

Figure 11: Panel (a) presents the energy density from the interaction process for three multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.35), (A 2 = 1, κ2 = 0.4) and (A3 = 1, κ3 = 0.6). Panel (b) presents the energy density from the interaction process for three multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.2), (A1 = 1, κ1 = 0.35) and (A2 = 1, κ2 = 0.6). In the figure, the black, blue and magenta colors represent the total energy, the potential energy and the kinetic energy densities of the system, respectively.

We depict a typical three-soliton feature as presented in Fig. 9 which describes the interaction between three amplitudes with the amplitude ratios r 21 = 0.3922, r31 = 0.333 and r32 = 0.85. As we can see in this figure, the three single solitary waves attract elastically each other while moving to the lhs of the C-axis. In the interacting zone the small solitons travel along the large one while exchanging amplitudes. The density plot presented by Fig. 9 clearly represents such a phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 8 (panel (b)). The relative stable part of the threesoliton profile is obtained for the potential-like energy density of the two-wave satisfying to −2.986 < V < 0. As depicted in Fig. 8, this stable part is given by 0 ≤ B ≤ 6.925. Another typical two-soliton solution feature is presented in Fig. 10 which describes the interaction between three amplitudes with the amplitude ratios r21 = 0.8750, r31 = 0.5833 and r32 = 0.667. As we can see in this figure, the three single solitary waves attract elastically each other while moving to the lhs of the C-axis. In the interacting zone the small solitons travel along the large one while exchanging amplitudes. Compared to the previous one, the interaction area is reduced. The density plot presented by Fig. 10 clearly represents such a 17

B

(a) 86 42

3 2 50

1 0

t

0

−1 −50

−2

C

−3

Figure 12: Interaction process of three multivalued waveguide channels with (A 1 = 1, κ1 = 0.2), (A2 = 1, κ2 = 0.35) and (A 3 = 1, κ3 = 0.6) so that the amplitude ratios r 21 = 0.5714, r31 = 0.333 and r32 = 0.5833 (panel (a)). The density plot (panel (b)) presents the shifts in space of the individual excitations in interactions. We can see that only the smaller soliton is shifted forwards by the interaction. Also, the interaction area is estimated as t ∈ [−3, 3] and C ∈ [−37, 14] yielding a magnitude of about tC = 306.

phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 11 (panel (a)). The relative stable part of the three-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −0.5194 < V < 0. As depicted in Fig. 11, this stable part is given by 0 ≤ B ≤ 2.687. In the wake of the previous depiction, we consider another situation presented as follows. We depict a typical three-soliton feature as presented in Fig. 12 which describes the interaction between three amplitudes with the amplitude ratios r21 = 0.5714, r31 = 0.333 and r32 = 0.5833. As we can see in this figure, the three single solitary waves attract elastically each other while moving to the lhs of the C-axis. In the interacting zone the small solitons travel along the large one while exchanging amplitudes. Compared to the above one, the interaction area is reduced. The density plot presented by Fig. 12 clearly represents such a phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 11 (panel (b)). The relative stable part of the three-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −2.987 < V < 0. As depicted in Fig. 11, this stable part is given by 0 ≤ B ≤ 6.935. Finally, we consider one further typical three-soliton feature as presented 18

B

(a) 3 2 1 20

t

0 50 −20

0 −50

C

Figure 13: Interaction process of three multivalued waveguide channels with (A 1 = 1, κ1 = 0.2), (A2 = 1, κ2 = 0.3) and (A 3 = 1, κ3 = 0.6) so that the amplitude ratios r 21 = 0.666, r31 = 0.333 and r32 = 0.5 (panel (a)). The density plot (panel (b)) presents the shifts in space of the individual excitations in interactions. We can see that only the larger soliton is shifted backwards by the interaction. Also, the interaction area is estimated as t ∈ [−27, 27] and C ∈ [−64, 48] yielding a magnitude of about tC = 6048.

in Fig. 13 which describes the interaction between three amplitudes with the amplitude ratios r21 = 0.9107, r31 = 0.85 and r32 = 0.9334. As we can see in this figure, the three single solitary waves attract elastically each other while moving to the lhs of the C-axis. In the interacting zone the small solitons travel along the large one while exchanging amplitudes. Compared to the previous ones, the interaction area here has increased significantly. The density plot presented by Fig. 13 clearly represents such a phenomenon. Besides, the computation of the energy densities of these waves is also depicted in Fig. 14 (panel (a)). The relative stable part of the three-soliton profile is obtained for the potential-like energy density of the two-wave satisfying to −0.08449 < V < 0. As depicted in Fig. 14, this stable part is given by 0 ≤ B ≤ 1.15. As a result from the analysis of the previous three-features, the interactions are always attractive and elastic. It also appears that when there are many dissimar amplitudes, the interaction area enlarges. We showed previously that the magnetization in the ferrite can take different values on a localization point. However, the computation of the energy functional of the system reveals the stable amplitudes of the observable which actually corresponds to a broken shape where the upper part of the loop-profile is discarded. What is currently known about the interactions between loop-shaped soli19

4

(a)

Energy density

3

2

1

0

−1 0

0.5

1 B

1.5

2

Figure 14: Panel (a) presents the energy density from the interaction process for three multivalued waveguide channels solutions to Eq. (1) with (A 1 = 1, κ1 = 0.2), (A2 = 1, κ2 = 0.3) and (A3 = 1, κ3 = 0.6). In the figure, the black, blue and magenta colors represent the total energy, the potential energy and the kinetic energy densities of the system, respectively. Panel (b) presents the shift of the larger soliton (Φ 1 ) in a two-scattering feature. From this panel, the larger soliton is always shifted backwards.

tons [38–40] is that they present two base features of scattering, namely, the attraction and the repulsion. The attraction happens when the amplitudes are “dissimilar” and the repulsion occcurs when the amplitudes are “similar”. These two features actually depend upon some characteristic parameter related to the amplitude ratio of the interacting waves, at which the scattering of these structures changes one to another. Nonetheless, in the present work we obtained some typical loop-shaped solitons with the lone property of attraction where the “similarly”/“dissimilarly” amplitudes of interacting waves are strongly related to the “enlargening”/“shortening” of the area of scattering. Now, in the wake of the above depictions, let us discuss properly the previous scattering features within the angle of the computation of the shifts of the individual solitons in interactions. 4. Discussion of the results 4.1. The two-soliton interaction We start-off with the head-on collisions within the two-soliton solution which actually represents the nonlinear interactions between two-individual solitary 20

waves. We need to determine the shifts of the individual waves in this scattering picture. Therefore, the problem is addressed in terms of an asymptotical consideration of the expression of the two-soliton solution given previously. Thus, without any loss of generality, we assume that 0 < κ 1 < κ2 meaning that the larger soliton is moving faster with the velocity c 1 than the smaller soliton of velocity c2 < c1 . Then, we derive the following expressions. (a). We consider that only the phase η2 varies. Naturally, the phase η1 is constant. • In the limit t → −∞, i.e. η2 → +∞, it comes: A1 √ sech (η1 + φ1 ) , 2 B11 C → x − 4/κ2 − (2/κ1 )[tanh(η1 + φ1 ) + 1] + C0 , B →

(30a) (30b)

where φ1 = (1/2) ln(E 12 /B22). • In the limit t → +∞, i.e. η2 → −∞, it comes: A1 √ sech (η1 + (1/2) ln B11 ) , 2 B11 C → x − (2/κ1 )[tanh(η1 + (1/2) ln B11 ) + 1] + C0 . B →

(31a) (31b)

The shift of the previous soliton is hence Φ1 = −(1/2κ1 ) ln(E 12 /B11 B22 ) − 4/κ2 .

(32)

(b). We now consider that only the phase η1 varies. • In the limit t → −∞, i.e. η1 → −∞, it comes: A2 √ sech (η2 + (1/2) ln B22 ) , 2 B22 C → x − (2/κ2 )[tanh(η2 + (1/2) ln B22 ) + 1] + C0 . B →

(33a) (33b)

• In the limit t → +∞, i.e. η1 → +∞, it comes: A2 √ sech (η1 + φ2 ) , 2 B22 C → x − 4/κ1 − (2/κ2 )[tanh(η2 + φ2 ) + 1] + C0 , B →

(34a) (34b)

where φ2 = (1/2) ln(E 12 /B11). The shift of the previous soliton is hence Φ2 = (1/2κ2 ) ln(E 12 /B11 B22 ) + 4/κ1 . 21

(35)

(b) −1

−1.5

0.8

Ψ1

−1 −2

−2

0.6 −2.5

−3 0.4 0.8 0.6

0.2

0.4

κ2

κ1

−3

0.2

Figure 15: Panel (a) presents the shift of the smaller soliton (Φ 2 ) in a two-scattering feature. From this panel, the smaller soliton is always shifted forwards by the interaction. Panel (b) instead presents the shift of the largest soliton (Ψ 1 ) in a three-scattering feature. From this panel, this soliton is always shifted backwards by the interaction.

From Eqs. (32) and (35), the first term of the shifts is due to two loops attracting each other, or traveling along another loop. The second terms show the shifts caused by the nonlinear interaction between the solitons. The plots of Φ1 and Φ2 as functions of the wave-numbers κ1 and κ2 are shown in Figs. 14 (panel (b)) and 15. It is shown that Φ1 < 0 meaning that the larger loop soliton is always shifted backwards by the interaction. However, for the shift Φ 2 , we find that the smaller loop soliton is always shifted forwards by the interaction. The illustrations are made upon the previous density plots of the two-wave features above. 4.2. The three-soliton interaction In order to determine the shifts of the individual waves, we assume 0 < κ 1 < κ2 < κ3 . Thus, we derive the following results. (a). We consider that only the phase η1 is constant. • In the limit t → −∞, i.e. η2 → +∞ and η3 → +∞, it comes: A1 √ sech (η1 + ψ1 ) , 2 B11 + − (2/κ1 )[tanh(η1 + ψ1 ) + 1] + C0 , C → x − 4/κ23 B →

where ψ1 = (1/2) ln(Q123 /E 23 ). 22

(36a) (36b)

(a)

(b)

1

0

4

0

4 3.5

−1

−1

−2 −3

−2

Ψ3

Ψ2

4.5

5

1

5

3

3

2

2.5

1

−4 −3

0.8 0.8

0.6

0.6

0.4

κ2

0.2

0.4 0.2

2 0.8

1.5 0.8

0.6 −4

κ1

0.6

0.4

κ2

0.2

0.4 0.2

1

κ1

Figure 16: Panel (a) presents the shift of the intermediate soliton (Ψ 2 ) in a three-scattering feature. From this panel, this soliton can be shifted either forwards or backwards by the interaction. Panel (b) instead presents the shift of the smallest soliton (Ψ 3 ) in a three-scattering feature. From this panel, this soliton is always shifted forwards by the interaction.

• In the limit t → +∞, i.e. η2 → −∞ and η3 → −∞, it comes: A1 √ sech (η1 + (1/2) ln B11 ) , 2 B11 C → x − (2/κ1 )[tanh(η1 + (1/2) ln B11 ) + 1] + C0 . B →

(37a) (37b)

Consequently, the shift of the previous soliton is + Ψ1 = −(1/2κ1 ) ln(Q123 /B11 E 23 ) − 4/κ23 .

(38)

(b). We now consider that only the phase η2 is constant. • In the limit t → −∞, i.e. η1 → −∞ and η3 → +∞, it comes: A2 √ sech (η2 + ψ2 ) , 2 B22 C → x − 4/κ3 − (2/κ2 )[tanh(η1 + ψ2 ) + 1] + C0 , B →

(39a) (39b)

where ψ2 = (1/2) ln(E 23 /B33 ). • In the limit t → +∞, i.e. η1 → +∞ and η3 → −∞, it comes:

A2  √ sech η2 + ψ¯ 2 , 2 B22 C → x − 4/κ1 − (2/κ2 )[tanh(η1 + ψ¯ 2 ) + 1] + C0 , 23 B →

(40a) (40b)

where ψ¯ 2 = (1/2) ln(E 12 /B11 ). Therefore, the shift of the previous soliton is − Ψ2 = (1/2κ2 ) ln(E 12 B33 /B11 E 23 ) + 4/κ13 .

(41)

(c). Finally, we consider that only the phase η3 is constant. • In the limit t → −∞, i.e. η1 → −∞ and η2 → −∞, it comes: A3 √ sech (η3 + (1/2) ln B33 ) , 2 B33 C → x − (2/κ3 )[tanh(η3 + (1/2) ln B33 ) + 1] + C0 . B →

(42a) (42b)

• In the limit t → +∞, i.e. η1 → +∞ and η2 → +∞, it comes: A3 √ sech (η2 + ψ3 ) , 2 B33 + − (2/κ3 )[tanh(η1 + ψ3 ) + 1] + C0 , C → x − 4/κ12 B →

(43a) (43b)

where ψ3 = (1/2) ln(Q123 /E 12 ). Hence, the shift of the previous soliton is + Ψ3 = (1/2κ3 ) ln(Q123 /B33 E 12 ) + 4/κ12 .

(44)

From Eqs. (38), (41) and (44), the first term is due to three loops attracting each other, or traveling along another loop. The second terms show the shifts caused by the nonlinear interaction between the solitons. The representation of the previous shifts are graphically depicted in Figs. 15 (panel (b)) and 16. From these pictures, it appears that Ψ1 < 0 and Ψ3 > 0 meaning that the largest soliton is always shifted backwards by the interaction while the smallest one is shifted forwards. Since the shift Ψ2 can be positive or negative-definite, the related soliton can be shifted forwards or shifted backwards by the interactions. All these cases are illustrated above while depicting the different scattering features of the three-soliton system. From the previous analysis, the elasic interactions investigated above show that a ferrite material can actually support the propagation and the interaction of localized excitations. These excitations practically refer to data inputs which undergo some fast remagnetization process within magnetic memory devices. Thus, while empowering these systems with great and efficient storage capacity, many inputs can be stored without distorting the initial signals. 24

5. Summary Throughout this work, we have investigated the soliton structure of a new evolution system describing the fast near-adiabatic magnetization dynamics of a nonconducting ferromagnetic slab magnetized to saturation by an in-plane external field in the absence of any eddy currents and with inhomogeneous exchange. After reviewing briefly the physical ground of the system, we have transformed the equation to a bilinear form more suitable for survey. Then, we have constructed the one-soliton-, two-soliton-, and three-soliton solutions to the system. In the same process, we have computed the energy densities of these waveguides. Accordingly, in the case of one-soliton solution, we have shown that the magnetization evolves as a localized multivalued wave. The computation of its energy functionals has shown the stable part of the profile, and this stable part is actually a broken loop with the upper-part discarded. Investigating the nonlinear interactions of such traveling waves, we have found that the scattering process though being complex yields at the final analysis an attraction interaction with well-preserved properties of the initial waves. With the computation of the energy functionals of these structures, we have determined the stable amplitudes of the two-wave and three-wave interaction features. As a result from the analysis of the previous two- and three-features, while evaluating the scattering area, we found that when there are many similar amplitudes in head-on process, the interaction area enlarges. However, these features actually contrast with what is currently known about the interactions between loop-shaped solitons [38–40] as referring to two base features of scattering, namely, the attraction and the repulsion which strongly depend upon some characteristic parameter. The attraction discussed previously occurs between “similar”/“dissimilar” amplitudes of interacting waves whereby resorting to the enlargening/shortening of the area of scattering. Following the depiction of the previous elastic features, we discussed the results within the viewpoint of the computation of the shifts of each individual wave. We found that these shifts comprise essentially two parts: the first term due to attraction and traveling along one another loop and the second term caused by the nonlinear interaction. From the viewpoint of physical implication, the elasic interactions investigated above have shown that a ferrite material can actually support the propagation and the interaction of localized excitations, which stand for data inputs undergoing some fast remagnetization process within magnetic memory devices. Thus, while empowering these systems with great and efficient storage capacity, many inputs can be stored without distorting the initial signals. The results pre25

sented above can hence be useful in the understanding of the data storage process elements in magnetic devices. More applications can also be found in soliton theory [35–37] with elastic and inelastic processes induced by the magnetization dynamics. Through the description of a magnetic moment coupled to a heat bath, Brown [41] established the range of validity of the LLG equation. Many models of the generalized LLG equation are currently under investigations. As a matter of illustration, enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum, a new system has been derived [42]. Besides, it has been shown previously that the inclusion of the spin torque term in the LLG equation provides more information about the magnetization relaxation processes described by the radiation-spin interaction where the radiation field is produced by the magnetization precessional motion itself [43]. For more perspective, we arguably beleive that it would be interesting to consider those generalizations to investigate more about the magnetization dynamics. References [1] V.K. Kuetche, F.T. Nguepjouo, T.C. Kofane, J. Magn. Magn. Mater. 3374 (2015) 1. [2] B. Hillebrands, K. Ounadjela, Spin Dynamics in Confined Magnetic Structures, Springer, Berlin, 2002. [3] D.C. Mattis, Theory of Magnetism: I. Statics and Dynamics, Springer, Berlin, 1988 [4] D.C. Mattis, Theory of Magnetism: II. Thermodynamics and Statistical Mechanics, Springer, Berlin, 1985. [5] M. Johnson, R.H. Silsbee, Phys. Rev. Lett. 55 (1985) 1790. [6] A. Brataas, A.D. Kent, H. Ohno, Nature Mater. 11 (2012) 372. [7] Y.S. Kivshar, B.A. Malomed, Rev. Mod. Phys. 61 (1989) 763. [8] D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson, D. Petit, R.P. Cowburn, Science 309 (2005) 1688. [9] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von MolnAr, M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [10] S. Parkin, M. Hayashi, L. Thomas, Science 320 (2008) 190. [11] S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, N. Ishiwata, J. Appl. Phys. 103 (2008) 07E718. [12] A. Pushp, T. Phung, C. Rettner, B. Hughes, L. Thomas, S.-H. Yang, S. Parkin, Nat. Phys. 9 (2013) 505. [13] G.S.D. Beach, M. Tsoi, J.L. Erskine, J. Magn. Magn. Mater. 320 (2008) 1272. [14] T.A. Moore, M. Klui, L. Heyne, P. M¨ohrke, D. Backes, J. Rhensius, U. R¨udiger, L.J. Heyderman, J.-U. Thiele, G. Woltersdorf et al., Phys. Rev. B 80 (2009) 132403. [15] S. Lepadatu, J.S. Claydon, C.J. Kinane, T.R. Charlton, S. Langridge, A. Potenza, S.S. Dhesi, P.S. Keatley, R.J. Hicken, B.J. Hickey et al., Phys. Rev. B. 81 (2010) 020413. [16] G.E.W. Bauer, S. Bretzel, A. Brataas, Y. Tserkovnyak, Phys. Rev. B 81 (2010) 024427. [17] K.-S. Ryu, L. Thomas, S.-H. Yang, S. Parkin, Nat. Nanotechnol. 8 (2013) 527.

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