Singular optics of spin waves in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy Yu. I. Gorobets, and O. Yu. Gorobets Citation: Low Temperature Physics 43, 564 (2017); View online: https://doi.org/10.1063/1.4985212 View Table of Contents: http://aip.scitation.org/toc/ltp/43/5 Published by the American Institute of Physics Articles you may be interested in Features of the magnetic state of ensembles of nanoparticles of substituted manganites: Experiment and model calculations Low Temperature Physics 43, 570 (2017); 10.1063/1.4985215 Magnetic-field induced phase transitions in intermetallic rare-earth ferrimagnets with a compensation point Low Temperature Physics 43, 551 (2017); 10.1063/1.4985214 Ultrasound propagation in bond frustrated HgCr2S4 spinel in magnetic fields Low Temperature Physics 43, 559 (2017); 10.1063/1.4985206 Magnetism of polyanionic compounds of transition metals (Review Article) Low Temperature Physics 43, 529 (2017); 10.1063/1.4985210 Comparative Raman scattering study of Ba3MSb2O9 (M = Zn, Co and Cu) Low Temperature Physics 43, 543 (2017); 10.1063/1.4985205 Low-temperature absorption spectra and electron structure of HoFe3(BO3)4 single crystal Low Temperature Physics 43, 610 (2017); 10.1063/1.4985208 LOW TEMPERATURE PHYSICS VOLUME 43, NUMBER 5 MAY 2017 Singular optics of spin waves in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy Yu. I. Gorobetsa) Institute of Magnetism, National Academy of Sciences and Ministry of Education and Science of Ukraine, Kiev 03142, Ukraine O. Yu. Gorobetsb) Institute of Magnetism, National Academy of Sciences and Ministry of Education and Science of Ukraine, Kiev 03142, Ukraine and National Technical University of Ukraine “KPI,” Kiev 03056, Ukraine (Submitted September 30, 2016) Fiz. Nizk. Temp. 43, 707–713 (May 2017) Based on exact 3D solutions of the Landau-Lifshitz equations in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy, the existence of nonlinear spin waves with singular points on the wavefront is predicted. These waves are spin-wave analogs of optical singularities. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4985212] recently been experimentally demonstrated by creating radiofrequency vortices with two different states of orbital angular momentum.13 This latest technology, in principle, allows the transmission of an unlimited number of channels within a given fixed frequency band.13 In view of new technical possibilities for detecting the phase and observing the spin-wave front, this paper theoretically shows the possibility of propagation of nonlinear spin waves with modulated fronts and threedimensional solitons in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy based on three-dimensional nonlinear solutions of the Landau-Lifshitz equations. In particular, the modulation of the front of such nonlinear spin excitations can contain wavefront singularities of the type of optical screw dislocations of different orders,14,15 as well as the plane singular wavefield points of the circulation, source, and sink types.16 Introduction Recently, high resolution, both spatially and temporally, techniques for measuring the intensity distribution of spin waves have been successfully developed.1–4 Using these techniques, the behavior of one- and two-dimensional spin-wave packets—envelope and bullet spin-wave solitons—has been studied, including their free propagation, collisions, parametric excitation and wavefront reversal.5–7 Lately, experimental approaches have been developed that allow to obtain information about the phase of wave packets, therefore enabling the observation of a spin-wave front with a total two-dimensional phase resolution and detection of the phase of linear and nonlinear spin-wave packets.8 At present, the inversion of the wavefront of spin-wave pulses is effectively used for processing ultra-high frequency signals.9 The use of peculiarities of the wavefront of spin waves for the transmission of information has mainly been investigated in ferromagnetic materials and ferrites. However, new promising materials for spin-wave electronics—antiferromagnets—have also been considered since they allow operating at higher frequencies.10 In addition, in connection with the technical capabilities of recording the phase and observing the wavefront of spin waves, it seems relevant to apply the methods developed and realized in singular optics (i.e., in the wave optics of helical fields)11 to create singularities on the front of linear and nonlinear spin waves for recording and transmitting information. To date, the ideas of singular optics have been successfully used for a wide range of electromagnetic wave lengths, as well as for wave fields of a different nature. For instance, such examples include singular electron optics12 and vorticity of radio waves.13 In the latter case, the possibility of coding and simultaneous transmission of several radio channels at the same frequency has Main section Let us consider a two-sublattice antiferromagnet with uniaxial magnetic anisotropy and sublattice magnetizations M1 ¼ M2, jM1j ¼ jM2j ¼ M0, where the magnetization modulus of both sublattices is M0 ¼ const. Taking into account the conditions for the constancy of the modulus of the antiferromagnetism vector jLj ¼ L0 ¼ const, we select a parametrization Lx ¼ 2M0 sin h cos u; Lz ¼ 2M0 cos h; Ly ¼ 2M0 sin h sin u; where h and u are the polar and azimuth angles for the antiferromagnetism vector and Lx, Ly, and Lz are the Cartesian coordinates of the antiferromagnetism vector. Then the Landau-Lifshitz equations describing the dynamics of the vector L have the form17 8 > @ @u > 2 > c2 div ðruÞ sin2 h ¼ 0 > < @t sin h @t xH " 2 # 2 > @ h @u > 2 2 2 2 2 > > sin h cos h ¼ 0; : @t2 c r h þ x0 sgnðb1 Þ þ c ðruÞ @t xH 1063-777X/2017/43(5)/6/$32.00 (1) 564 (2) Published by AIP Publishing. Low Temp. Phys. 43 (5), May 2017 Yu. I. Gorobets and O. Yu. Gorobets 565 where xH ¼ gH0, g ¼ 2lh 0 (l0 is the Bohr magneton, ¯ is the Planck constant, and H0 is the external magnetic field strength), ( sgnðb1 Þ ¼ 1; b1 > 0; 1; b1 < 0; c¼ 4l0 M0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4l M0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Aa1 ; x0 ¼ 0 Ajb1 j; h h (3) where A is the homogeneous exchange energy constant, a1 is the inhomogeneous exchange constant, and b1 is the uniaxial magnetic anisotropy constant. Equation (2) has the following particular three-dimensional nonlinear solutions18–21 [the derivation of Eq. (2) and the method of obtaining particular solutions are considered in Ref. 22]: 8 z vP t x y > > ; þf > < h ¼ 2arctanfH Pð x; y; zÞ g; Pð x; y; zÞ ¼ p l0 l0 l0 Ð > z vQ t x y > > þ xH ðtÞdt; ; þg :u ¼ q l0 l0 l0 where x, y, and z are the Cartesian coordinates of the radius vector to an arbitrary point in the antiferromagnet, v is the qﬃﬃﬃﬃﬃﬃ velocity of a nonlinear spin wave, l0 ¼ jba1 j ¼ xc00 and the 1 parameters p and q are determined by the expressions sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sgnðb1 Þ p ¼ 0; q ¼ 6 1 v2Q =c2 (5) or sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sgnðb1 Þ p¼6 ; q ¼ 0; 1 v2P =c2 (6) and the function H(P) has the form H ð PÞ ¼ b0 pﬃﬃﬃﬃﬃﬃﬃﬃ ; dn c0 jC1 jP; k1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 1þ4C1 1þ2C1 1þ4C1 ¼ where c0 ¼ 1þ2C12jC , b , 0 2jC1 j 1j qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1þ4C 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1þ2C , 14 < C1 < 0, and 0 < k1 1, or 1 þ 1þ4C1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 snðP; k2 Þ H ð PÞ ¼ ; 1 þ snðP; k2 Þ (7) k1 (8) 1 , C1 > 0 and 0 < k2 1. The functions where k2 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ4C 1 f(X, Y) and g(X, Y) have the form 8 X X 2 > > n~~ lnðjr r0i jÞ þ k1;2 K ðk1;2 Þ n~i ai þ C2 f ð X; Y Þ ¼ > > > p > i i > > > > X X AðiÞ ð Þ > ð iÞ > i n > þ B cos na þ C sin na ; > i i n < jr r0i jn n n i X X 2 > > > ai n~~i þ C3 n~i lnðjr r0i jÞ þ gð X; Y Þ ¼ k1;2 K ðk1;2 Þ > > p > i i > > > > X X AðiÞ ð Þ > > ð iÞ i n > þ C cos na þ B sin na ; > i i n n n : jr r0i j n i (9) where the following notation is introduced: X¼ (4) x y ; Y¼ ; l0 l0 (10) where r is a two-dimensional vector with the coordinates r ¼ (X, Y) in the XOY plane, r0i is a two-dimensional vector with the coordinates in the XOY plane perpendicular to the propagation direction of the spin wave—r0i ¼ (X0i, Y0i) where X0i and Y0i are some dimensionless constants, YY0i , i, n, n~i , n~~ i are integers, and ai ¼ arctan XX 0i 8 > < 0; HðnÞ ¼ > : 1; p n0 n>0 ð2 ; dn K ðkÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 1 k2 sin2 n (11) 0 Note that the expression for the function f(X, Y) in Eq. (9) is an expansion in powers of jr r0i j of an arbitrary harmonic function of two variables X and Y, and the expression for the function g(X, Y) in Eq. (9) represents an expansion in powers of jr r0i j of a harmonic function of the same two variables X and Y, which is the conjugate function with respect to the function g(X, Y). This means that the functions f(X, Y) and g(X, Y) are connected by the Cauchy-Riemann conditions22 and are the eigenfunctions of the twodimensional Laplace operator. It should be noted that in the static case (i.e., for @h @t ¼ 0 ¼ 0), Eq. (2) describing the coordinate dependences and @u @t of the polar and azimuth angles of the antiferromagnetism vector in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy coincide with the corresponding equations for the spatial distribution of the polar and azimuth angles of the magnetization vector in a ferromagnet with uniaxial magnetic anisotropy in the exchange approximation17 (i.e., in the case when the magnetostatic energy of the ferromagnet can be neglected17) Therefore, the functional form of all the solutions of Eq. (2) obtained in the present paper for vP ¼ 0 and vQ ¼ 0 also extends to the case of static magnetization distributions of a ferromagnet with an easy-axis anisotropy or “easy plane.” In this sense of the word, in analyzing the particular cases of solutions (4), we will also discuss the known analogous solutions in a ferromagnet. Let us analyze the obtained three-dimensional nonlinear solutions (4) of the Landau-Lifshitz equations for a 566 Low Temp. Phys. 43 (5), May 2017 Yu. I. Gorobets and O. Yu. Gorobets two-sublattice antiferromagnet with uniaxial magnetic anisotropy (2). For example, let us choose the function H(P) in the form (8) and consider the external magnetic field H0 ¼ 0. Also, for definiteness, we consider the case of magnetic anisotropy of the easy-axis type with the parameters p and q determined by Eq. (6). Then the projections of the antiferromagnetism vector on the axes of the Cartesian coordinate system take the form 8 ! > 1 z vP t x y x y > > p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ > ; k ; ¼ 2M cn ; ; L þ f cos g x 0 2 > > l0 l0 l0 l0 l0 1 v2P =c2 > > > > > > ! < 1 z vP t x y x y ﬃ þ f ; k ; Ly ¼ 2M0 cn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ; 2 sin g l0 l0 l0 l0 l0 > 1 v2P =c2 > > > > > ! > > 1 z vP t x y > > ﬃ L ¼ 2M0 sn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ; k2 : þf > > : z l0 l0 l0 1 v2P =c2 Expression (12) represents a nonlinear propa spin wave gating along the Oz axis. For f x y l0 ; l0 ¼g x y l0 ; l0 ¼ 0, solu- tion (12) is a nonlinear plane wave. It is known that the wavefront of the wave beams that are similar in properties to a plane wave looks like a family of disjoint surfaces. The distance between the adjacent surfaces wave is equal to the x y x y length. Furthermore, the functions f l0 ; l0 and g l0 ; l0 represent the modulation of the plane wavefront. In optics, the deviations of wavefronts from planarity occurring in real beams are called optical aberrations. However, all the aberrations considered in the classical theory deform a wavefront without changing its topology. A different picture is observed in the presence of optical vortices in a monochromatic wave.11–15 If such vortices appear, then singular points arise on the wavefront surface, which are in many respects analogous to crystal lattice defects known in solid state physics—screw dislocations—and have the same name.11–15 At the very singular point, the amplitude of light oscillations is zero, and the phase value is not defined. In its vicinity, there are sharp collapsing phase changes. Because of this singularity, the phase distribution function belongs to the class of singular functions, which is the reason for the term “singular optics” mentioned above. As will be shown below, a spin wave of the type (12) with a specific choice of the functions x y x y f l0 ; l0 and g l0 ; l0 can contain singular points, including screw dislocations, on the surface of the wavefront [if we use the terminology adopted for electromagnetic waves in the optical range for the spin waves of the type (4)]. Thus, in optics, wavefront points such that the phase of light oscillations along the path traced around them in a plane perpendicular to the propagation direction changes by 2pl, where l is a non-zero integer are called screw dislocations. The quantity l is called the dislocation order or the topological charge of the wavefront surface. The amplitude of the electric field E(r, a) near optical screw dislocations r ! 0 of the order l is described by the expression E(r, a) ¼ rlexp(6ila).14 Depending on the direction of the twist of the wave surface “screw,” screw dislocations are classified (12) as left (negative) and right (positive) dislocations. On the surface of the wavefront, there can be both a single screw dislocation and an entire system of dislocations. The appearance of screw dislocations radically changes the topology of the wave front. The equiphase surface ceases to be multi-sheeted, and a transition to a single surface with a specific helical structure takes place (Fig. 1).11–15 To describe analogous singularities on the front of a spin wave in Eq. (12), we consider the oscillations of the antiferromagnetism vector in the vicinity of a singular point in the plane perpendicular to the propagation direction of the spin wave z ¼ vpt. Here and below, the presence of the index i in the respective coefficients actually means the possibility of describing the system of singular points with radius vectors r ¼ r0i in a plane perpendicular to the propagation direction of the spin wave. In this case, we take the asymptotes for jr r0ij ! 0 in Eq. (9) for the full expansion of the harmonic function f x y l0 ; l0 and its conjugate harmonic function g x y l0 ; l0 in powers of jr r0ij. Let us consider such a perturbation of the wave front, for which the coefficients at ln(jr r0ij) and negative integer powers of jr r0ij vanish. Substituting this asymptotes into Eq. (12), we obtain: Fig. 1. Wavefront structure in the absence (a) and in the presence of a screw dislocation (b); k is the wavelength.11–15 Low Temp. Phys. 43 (5), May 2017 Yu. I. Gorobets and O. Yu. Gorobets 8 ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ > > L ¼ 2M0 cnðAl jr r0i jl ðBl cos nai þ Cl sin lai Þ; k2 Þ cos ðAl jr r0i jl ðCl cos lai Bl sin lai ÞÞ; > < x ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ Ly ¼ 2M0 cnðAl jr r0i jl ðBl cos lai þ Cl sin lai Þ; k2 Þ sin ðAl jr r0i jl ðCl cos lai þ Bl sin lai ÞÞ; > > > : L ¼ 2M cnðAðiÞ jr r jl ðBðiÞ cos la þ CðiÞ sin la Þ; k Þ: z 0 0i l i l l In Eq. (13), from the whole series (9) of the expansion of a harmonic function in powers of jr r0ij for jr r0ij ! 0, only the term with n ¼ l remains, where l > 0 is the minimum positive power of jr r0ij in the respective series. Also, taking into account the condition jr r0ij ! 0, in Eq. (13) the trigonometric as well as elliptic sine and cosine can be expanded in a Taylor series up to linear terms in the argument of these functions 8 > > < Lx 2M0 ; ðiÞ ðiÞ ðiÞ Ly ¼ 2M0 Al jr r0i jl ðCl cos lai Bl sin lai Þ; (14) > > : Lz ¼ 2M0 AðiÞ jr r0i jl ðBðiÞ cos lai þ CðiÞ sin lai Þ: l l l The expansion (14) is valid if ðiÞ ðiÞ ðiÞ jAl jr r0i jl ðBl cos lai þ Cl sin lai Þj < jKðk20 Þj ðiÞ jCl and ðiÞ Bl cos lai sin lai j < 1; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where k0 ¼ 1 k2 . If we choose a wavefront perturbation for which the ðiÞ ðiÞ ðiÞ coefficient Bl ¼ 0 and denote CðiÞ ¼ Al Cl , then it follows from Eq. (14): 8 > < Lx 2M0 ; Ly 2M0 CðiÞ jr r0i jl cos lai ; (15) > : Lz 2M0 CðiÞ jr r0i jl cos lai : It is evident from Eq. (15) that in the vicinity of a singular point with a radius vector r r0i in the plane perpendicular to the propagation direction of the spin wave, the magnitude of the deviation of the antiferromagnetism vector from the uniform direction tends to zero as jr r0ij ! 0, and for a path enclosing the singular point, the phase increases by 2pl. This allows us to treat Eq. (15) as a spinwave analogy of an optical screw dislocation. Depending on the rotation direction of the antiferromagnetism vector, which is determined by the sign of the topological charge l, Eq. (15) describes left (negative) or right (positive) screw dislocations (Fig. 2). If, in solution (12), we do not assume vanishing coefficients at ln(jr r0ij) or at integer negative powers of jr r0ij, singularities with a nonzero amplitude on the front of a spin wave can be obtained. The following example of the spin-wave singularity illustrates the case for which the amplitude of the deviation of the antiferromagnetism vector from the uniform direction is finite for jr r0ij ! 0, and for a path enclosing the singular point, the phase increases by 2pl: 8 > ~ ~ > < Lx ¼ 2M0 cnðn~ i lnðjr r0i jÞ; k2 Þ cos ðai n~ i þ C3 Þ; ~ Ly ¼ 2M0 cnðAn~ i lnðjr r0i jÞ; k2 Þ sin ðai n~~ i þ C3 Þ; (16) > > : L ¼ 2M snðn~~ lnðjr r jÞ; k Þ: z 0 i 0i 2 i 567 (13) 2 Despite the fact that, except for special cases, the superposition principle is not valid for solutions of nonlinear equations, in solutions (4) the superposition principle for harmonic functions f ðlx0 ; ly0 Þ and gðlx0 ; ly0 Þ under the sign of the function H(P) is realized. Specifically, the sum of harmonic functions is also harmonic, and their expansions into series (9) are possible due to the linearity of the Laplace operator, the eigenfunctions of which they are. Therefore, for a nonlinear spin wave in an antiferromagnet considered in this paper, both a single singular point as well as an entire system of singularities with an arbitrary arrangement on the wavefront surface can appear. Several interesting particular cases of solutions of the type (4) should also be mentioned. It is well known that for the modulus of the elliptic function equal to one, the elliptic functions degenerate into hyperbolic functions, and Eq. (8) for the function H(P) is considerably simplified: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 tanhðPÞ : (17) H ð PÞ ¼ 1 þ tanhðPÞ In this case, in the parametrization (1), the usual trigonometric sine and the cosine of the polar angle for the antiferromagnetism vector are given by the expressions 8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > ¼ 2M sin h cos u ¼ 2M 1 tanh2 ðPÞ cos u; L 0 0 < x qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (18) Ly ¼ 2M0 sin h sin u ¼ 2M0 1 tanh2 ðPÞ sin u; > > > : L ¼ 2M cos h ¼ 2M thðPÞ; z 0 0 where P and u are given by Eq. (4). The expression Lz ¼ 2M0 tanhðPÞ in the last equation resembles a kink-like soliton (monopole).24 Formally, a kink can be introduced as a solution of the Korteweg-de Vries,25 the nonlinear Schr€odinger,26 or the sin-Gordon27 equations, described by a hyperbolic tangent. Inverting the sign of the kink-type solution produces an anti-kink. In the general case, Fig. 2. Two types of singularities for the rotating components of the antiferromagnetism vector: circulation23 according to Eq. (16) for C3 ¼ 6(p/2) and n~~ i ¼ 1 (a); source or sink23 according to Eq. (15) for l ¼1 or Eq. (16) for n~~ j ¼ 1 and C3 ¼ 0 or p (b). 568 Low Temp. Phys. 43 (5), May 2017 Yu. I. Gorobets and O. Yu. Gorobets Lz ¼ 2M 0 tanhðPÞ is a three-dimensional kink, and for x y f l0 ; l0 ¼ 0 and g lx0 ; ly0 ¼ 0 it becomes one-dimensional. Furthermore, Eq. (8) for the function allows a significant simplification when the modulus of the elliptic function is zero since in this case the elliptic functions degenerate into ordinary trigonometric sine and cosine. Taking into account the above limiting case, Eq. (8) for the function H(P) takes the form sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 sin ðPÞ : (19) H ð PÞ ¼ 1 þ sin ðPÞ At the same time, in the parametrization (1), the usual trigonometric sine and cosine of the polar angle for the antiferromagnetism vector are also expressed in terms of trigonometric sine and cosine: 8 > < Lx ¼ 2M0 sin h cos u ¼ 2M0 cosðPÞ cos u; (20) Ly ¼ 2M0 sin h sin u ¼ 2M0 cosðPÞ sin u; > : Lz ¼ 2M0 cos h ¼ 2M0 sin ðPÞ; where P and u are also given by Eq. (4). From the above correspondence between the solutions (4) and analogous solutions for the magnetization distributions in a ferromagnet with uniaxial magnetic anisotropy, it is worth noting that for vP ¼0 inthe one-dimensional case (i.e., for f x y l0 ; l0 ¼ 0 and g x y l0 ; l0 ¼ 0, P ¼ PðzÞ ¼ z l0 ), the substitution of Eq. (18) into Eq. (1) yields a flat domain boundary.28,29 In the same sense, the substitution of Eq. (12) into Eq. (1) in the one-dimensional case for vP ¼ 0 describes the magnetization distribution in a ferromagnet, obtained for the first time by Shirobokov30 8 > z > > Lx ¼ 2M0 cn ; k2 ; > > l0 < Ly ¼ 0; (21) > > z > > > : Lz ¼ 2M0 sn l0 ; k2 : This and the following equations can be used to draw an analogy between the solutions (4) and static distributions of the magnetization of a ferromagnet if we set L ¼ M—the magnetization vector of a ferromagnet, and MS ¼ 2M0 —the saturation magnetization of a ferromagnet. However, these equations simultaneously describe nonlinear static solutions of Eq. (2) in a two-sublattice antiferromagnet. Let us also consider the above correspondence between the solutions (4) and analogous solutions for the magnetization distributions in a ferromagnet with uniaxial magnetic anisotropy in the two-dimensional case (i.e., in the absence of the dependence of the angles h and u on the coordinate z, which occurs in a magnet without anisotropy, i.e., for b1 ¼ 0). Then for vP ¼ 0 and for the following choice of the functions f(X, Y) and g(X, Y) ( f ðX; YÞ ¼ n~~ lnðrÞ; (22) gðX; YÞ ¼ an~~ ; the substitution of Eq. (18) into Eq. (1) yields the well-known two-dimensional Belavin-Polyakov soliton in an isotropic ferromagnet as a special case of the solutions (4)31 8 n~~ > < h 1 ¼ tan ; (23) 2 r > : ~ u ¼ n~ a; or, in a different notation 8 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 u > > u n~~ n~~ > r r > t > > Lx ¼ 2M0 1 ~ cos an~~ ; > > n~ þ r n~~ > r > > > vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > !2 < u ~ ~ u r n~ r n~ t ~ > > Ly ¼ 2M0 1 r n~~ þ r n~~ sin a n~ ; > > > > ! > > > n~~ n~~ > r r > > : > : Lz ¼ 2M0 n~~ r þ rn~~ (24) Thus, for particular soliton cases, the superposition of individual terms in the expansion of the functions f(X, Y) and g(X, Y) in powers of jr r0ij actually represents a “superposition” of the shape modulations of threedimensional moving solitons. Conclusion Special cases of the considered type of the solutions of the Landau-Lifshitz equations for an antiferromagnet with uniaxial magnetic anisotropy, when put side by side with the static solutions for the magnetization distributions of a uniaxial ferromagnet, include well-known solutions of the type of the two-dimensional Belavin-Polyakov solitons,31 onedimensional Shirobokov solitons,30 Bloch domain boundary,28 Khodenkov solitons,32 target soliton,33 as well as some other known nonlinear solutions. Furthermore, this class of solutions of the Landay-Lifshitz equations for an antiferromagnet with uniaxial magnetic anisotropy demonstrates the fundamental possibility of realizing vortex optics based on nonlinear spin waves.34 This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 644 348 (MagIC). a) Email: [email protected] Email: [email protected] b) 1 M. Bauer, O. B€ uttner, S. 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