刘伍明-中国科学院大学-UCAS.pdf
刘伍明-中国科学院大学-UCAS.pdf
PHYSICAL REVIEW B VOLUME 60, NUMBER 18 1 NOVEMBER 1999-II Nonlinear magnetization dynamics of the classical ferromagnet with two single-ion anisotropies in an external magnetic field Wu-Ming Liu Department of Physics, The University of Texas, Austin, Texas 78712; Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100 080, China;* and Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032† Wu-Shou Zhang and Fu-Cho Pu Institute of Physics, Chinese Academy of Sciences, P.O. Box 603-99, Beijing 100 080, China Xin Zhou Department of Mathematics, Duke University, Durham, North Carolina 27706 共Received 6 April 1998; revised manuscript received 17 September 1998兲 By using a stereographic projection of the unit sphere of magnetization vector onto a complex plane for the equations of motion, the effect of an external magnetic field for integrability of the system is discussed. The properties of the Jost solutions and the scattering data are then investigated through introducing transformations other than the Riemann surface in order to avoid double-valued functions of the usual spectral parameter. The exact multisoliton solutions are investigated by means of the Binet-Cauchy formula. The results showed that under the action of an external magnetic field nonlinear magnetization depends essentially on two parameters: its center moves with a constant velocity, while its shape changes with another constant velocity; its amplitude and width vary periodically with time, while its shape is also dependent on time and is unsymmetric with respect to its center. The orientation of the nonlinear magnetization in the plane orthogonal to the anisotropy axis changes with an external magnetic field. The total magnetic momentum and the integral of the motion coincident with its z component depend on time. The mean number of spins derivated from the ground state in a localized magnetic excitations is dependent on time. The asymptotic behavior of multisoliton solutions, the total displacement of center, and the phase shift of the jth peak are also analyzed. 关S0163-1829共99兲07121-0兴 I. INTRODUCTION Nonlinear magnetization dynamics of the classical ferromagnet with two single-ion anisotropies in an external magnetic field can provide an approximative description of various kinds of behavior of magnetic materials as well as the natural starting point for analyzing the anomalous hydrodynamical behavior of low-dimensional magnetic systems. Such fascinating nonlinear dynamic problem exhibits both coherent and chaotic structures depending on the nature of the magnetic interactions, and it is of considerable interest from the point of view of condensed-matter physics, statistical physics, and soliton theory. Nonlinear magnetization dynamics of the classical ferromagnet can be described by the Landau-Lifschitz equation,1 special solutions of which have been derived by many authors: Makamura and Sasada2 found analytic expressions for the permanent profile solitary waves and periodic wave trains; Laksmanan, Ruijgrok, and Thompson3 discussed the spin-wave spectrum and derived also the solitary wave solution. Tjon and Wright4 found that a single-solitary wave is stable with respect to small perturbations and that two colliding ones preserve their identity, thus providing evidence that the solitary wave is a bona fide soliton. Kosevich, Ivanov, and Kovalev5 found a solution by reducing the equation to an appropriate form. Mikeska6 obtained a solution by reducing the equation of motion to a sine-Gordon equation 0163-1829/99/60共18兲/12893共19兲/$15.00 PRB 60 for a ferromagnet with an easy plane. Long and Bishop7 proposed another solution which does not tend to the wellknown solution of an isotropic ferromagnet when an anisotropy parameter vanishes. Zakharov and Takhtajan8 found the equivalence of a nonlinear Schrodinger equation and Landau-Lifschitz equation of an anisotropic ferromagnet. Ivanov, Kosevich, and Babich9 obtained a solution by taking into account only the first-order approximation. Using the Hirota method, Bogdan and Kovalev10 attempted to construct exact multisoliton solutions of an anisotropic ferromagnet. Svendsen and Fogedby11 derived the complete spectrum of the Landau-Lifshitz equation by the Hirota method. Using the variation method, Nakumura and Sasada12 obtained a solution which does not satisfy the equation if it is substituted into the equation of motion.13 By separating variables in moving coordinates, Quispel and Capel14 obtained a solution of the Landau-Lifschitz equation of a ferromagnet with an easy plane. Potemina15 and Kivshar16 elaborated on the perturbation theory for the Landau-Lifschitz equation describing a biaxial anisotropic ferromagnet. The general solution of Landau-Lifschitz equation for the special initial condition has been considered by several investigators. Lakshmann17 shown that the energy and current densities are given by the solutions of a completely integrable nonlinear Schrodinger equation. Takhtajan18 concluded that the Landau-Lifschitz equation admits a Lax representation and, consequently, falls within the scope of an 12 893 ©1999 The American Physical Society 12 894 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU inverse scattering transformation. Fogedby19 reviewed the permanent profile solutions of a continuous classical Heisenberg ferromagnet and expounded on the application of an inverse scattering transformation. Sklyanin20 and Borisov21 found the Lax pair of Landau-Lifschitz equations for a complete anisotropic ferromagnet, respectively. Mikhailov22 and Rodin23 reduced the problem to the Riemann boundary-value problem on a torus, then obtained some results which are expressed by the elliptic functions. Borovik and Kulinich24,25 derivated the Marchenko equation by an inverse scattering transformation. Pu, Zhou, and Li26 reported the multisoliton solutions of the Landau-Lifschitz equation in an isotropic ferromagnet in a magnetic field. Chen, Huang, and Liu27 obtained soliton solutions of the Landau-Lifschitz equation for a spin chain with an easy axis. Yue, Chen, and Huang28 investigated solitons of the Landau-Lifschitz equation for a spin chain with an easy plane. By means of the Darboux transformation, Huang, Chen, and Liu29 found the soliton solutions of the Landau-Lifschitz equation for a spin chain with an easy plane. Liu et al.30 studied solitons in a uniaxial Heisenberg spin chain with Gilbert damping in an external magnetic field. Using the method of the Riemann problem with zeros, Yue and Huang31 investigated solitons for a spin chain with an easy plane. There are some difficulties in the study of nonlinear magnetization dynamics of a ferromagnet with an anisotropy in an external magnetic field. Its equations of motion, which differ from those of an isotropic ferromagnet, could not be solved by the method of separating variables in moving coordinates.4 Then, this equation could also not solved by an usual form of inverse scattering transformation since the double-valued function of the spectral parameter is required to introduce a Riemann surface. The reflection coefficient at the edges of cuts in the complex plane could not be neglected even in the case of nonreflection. Thirdly, it is impossible to use Darboux transformation to include the contribution due to the continuous spectrum of the spectral parameter. If we consider the exact solutions of the Landau-Lifschitz equation under various external actions such as an external field in the present paper, a general theory with terms of the continuous spectrum as a starting point is necessary. Finally, an external magnetic field will affect the integrability of the system. This field will change the initial condition of the Landau-Lifschitz equation of a ferromagnet with an anisotropy. It would be instructive if the effect of a magnetic field is discussed. Introducing the coherent-state ansatz, the time-dependent variational principle, and the method of multiple scales, Liu and Zhou investigated the equation of motion and obtained multisolitons in the pure32 and the biaxial33 anisotropic antiferromagnets in an external field. Up to date, the effect of an external magnetic field for magnetic systems with anisotropy is treated as various perturbations. The exact solutions of the Landau-Lifschitz equation of the classical ferromagnet with two single-ion anisotropies in an external field have not been obtained yet. On the experimental side,34,35 a ferromagnet with an easy plane in a symmetry-breaking external transverse field has received continuing interest, though most theoretical treatments have been based on the approximate mapping6 to a sine-Gordon equation. This paper focuses on the integrability and nonlinear magnetization dynamics of the classical ferromagnet with two PRB 60 single-ion anisotropies in an external magnetic field. This is an important problem which has been treated to a large extent for the vanishing magnetic field by Sklyanin in a famous, but unpublished preprint, cited in Ref. 20. For magnetic fields with rotational symmetry 共an easy-plane or an easy-axis case兲, the analysis of Ref. 20 can be generalized by transformation to a rotating coordinate frame. For the general direction of the magnetic field the results will be generally investigated in the following content. The plan of this paper is as follows. In Sec. II by using stereographic projection of the unit sphere of the magnetization vector onto a complex plane for the equations of motion, the effect of a magnetic field for integrability of the system will be discussed. Then, though introducing transformations other than the Riemann surface, the properties of the Jost solutions and the scattering of data will be investigated in detail. In Sec. III will be derived the Gel’fand-Levitan-Marchenko equation to construct solutions from the scattering data. The exact multisoliton solutions will be investigated by means of the Binet-Cauchy formula. The total magnetic momentum and its z component will be obtained. Section IV will be devoted to the asymptotic behavior of multisoliton solutions as well as the total displacement of center and the phase shift of the jth peak. Finally, Sec. V will given our concluding remarks. II. THE EQUATIONS OF MOTION When we use a macroscopic description, dynamics of the classical ferromagnet is determined by giving at each point of the magnetization vector M⫽(M x ,M y ,M z ). The energy of a ferromagnet in this approach called, generally, micromagnetism, is written as the magnetization function. The magnetic energy E of the classical ferromagnet with two single-ion anisotropies in an external magnetic field, including an exchange energy E ex , an anisotropic energy E an, and a Zeeman energy E Z can be written as E⫽E ex⫹E an⫹E Z 1 ⫽ ␣ 2 ⫺  冕 冕 兺  冕 冕 1 2 k z 1 M M 3 d x⫺ xk xk 2 M z2 d 3 x⫺ B x M 2x d 3 x M–Bd 3 x, 共1兲 where B is the Bohn magneton. Equation 共1兲 has an integral of motion 具 M2 典 ⬅M20 ⫽const. In the ground state, the quantity M 0 coincides with a so-called spontaneous magnetization M 0 ⫽(2 B S)/a 3 , where S is the atomic spin and a is the interatomic spacing. In the limit  x ⫽0, a biaxial anisotropic ferromagnet reduces into an uniaxial anisotropic ferromagnet with an anisotropy axis coincident with the z axis: when  z ⬎0, an anisotropy is of an easy-axis type and its magnetization vector in the ground state is directed along the z axis; when  z ⬍0 it is of an easy-plane type, its vector M in the ground state lies in the easy plane in the absence of an external magnetic field and can be directed arbitrarily in this plane. If E an⫽0, a crystal is called an isotropic ferromagnet. As a function of space coordinates and time, the magnetization vector of the classical ferromagnet M(x,t) is a solution of the Landau-Lifschitz equation PRB 60 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . M 2B ␦E ⫽ M⫻ . t ប ␦M 共2兲 If we measure the space coordinate x and time t in unit of l 0 ⫽( ␣ /  z ) 1/2 and 0 ⫽(2 B  z M 0 )/ប, respectively, then according to Eqs. 共1兲 and 共2兲, we can obtain the following equation of motion: 冋 册 M M ⫹JM⫹ B B , ⫽Mⴛ t x2 2 共3兲 where the matrix J⫽diag(J x ,J y ,J z ) is related to the anisotropic constants. Equation 共3兲 with B⫽0 is exactly integrable by Sklyanin in a famous, but unpublished paper 关20兴. The additional terms on the right-hand side of Eq. 共3兲 describes various external actions, e.g., a magnetic field in the present paper, magnetic impurities, dissipative loses, etc. When oscillations of the magnetization vector M are localized near an easy plane yz, Eq. 共3兲 with B⫽0 could be transformed into a sine-Gordon equation in the limit J x ⰆJ y ⬍J z . Similarly, this equation with B⫽0 also becomes a nonlinear Schrodinger equation in the limit J x ⬇J y ⰆJ z when oscillations of the magnetization vector M are localized in the vicinity of the vacuum state M(x,t)⫽(0,0,M 0 ). In the special case  z ⫽0, an isotropic ferromagnet in an external magnetic field is also completely integrable.26 When a magnetic field is zero, Eq. 共3兲 is equivalent to a nonlinear Schrodinger equation.8 Thus Eq. 共3兲 is the most general equation describing the classical ferromagnet with two single-ion anisotropies in an external magnetic field, but its exact solutions have not been obtained so far because the additional terms such as an external magnetic field in the present paper on the right-hand side of Eq. 共3兲 are determined by various perturbations.33 We first consider the effect of an external magnetic field on integrability of the system. For magnetic fields with rotational symmetry 共an easy-plane or an easy-axis case兲, the analysis of Ref. 20 can be generalized by going over to a rotating coordinate frame. For the general direction of the magnetic field, we first use a stereographic projection of the unit sphere of magnetization vector onto a complex plane17,36 P 共 x,t 兲 ⫽ M x ⫹iM y . 1⫹M z 共4兲 Substituting Eq. 共4兲 into Eq. 共3兲, we can find 共 1⫺ P * 2 兲 ⌽ x 共 P, P * 兲 ⫺ 共 1⫺ P 2 兲 ⌽ * x 共 P, P * 兲 ⫽0, ⫺i 共 1⫹ P * 2 兲 ⌽ y 共 P, P * 兲 ⫺i 共 1⫹ P 2 兲 ⌽ * y 共 P, P * 兲 ⫽0, 共5兲 P * ⌽ z 共 P, P * 兲 ⫺ P⌽ z* 共 P, P * 兲 ⫽0, where ⌽ x , ⌽ y and ⌽ z can be written as 冉 冊 P 2P 2P ⌽ i 共 P, P * 兲 ⫽i 共 1⫹ 兩 P 兩 兲 ⫹ 共 1⫹ 兩 P 兩 2 兲 2 ⫺2 P * t x x2 2 ⫹2⌬J i P 共 1⫺ 兩 P 兩 2 兲 ⫹ B 共 1⫹ 兩 P 兩 2 兲 ⫻ 冋 册 1 x 1 B 共 1⫺ P 2 兲 ⫹ iB y 共 1⫹ P 2 兲 ⫺B z P , 2 2 2 共6兲 12 895 where i⫽x,y,z, ⌬J x ⫽J z ⫺J y , ⌬J y ⫽J x ⫺J z , ⌬J z ⫽J y ⫺J x , respectively. The consistency of Eq. 共6兲 implies ⌽ i ( P, P * )⫽0 and * ⌽ i ( P, P * )⫽0, therefore the evolution equation for the stereographic projection P(x,t) in the presence of the general direction of an external magnetic field becomes 冉 冊 P 2P 2P i 共 1⫹ 兩 P 兩 兲 ⫹ 共 1⫹ 兩 P 兩 2 兲 2 ⫺2 P * t x x2 2 ⫹2⌬J i P 共 1⫺ 兩 P 兩 2 兲 ⫹ B 共 1⫹ 兩 P 兩 2 兲 ⫻ 冋 2 册 1 1 x B 共 1⫺ P 2 兲 ⫹ iB y 共 1⫹ P 2 兲 ⫺B z P ⫽0. 共7兲 2 2 According to Eq. 共7兲, we can analyze the effect of an external magnetic field on the integrability of the system. When an external field is directed along an anisotropic axis, e.g., B⫽ 关 0,0,B z (t) 兴 , the magnetic field term in Eq. 共7兲 can be removed by the following gauge transformation P→ P̃ ⫽ P exp关iB兰dtBz(t)兴, and the system becomes integrable. However, if the magnetic field is transverse, e.g., B ⫽ 关 0,B y (t),0 兴 , the magnetic field term is not removable by previous gauge transformation and none of the magnetization components remain conserved quantities. Consequently, the combined Galilean plus gauge invariance of the LandauLifschitz equation is broken, no Lax pairs seem to exist, and the system appears to be nonintegrable. The influence of the magnetic field on the classical ferromagnet with an easy axis amounts to a change of the precession frequency of the magnetization vector M by B ⫽ B B. Therefore, if we can introduce an angular variable ˜ ⫽ ⫺ B t in the polar coordinates ( , ), then in terms of the angular variables and ˜ Eq. 共3兲 will not depend on B. However, the magnetization dynamics of the classical ferromagnet with an easy plane is very sensitive to an external magnetic field. Even a weak magnetic field alters the character of the ground state and therefore the form of localized solutions. When an external magnetic field is perpendicular to an easy plane, it does not alter the axial symmetry associated with the z axis, and the form of the ground state depends on the strength of an external field. The critical value is B c ⫽ 关 (J x ⫺J z )M 兴 / B . When an external magnetic field B z ⬍B c the magnetization vector M in the ground state deviates from an easy plane, and it is characterized by an inclination ⫽ 0 to the z axis, where 0 ⫽arccos(B z /B c ). The angle remains arbitrary. For brevity, such a ground state is referred to as an easy cone. As an external magnetic field increases, the angular opening of the easy cone becomes smaller, especially in the case of B z ⰇB c , and the magnetization vector M in a nonexcited ferromagnet with an easy plane lies along the z axis. In the context of the experiments,34,35 the situation where an external magnetic field lies in an easy plane, e.g., B ⫽ 关 B x (t),0,0 兴 , or B⫽ 关 0,B y (t),0 兴 , seems quite topical. In experiments on samples of a ferromagnet with an easy plane, CsNiF3 and (C6 H11NH3 )CuBr3 , an external field is applied as a rule in an easy plane. The presence of an external field, which lies in an easy plane, makes finding soliton solutions of the Landau-Lifschitz equation essentially more difficult. 12 896 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU The magnetic-field term in Eq. 共7兲 is not removable by previous gauge transformation. Thus, we can conclude that a ferromagnet with a uniaxial anisotropy in a transverse magnetic field is, in general, nonintegrable and becomes integrable only in the absence of either an anisotropic interaction or an external field. Equation 共3兲 may be represented as a compatibility condition t L⫺ x A⫹ 关 L,A 兴 ⫽0 of two equations for 2⫻2 matrices ⌿(x,t; ,): PRB 60 ⌿ 共 x,t; , 兲 ⫽L 共 , 兲 ⌿ 共 x,t; , 兲 , x ⌿ 共 x,t; , 兲 ⫽A 共 , 兲 ⌿ 共 x,t; , 兲 , t 共8兲 while L 共 , 兲 ⫽⫺i ns 共 , 兲 M x x ⫺i ds 共 , 兲 M y y ⫺i cs 共 , 兲 M z z , A 共 , 兲 ⫽i2 2 ds 共 , 兲 cs 共 , 兲 M x x ⫹i2 2 ns 共 , 兲 cs 共 , 兲 M y y ⫹i2 2 ns 共 , 兲 ds 共 , 兲 M z z ⫺i ns 共 , 兲 冉 ⫻ My 冊 冉 where i (i⫽x,y,z) are the Pauli metrics, ns( ,),ds( ,), and cs( ,) are elliptical functions, while and are defined as ⫽(J y M ⫺J x M ) 1/2/2 , ⫽1/2(J z M ⫺J x M ) 1/2. The coefficients in the Lax pairs include two parameters and instead of the three J i (i ⫽1,2,3), because adding a constant to all the J i does not change Eq. 共3兲. Since the coefficients are double-periodic functions of the parameter , it is sufficient to consider inside the rectangle 兩 Re 兩 ⭐2K, 兩 Im 兩 ⭐2K ⬘ , where K( ) is a complete elliptic integral of the first kind and K ⬘ ( ) ⫽K 关 (1⫺ 2 ) 1/2兴 . For an uniaxial anisotropic ferromagnet in an external magnetic field, the Lax pairs can be written as L 共 , 兲 ⫽⫺i M x x ⫺i M y y ⫺iM z z , A 共 , 兲 ⫽i2 M x x ⫹i2 M y y ⫹i2 2 M z z 冉 ⫺i M y ⫺M z 冊 冉 Mz My Mx ⫺M z x ⫺i M z x x x 冊 冉 冊 Mz My Mx y ⫺i M z z , ⫺M z x x x 共10兲 where the spectral parameters and satisfy the following relation: ⫽ 2 再 2 ⫹4 2 , for  z ⬍0 共 an easy plane兲 ; 2 ⫺4 2 , for  z ⬎0 共 an easy axis兲 , 共11兲 and where is defined as ⫽ 再 冊 冉 冊 Mz My Mx Mz My Mx ⫺M z ⫺M x ⫺M y x ⫺i ds 共 , 兲 M z y ⫺i cs 共 , 兲 M x z , x x x x x x If one of two parameters in Eq. 共11兲 is taken as an independent parameter, then another is the double-value function of the first, therefore it is necessary to introduce a Riemann surface. In order to avoid the complexity brought about a Riemann surface, introducing another parameter k called the affine parameter, we will consider (k) and (k) as a single-valued function of k, ⫽ 冦 2 共 k 2 ⫹1 兲 k ⫺1 2 k 2⫺ 2 , k for  z ⬍0 共 an easy plane兲 ; 1 关共 J z ⫺J x 兲 M 兴 1/2, 4 for  z ⬎0 共 an easy axis兲 . 共12兲 , ⫽ 冦 4k , for an easy plane, k 2⫹ 2 , k for an easy axis. k 2 ⫺1 共13兲 There are two different types of physical boundary conditions in Eq. 共3兲. The boundary condition of the first type corresponds to a breatherlike solution, which is usually called a magnetic soliton. For the classical ferromagnet with two single-ion anisotropies in an external magnetic field, in terms of analysis for integrability of Eq. 共7兲, we will study soliton solutions of possessing asymptotes M→M0 ⫽(0,0,M 0 ), as x→⫾⬁. The corresponding Jost solutions ⌿ 0⫾ (x,k) of Eq. 共8兲 may be chosen as ⌿ 0⫾ (x,k)→E(x,k) as x→⫾⬁, where E(x,k)⫽exp关⫺ics(k)M 0xz兴, while 冋 再 ⌿ 0 共 x,k 兲 ⫽exp ⫺i cs 共 k 兲 M 0 x⫺ 册冎 2 ns 共 k 兲 ds 共 k 兲 t z cs 共 k 兲 for Im k⫽0,2K ⬘ . There are two independent solutions E 1 (x,k) and E 2 (x,k) in E(x,k), with every solution having two components, E 1 共 x,k 兲 ⫽ 1 关共 J x ⫺J z 兲 M 兴 1/2, 4 共9兲 冉 E 11共 x,k 兲 E 21共 x,k 兲 冊 , E 2 共 x,k 兲 ⫽ 冉 E 12 共 x,k 兲 E 22 共 x,k 兲 冊 . ⌿ 0⫹ (x,k), ⌿ 0⫺ (x,k), and ⌿ 0 (x,k) have also two independent solutions ⌿ 0⫹1 (x,k) and ⌿ 0⫹2 (x,k), ⌿ 0⫺1 (x,k) and ⌿ 0⫺2 (x,k), ⌿ 01(x,k) and ⌿ 02(x,k), respectively. Under an external magnetic field the magnetization vector M in the ground state of a ferromagnet with an easy plane PRB 60 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . deviates from an easy plane, and it is characterized by an inclination 0 to the z axis and 0 to the x axis, where the asymptotic magnetization vector M lies on the surface of an easy cone. The simplest solution of Eq. 共3兲 can be written as M→M0 ⫽ (M 0 sin 0 cos 0 ,M 0 sin 0 sin 0 ,M 0 cos 0), as p (x,k) of Eq. x→⫾⬁, the corresponding Jost solutions ⌿ 0⫾ p p 共8兲 may be chosen as ⌿ 0⫾ (x,k)→E (x,k) as x→⫾⬁, where 冋 E 共 x,k 兲 ⫽exp ⫺i p ⫺i ⫺i while ⌿ 0p 共 x,k 兲 ⫽exp 冋 再 ⫺i ⫻ x⫺ ⫺i 2k k 2 ⫺1 2k 共 k 2 ⫹1 兲 k 2 ⫺1 2k k 2 ⫺1 k 2 ⫺1 2k k 2 ⫺1 M 0 sin 0 x cos 0 x 册 M 0 cos 0 x z , 册 共 k ⫹1 兲 冋 2 ⫺i 冋 k 2 ⫺1 M 0 cos 0 x⫺ 2 共 k 2 ⫹1 兲 8k k 2 ⫺1 k 4 ⫺1 册冎 册 t y t z . When an external magnetic field increases, magnetization will be far from an easy plane, and in the case of B z ⰇB c , magnetization will lie along the z axis. When a magnetic fields vanishes, magnetization will lie on an easy plane and can be written as M0⫽(M 0 cos 0 ,M 0sin 0 ,0). There are two independent solutions E 1p (x,k) and E 2p (x,k) in E p (x,k), with every solution having two components, E 1p 共 x,k 兲 ⫽ E 2p 共 x,k 兲 ⫽ 冉 冉 p E 11 共 x,k 兲 p E 21共 x,k 兲 p E 12 共 x,k 兲 p E 22 共 x,k 兲 冊 冊 , . 冋 册 k ⫺ M 0x z , E a 共 x,k 兲 ⫽exp ⫺i 2k 2 冊 a E 21 共 x,k 兲 E a2 共 x,k 兲 ⫽ , 冉 a E 12 共 x,k 兲 a E 22 共 x,k 兲 冊 . 冕 冕 ⬁ x dyK ⫹ 共 x,y 兲 E 共 y,k 兲 , 共14兲 x ⫺⬁ dyK ⫺ 共 x,y 兲 E 共 y,k 兲 , where the kernels K ⫹ (x,y) and K ⫺ (x,y) depend functionally on magnetization M (x) but are independent of the eigenvalue , and K ⫾ (x,⫾⬁)⫽0. For a ferromagnet with an easy plane in an external magnetic field, we can also obtain 共 k 2 ⫹1 兲 p ⌿⫹ 共 x,k 兲 ⫽E p 共 x,k 兲 ⫹ ⫹ 2k k ⫺1 2 冕 2k k 2 ⫺1 ⬁ x p,d dyK ⫹ 共 x,y 兲 E p 共 y,k 兲 p,nd dyK ⫹ 共 x,y 兲 E p 共 y,k 兲 , k ⫺1 2 冕 ⬁ x 共 k 2 ⫹1 兲 p ⌿⫺ 共 x,k 兲 ⫽E p 共 x,k 兲 ⫹ ⫹ k ⫺1 2 冕 x ⫺⬁ 冕 x ⫺⬁ p,d dyK ⫺ 共 x,y 兲 E p 共 y,k 兲 p,nd dyK ⫺ 共 x,y 兲 E p 共 y,k 兲 , 共15兲 p (x,⫾⬁)⫽0, the superscripts d and nd denote the where K ⫾ diagonal and nondiagonal parts of the matrix, respectively. While for a ferromagnet with an easy axis in an external magnetic field, p p The solutions ⌿ 0⫹ (x,k), ⌿ 0⫺ (x,k) and ⌿ 0p (x,k) have also p p two independent solutions ⌿ 0⫹1 (x,k) and ⌿ 0⫹2 (x,k), p p p p ⌿ 0⫺1 (x,k) and ⌿ 0⫺2 (x,k), ⌿ 01(x,k) and ⌿ 02(x,k), respectively. Since the z axis is an easy axis in a ferromagnet, the boundary condition is chosen as M→M0⫽(0,0,M 0 ) as x→ a (x,k) of Eq. ⫾⬁, and the corresponding Jost solutions ⌿ 0⫾ a a 共8兲 may be chosen as ⌿ 0⫾ (x,k)→E (x,k) as x→⫾⬁, where 2 冉 a E 11 共 x,k 兲 ⌿ ⫺ 共 x,k 兲 ⫽E 共 x,k 兲 ⫹ 2 册冎 Similarly, E a (x,k) also has two independent solutions E a1 (x,k) and E a2 (x,k), with every solution having two components, ⌿ ⫹ 共 x,k 兲 ⫽E 共 x,k 兲 ⫹ t x M 0 sin 0 sin 0 x⫺ 冋 k 2⫺ 2 共 k 2⫹ 2 兲2 M 0 x⫺ t z . 2k k共 k 2⫺ 2 兲 a a (x,k), ⌿ 0⫺ (x,k) and ⌿ a0 (x,k) have also two indepen⌿ 0⫹ a a a (x,k) and ⌿ 0⫹2 (x,k), ⌿ ⫺1 (x,k) and dent solutions ⌿ 0⫹1 a a a ⌿ 0⫺2 (x,k), ⌿ 01(x,k) and ⌿ 02(x,k), respectively. By means of the standard procedures of characteristic theory, we can obtain the following integral representation: M 0 sin 0 cos 0 2 共 k 2 ⫹1 兲 再 ⌿ a0 共 x,k 兲 ⫽exp ⫺i E a1 共 x,k 兲 ⫽ M 0 sin 0 x sin 0 y k 2 ⫺1 while 12 897 a ⌿⫹ 共 x,k 兲 ⫽E a 共 x,k 兲 ⫹ k 2⫹ 2 ⫹ 2k a ⌿⫺ 共 x,k 兲 ⫽E a 共 x,k 兲 ⫹ ⫹ k 2⫹ 2 2k a (x,⫾⬁)⫽0. where K ⫾ k 2⫺ 2 2k 冕 ⬁ x ⬁ x a,d dyK ⫹ 共 x,y 兲 E a 共 y,k 兲 a,nd dyK ⫹ 共 x,y 兲 E a 共 y,k 兲 , k 2⫺ 2 2k 冕 冕 x ⫺⬁ 冕 x ⫺⬁ a,d dyK ⫺ 共 x,y 兲 E a 共 y,k 兲 a,nd dyK ⫺ 共 x,y 兲 E a 共 y,k 兲 , 共16兲 12 898 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU III. SOLITONS By means of the results obtained in the previous section, we will investigate soliton solutions. The reconstruction of magnetization, i.e., the ‘‘potential’’ M (x,t), from the timedependent scattering data is called the ‘‘inverse scattering problem’’ and is achieved by means of a linear integral equation, the Gel’fand-Levitan-Marchenko equation. It is well known that the pure soliton solutions correspond to the reflectionless case. In the reflectionless case, the reflectional coefficient r(k,t)⫽0, the Gel’fand-Levitan-Marchenko equation can be written as K 11共 x,t 兲 ⫹K 12共 x,t 兲 N ⬙ 共 x,t 兲 ⫽0, K 11共 x,t 兲 ⫽i K 12共 x,t 兲 ⫽ det关 I⫹N ⬙ 共 x,t 兲 M ⬘ 共 x,t 兲兴 det关 I⫹N ⬙ 共 x,t 兲 N ⬘ 共 x,t 兲兴 冎 ⫺1 , det关 I⫹N ⬙ 共 x,t 兲 N ⬘ 共 x,t 兲 ⫹H 共 x 兲 T G 共 x,t 兲兴 det关 I⫹N ⬙ 共 x,t 兲 N ⬘ 共 x,t 兲兴 ⫺1, 共18兲 where M ⬘ (x,t)⫽N ⬘ (x,t)⫹iH(x) T G(x,t), while N ⬘ (x,t) and N ⬙ (x,t) are N⫻N matrices. In order to obtain K 11(x,t) and K 12(x,t), we will calculate det关 I⫹N ⬙ (x,t)N ⬘ (x,t) 兴 , det关 I⫹N ⬙ (x,t)M ⬘ (x,t) 兴 and det关 I⫹N ⬙ (x,t)M ⬘ (x,t)⫹H(x) T G(x,t) 兴 by the BinetCauchy formula, respectively. Setting 共17兲 K 12共 x,t 兲 ⫺G 共 x,t 兲 ⫺K 11共 x,t 兲 N ⬘ 共 x,t 兲 ⫽0, ⌫ 0 ⫽det共 I⫹N ⬙ N ⬘ 兲 , where K 11(x,t) and K 12(x,t) can be expressed by ⌫ 0 ⫽1⫹ 再 PRB 60 共19兲 and using the Binet-Cauchy formula, we can obtain 兺 1⭐n ⬍n 兺 兺 ⬍•••⬍n ⭐N 1⭐m ⬍m ⬍•••⬍m ⭐N r⫽1 1 2 r 1 2 r ⫻ ␥ 0 共 n 1 ,n 2 , . . . ,n r ;m 1 ,m 2 , . . . ,m r 兲 . 共20兲 For a ferromagnet with two single-ion anisotropies in an external magnetic field, ␥ 0 共 n 1 ,n 2 , . . . ,n r ;m 1 ,m 2 , . . . ,m r 兲 ⫽ 共 ⫺1 兲 r 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m␣ n␣ m ⬘ 2 关 cs 共 k n 兲 ⫺cs 共 k n ⬘ 兲兴 2 关 cs 共 k m 兲 ⫺cs 共 k m ⬘ 兲兴 2 关 cs 共 k m 兲 ⫺cs 共 k m 兲兴 2 ⬘ 共21兲 , where ␣ n⫽ cs 共 k n 兲 ⫺cs 共 k m 兲 兿 m⫽n 关 cs 共 k m 兲 ⫺cs 共 k m 兲兴关 cs 共 k n 兲 ⫺cs 共 k n 兲兴 N f n⫽ , cs 共 k 兲 l b n H 2n . 兿 l⫽1 cs 共 k l 兲 For an uniaxial anisotropic ferromagnet in an external magnetic field, we can find ␥ 0 共 n 1 ,n 2 , . . . ,n r ;m 1 ,m 2 , . . . ,m r 兲 ⫽ where 冦 2 共 ⫺1 兲 r 4 2 k m 共 k n 4 ⫺1 兲共 k m 2 ⫺1 兲共 k n ⬘ 2 ⫺k n 2 兲 2 共 k m ⬘ ⫺k 2n 兲 2 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m ␣ n ␣ m k 共 k 2 ⫹1 兲共 k 2 ⫺1 兲 2共 k 2 ⫺1 兲 2共 k 2 ⫺1 兲 2共 k 2 ⫺k 2 兲 , ⬘ ⬘ m m n n⬘ m⬘ m n for an easy plane; 共 ⫺1 兲 r 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m␣ n␣ m ⬘ ⬘ 2 2 2 4 2k n共 k m ⫹1 兲关 k n 共 k n ⬘ 2 ⫺1 兲 ⫺k n ⬘ 2 共 k n 2 ⫺1 兲兴 2 关 k m 共 k m ⬘ ⫺1 兲 ⫺k m ⬘ 共 k m ⫺1 兲兴 2 2 k m 共 k n 2 ⫹1 兲关 k n 共 k m ⫺1 兲 ⫺k m 共 k n 2 ⫺1 兲兴 2 , for an easy axis. 共22兲 PRB 60 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . ␣ n⫽ 冦 2 ⫺k 2n 兲 共 k 2n ⫺1 兲共 k n 2 ⫺1 兲共 k m 2 ⫺1 兲共 k m 兿 2 2共 km ⫺1 兲共 k n 2 ⫺k 2n 兲共 k m 2 ⫺k 2n 兲 m⫽n 12 899 ; 2 ⫺1 兲 ⫺k m 共 k 2n ⫺1 兲兴 共 k 2n ⫺1 兲共 k n 2 ⫺1 兲共 k m 2 ⫺1 兲关 k n 共 k m 兿 2 2 2 m⫽n 4 共 k m ⫺1 兲关 k n 共 k m 2 ⫺1 兲 ⫺k m 2 共 k n ⫺1 兲兴关 k n 共 k n 2 ⫺1 兲 ⫺k n 共 k n ⫺1 兲兴 f n⫽ 冦 N 兿 共 k 2l ⫹1 兲共 k l 2 ⫹1 兲 2 l⫽1 共 k l ⫺1 兲共 k l 2 ⫺1 兲 N 兿 k l 共 k 2l ⫺1 兲 l⫽1 k l 共 k l 2 ⫺1 兲 . b n H 2n , for an easy plane; 共23兲 b n H 2n , for an easy axis. Setting ⌫ 1 ⫽det共 I⫹N ⬙ M ⬘ 兲 , 共24兲 then ⌫ 1 can be written as ⌫ 1 ⫽1⫹ ␥ 1 共 n 1 ,n 2 , . . . ,n r ;m 1 ,m 2 , . . . ,m r 兲 . 兺 1⭐n ⬍n 兺 兺 ⬍•••⬍n ⭐N 1⭐m ⬍m ⬍•••⬍m ⭐N r⫽1 1 2 r 1 2 共25兲 r For the classical ferromagnet with two single-ion anisotropies in an external magnetic field, ␥ 1 共 n 1 ,n 2 ,•••,n r ;m 1 ,m 2 ,•••,m r 兲 ⫽ 共 ⫺1 兲 r 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m␣ n␣ m ⬘ 2 cs 共 k m 兲关 cs 共 k n 兲 ⫺cs 共 k n ⬘ 兲兴 2 关 cs 共 k m 兲 ⫺cs 共 k m ⬘ 兲兴 2 cs 共 k n 兲关 cs 共 k n 兲 ⫺cs 共 k m 兲兴 2 ⬘ . 共26兲 For an uniaxial anisotropic ferromagnet in an external magnetic field, ␥ 1 共 n 1 ,n 2 ,•••,n r ;m 1 ,m 2 ,•••,m r 兲 ⫽ 冦 2 共 ⫺1 兲 r 2 k m 共 k n 2 ⫺1 兲共 k n ⬘ 2 ⫺k n 2 兲 2 共 k m ⬘ ⫺k m2 兲 2 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m ␣ n ␣ m k 共 k 2 ⫺1 兲共 k 2 ⫺1 兲共 k 2 ⫺1 兲共 k 2 ⫺k 2 兲 2 , ⬘ ⬘ n n⬘ m m⬘ n m for an easy plane; 共 ⫺1 兲 r 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m␣ n␣ m ⬘ 2 2 2 16 2 共 k m ⫹1 兲共 k n 2 ⫺1 兲关 k n 共 k n ⬘ 2 ⫺1 兲 ⫺k n ⬘ 共 k n 2 ⫺1 兲兴 2 关 k m 共 k m ⬘ ⫺1 兲 ⫺k m ⬘ 共 k m ⫺1 兲兴 2 2 2 ⫺1 兲共 k n 2 ⫹1 兲关 k n 共 k m ⫺1 兲 ⫺k m 共 k n 2 ⫺1 兲兴 2 共km ⬘ for an easy axis. , 共27兲 Third, in order to obtain det关 I⫹N ⬙ (x,t)N ⬘ (x,t)⫹H(x) T G(x,t) 兴 in Eq. 共18兲, we will introduce a N⫻(N⫹1) matrix Q ⬙ and ⬙ ⫽N nm ⬘ , Q n0 ⬙ ⫽⫺iH n , Q nm ⬘ ⫽N nm ⬘ , Q n0 ⬙ ⫽iG n , n,m⫽1,2, . . . ,N, then det(I⫹Q ⬙ Q ⬘ ) can be a (N⫹1)⫻N matrix Q ⬘ , Q nm written as det共 I⫹Q ⬙ Q ⬘ 兲 ⫽1⫹ 兺 1⭐n ⬍n 兺 兺 ⬍•••⬍n ⭐N 0⭐m ⬍m ⬍•••⬍m ⭐N r⫽1 1 2 r 1 2 r ⫻Q ⬙ 共 n 1 ,n 2 , . . . ,n r ;m 1 ,m 2 , . . . ,m r 兲 Q ⬘ 共 m 1 ,m 2 , . . . ,m r ;n 1 ,n 2 , . . . ,n r 兲 , 共28兲 where the sum is decomposed into two parts: one is extended to m 1 ⫽0, the other to m 1 ⭓1. Except for the same extended to m 1 ⫽0, Eq. 共28兲 is just Eq. 共19兲, therefore, det共 I⫹Q ⬙ Q ⬘ 兲 ⫺det共 I⫹N ⬙ N ⬘ 兲 ⫽ 兺 1⭐n ⬍n 兺 兺 ⬍•••⬍n ⭐N 1⭐m ⬍m ⬍•••⬍m ⭐N r⫽1 1 2 r 1 2 r ⫻Q ⬙ 共 n 1 ,n 2 , . . . ,n r ;0,m 2 , . . . ,m r 兲 Q ⬘ 共 0,m 2 , . . . ,m r ;n 1 ,n 2 , . . . ,n r 兲 . 共29兲 ⌫ 2 ⫽det共 I⫹Q ⬙ Q ⬘ 兲 ⫺det共 I⫹N ⬙ N ⬘ 兲 , 共30兲 Setting 12 900 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU PRB 60 we can obtain ⌫ 2⫽ ␥ 2 共 n 1 ,n 2 , . . . ,n r ;0,m 2 , . . . ,m r 兲 . 兺 1⭐n ⬍n 兺 兺 ⬍•••⬍n ⭐N 1⭐m ⬍m ⬍•••⬍m ⭐N r⫽1 1 2 r 1 2 共31兲 r For the classical ferromagnet with two single-ion anisotropies in an external magnetic field, ␥ 2 共 n 1 ,n 2 , . . . ,n r ;0,m 2 , . . . ,m r 兲 ⫽ 共 ⫺1 兲 r⫹1 兿n 兿m n⬍n 兿 m⬍m 兿 ⬘ f n f m␣ n␣ m ⬘ 2 cs 共 k m 兲关 cs 共 k n 兲 ⫺cs 共 k n ⬘ 兲兴 2 关 cs 共 k m 兲 ⫺cs 共 k m ⬘ 兲兴 2 cs 共 k n 兲关 cs 共 k n 兲 ⫺cs 共 k m 兲兴 2 共32兲 . For an uniaxial anisotropic ferromagnet in an external magnetic field, ␥ 2 共 n 1 ,n 2 , . . . ,n r ;0,m 2 , . . . ,m r 兲 ⫽ 冦 2 共 ⫺1 兲 r⫹1 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m ␣ n ␣ m k 共 k 2 ⫺1 兲共 k 2 ⫺1 兲 2共 k 2 ⫺1 兲 2共 k 2 ⫺k 2 兲 2 , for an easy plane, ⬘ 共 ⫺1 兲 r⫹1 ⫻ 2 k m 共 k n 2 ⫺1 兲 2 共 k n ⬘ 2 ⫺k n 2 兲 2 共 k m ⬘ ⫺k m2 兲 2 ⬘ n n⬘ m m⬘ m n 兿n 兿m n⬍n 兿 m⬍m 兿 f n f m␣ n␣ m ⬘ 共33兲 ⬘ 2 2 ⫹1 兲共 k n 2 ⫺1 兲关 k n 共 k n ⬘ 2 ⫺1 兲 ⫺k n ⬘ 共 k n 2 ⫺1 兲兴 2 关 k m 共 k m ⬘ 2 ⫺1 兲 ⫺k m ⬘ 共 k m ⫺1 兲兴 2 16 2 共 k m 2 2 ⫺1 兲共 k n 2 ⫹1 兲关 k n 共 k m ⫺1 兲 ⫺k m 共 k n 2 ⫺1 兲兴 2 共km where f n can also be written as f n ⫽exp(⫺⌽1n⫹i⌽2n). Substituting Eqs. 共19兲, 共24兲, and 共30兲 into Eq. 共18兲, we can obtain K 11 and K 12 . Using the following relations: M„x,t…⫽ 关 iK 共 x,x,t 兲 ⫺ z 兴 z 关 iK 共 x,x,t 兲 ⫺ z 兴 ⫺1 , 共34兲 we can obtain the multisoliton solutions in the classical ferromagnet with two single-ion anisotropies in an external magnetic field, 共 Mn 兲 x ⫽Re 共 Mn 兲 y ⫽Im 冉 冉 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 共 Mn 兲 z ⫽M 0 ⫺ 冊 冊 兩 ⌫ 1兩 2⫺ 兩 ⌫ 2兩 2 兩 ⌫ 1兩 ⫹ 兩 ⌫ 2兩 2 2 , 共 Mn 兲 z ⫽ 冦 M 0 cos 0 ⫺ M 0⫺ 共 Mn 兲 x ⫽ 冦冉 冦冉 Re 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 冊 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 Im 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 冊 , 冉 , for an easy plane, , for an easy axis. Then taking the z axis as the polar axis in the polar coordinates, we can obtain the multisoliton solutions of the classical ferromagnet with two single-ion anisotropies in an external magnetic field, 共35兲 2 兩 ⌫ 2兩 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 , 共37兲 ⫽⫺arg ⌫ 2 ⫺arg ⌫ 1 . For an uniaxial anisotropic ferromagnet in an external magnetic field, the multisoliton solutions can be written as . 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 ⫹ 兩 ⌫ 2兩 2 2 冊 cos ⫽ , , M 0 sin 0 sin 0 ⫺Im 共 Mn 兲 y ⫽ 冉 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 兩 ⌫ 1兩 2⫺ 兩 ⌫ 2兩 2 For an uniaxial anisotropic ferromagnet in an external magnetic field, the multisoliton solutions can be written as M 0 sin 0 cos 0 ⫺Re 兩 ⌫ 1兩 2⫺ 兩 ⌫ 2兩 2 cos ⫽ , , for an easy axis, 2⌫ 1 ⌫ 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 冊 ⫽ , 共36兲 再 冦 cos 0 ⫺ 1⫺ 2 兩 ⌫ 2兩 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 2 兩 ⌫ 2兩 2 兩 ⌫ 1兩 2⫹ 兩 ⌫ 2兩 2 , , ⫺arg ⌫ 2 ⫺arg ⌫ 1 , for an easy plane, ⫺arg ⌫ 2 ⫺arg ⌫ 1 , for an easy axis, 共38兲 where ⌫ 1 and ⌫ 2 are expressed by Eqs. 共25兲 and 共31兲, respectively. When n⫽1, the single-soliton solutions of the classical ferromagnet with two single-ion anisotropies in an external magnetic field can be written as PRB 60 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . 共 M1 兲 x ⫽ 共 M1 兲 y ⫽ ns 共 k 1 兲 ⬙ 2 sinh ⌽ 1 sin ⌽ 2 ⫹ns 共 k 1 兲 ⬘ 2 cosh ⌽ 1 cos ⌽ 2 4ns 共 k 1 兲 ⬘ 2 cosh2 ⌽ 1 ⫹4ds 共 k 1 兲 ⬙ 2 sinh2 ⌽ 1 ⫹cs 共 k 1 兲 ⬙ 2 ds 共 k 1 兲 ⬙ 2 sinh ⌽ 1 cos ⌽ 2 ⫺ds 共 k 1 兲 ⬘ 2 cosh ⌽ 1 sin ⌽ 2 4ns 共 k 1 兲 ⬘ 2 cosh2 ⌽ 1 ⫹4ds 共 k 1 兲 ⬙ 2 sinh2 ⌽ 1 ⫹cs 共 k 1 兲 ⬙ 2 共 M1 兲 z ⫽M 0 ⫺ 12 901 , 共39兲 , 2cs 共 k 1 兲 ⬙ 2 4ns 共 k 1 兲 ⬘ 2 cosh2 ⌽ 1 ⫹4ds 共 k 1 兲 ⬙ 2 sinh2 ⌽ 1 ⫹cs 共 k 1 兲 ⬙ 2 , where ⌽ 1 ⫽2 cs 共 k 1 兲 ⬙ 共 x⫺V 1 t⫺x 10兲 , ⌽ 2 ⫽2 cs 共 k 1 兲 ⬘ 共 x⫺V 2 t⫺x 20兲 , 共40兲 and V 1 ⫽4cs 共 k 1 兲 ⬘ , V 2 ⫽ 2„cs 共 k 1 兲 ⬘ 2 ⫺cs 共 k 1 兲 ⬙ 2 ⫹4 2 … cs 共 k 1 兲 ⬘ 共41兲 . The single-soliton solutions of a uniaxial anisotropic ferromagnet in an external magnetic field can be written as 共 M1 兲 x ⫽M 0 sin 0 cos 0 ⫺ 共 M1 兲 y ⫽M 0 sin 0 sin 0 ⫺ 共 M1 兲 z ⫽M 0 cos 0 ⫺ 2k 1⬙ 2 关 4k 1⬘ 2 ⫹ 兩 k 21 ⫺1 兩 2 sin2 ⌽ 2 兴 兩 k 21 ⫺1 兩 2 关 k 1⬘ 2 cosh2 ⌽ 1 ⫹k ⬙1 2 sin2 ⌽ 2 兴 , 2k ⬘1 k ⬙1 关 4k 1⬘ k 1⬙ sinh ⌽ 1 cos ⌽ 2 ⫹ 共 兩 k 1 兩 4 ⫺1 兲 cosh ⌽ 1 sin ⌽ 2 兴 兩 k 21 ⫺1 兩 2 关 k 1⬘ 2 cosh2 ⌽ 1 ⫹k 1⬙ 2 sin2 ⌽ 2 兴 4k ⬘1 k ⬙1 关 k ⬙1 共 兩 k 1 兩 2 ⫹1 兲 sinh ⌽ 1 sin ⌽ 2 ⫺k ⬘1 共 兩 k 1 兩 2 ⫺1 兲 cosh ⌽ 1 cos ⌽ 2 兴 兩 k 21 ⫺1 兩 2 关 k 1⬘ 2 cosh2 ⌽ 1 ⫹k 1⬙ 2 sin2 ⌽ 2 兴 , , for an easy plane, 共42兲 and 共 M1 兲 x ⫽ 共 M1 兲 y ⫽ 共 M1 兲 z ⫽M 0 ⫺ 16k 1⬘ 2 k ⬙1 2 sinh ⌽ 1 sin ⌽ 2 ⫹ 共 兩 k 1 兩 4 ⫺1 兲 2 cosh ⌽ 1 cos ⌽ 2 共 兩 k 1 兩 4 ⫺1 兲 2 cosh2 ⌽ 1 ⫹16k 1⬘ 2 k ⬙1 2 sinh2 ⌽ 1 ⫹4k ⬙1 2 共 兩 k 1 兩 2 ⫹1 兲 2 16k ⬘1 2 k ⬙1 2 sinh ⌽ 1 cos ⌽ 2 ⫺ 共 兩 k 1 兩 4 ⫺1 兲 2 cosh ⌽ 1 sin ⌽ 2 共 兩 k 1 兩 4 ⫺1 兲 2 cosh2 ⌽ 1 ⫹16k 1⬘ 2 k ⬙1 2 sinh2 ⌽ 1 ⫹4k ⬙1 2 共 兩 k 1 兩 2 ⫹1 兲 2 2k ⬙1 2 共 兩 k 1 兩 2 ⫹1 兲 2 共 兩 k 1 兩 4 ⫺1 兲 2 cosh2 ⌽ 1 ⫹16k ⬘1 2 k ⬙1 2 sinh2 ⌽ 1 ⫹4k ⬙1 2 共 兩 k 1 兩 2 ⫹1 兲 2 , , , for an easy axis, 共43兲 where ⌽ 1⫽ ⌽ 2⫽ and 冦 冦 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 8 k ⬘1 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 共 x⫺V 1 t⫺x 10兲 , 共 x⫺V 1 t⫺x 10兲 , 共 x⫺V 2 t⫺x 20兲 , for an easy plane, 共44兲 共 x⫺V 2 t⫺x 20兲 , for an easy axis, WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU 12 902 PRB 60 FIG. 1. Some graphical illustrations of the motion of the center and the change of shape of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共42兲 in a ferromagnet with an easy plane, where 0 ⫽300 , ⫽0.1, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. V 1⫽ V 2⫽ 冦 冦 2 关共 兩 k 1 兩 4 ⫺1 兲 ⫹4k 1⬘ 2 共 兩 k 1 兩 2 ⫺1 兲兴 兩 k 21 ⫺1 兩 2 16 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 2 关共 兩 k 1 兩 2 ⫺1 兲 2 ⫺4k 1⬙ 2 共 兩 k 1 兩 2 ⫹1 兲兴 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 , , , for an easy plane, 2 关 4k ⬘1 2 共 兩 k 1 兩 2 ⫺1 兲 2 ⫺4k ⬙1 2 共 兩 k 1 兩 2 ⫹1 兲 2 ⫹ 兩 k 1 ⫺1 兩 4 兴 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 共45兲 , for an easy axis. These results show that under the action of an external magnetic field, the nonlinear magnetization of the classical ferromagnet with an anisotropy depends essentially on two parameters, namely, two velocities V 1 and V 2 in Eqs. 共41兲 and 共45兲; the center of nonlinear magnetization moves with a constant velocity V 1 , while its shape also changes with another velocity V 2 . Figures 1–4 give some graphical illustrations of the motion of the center and the change of shape of the z component of nonlinear magnetization (M1 ) z , expressed by Eq. 共42兲 in a ferromagnet with an easy plane and by Eq. 共43兲 in a ferromagnet with an easy FIG. 2. Some graphical illustrations of the motion of the center and the change of shape of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共43兲 in a ferromagnet with an easy axis, where ⫽0.1, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . PRB 60 12 903 FIG. 3. Some graphical illustrations of the motion of the center and the change of shape of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共42兲 in a ferromagnet with an easy plane, where 0 ⫽300 , ⫽0.3, k 1⬘ ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. axis, as an anisotropic parameter. Also, there is an external magnetic field increase from ⫽0.1 in Figs. 1 and 2 to ⫽0.3 in Figs. 3 and 4, where k ⬘1 ⫽0.1, k 1⬙ ⫽0.2, x 10⫽0, x 20⫽0, 0 ⫽300 . If we take the z axis as the polar axis in the polar coordinates, the single-soliton solutions of the classical ferromagnet with two single-ion anisotropies in an external magnetic field can be written as cos ⫽1⫺ tan ⫽ 2cs 共 k 1 兲 ⬙ 2 4ns 共 k 1 兲 ⬘ 2 cosh2 ⌽ 1 ⫹4ds 共 k 1 兲 ⬙ 2 sinh2 ⌽ 1 ⫹cs 共 k 1 兲 ⬙ 2 ds 共 k 1 兲 ⬙ 2 sinh ⌽ 1 cos ⌽ 2 ⫺ds 共 k 1 兲 ⬘ 2 cosh ⌽ 1 sin ⌽ 2 ns 共 k 1 兲 ⬙ 2 sinh ⌽ 1 sin ⌽ 2 ⫹ns 共 k 1 兲 ⬘ 2 cosh ⌽ 1 cos ⌽ 2 , 共46兲 . The single-soliton solutions of an uniaxial anisotropic ferromagnet in an external magnetic field can be written as cos ⫽ cos 0 ⫺ tan ⫽ 2 关 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 sinh ⌽ 1 sin ⌽ 2 ⫹k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 cosh ⌽ 1 cos ⌽ 2 兴 兩 k 21 ⫺1 兩 2 关 k ⬘1 2 cosh2 ⌽ 1 ⫹k ⬙1 2 sin2 ⌽ 2 兴 sin 0 sin 0 ⫺8k 1⬘ k 1⬙ sinh ⌽ 1 cos ⌽ 2 ⫺2 共 兩 k 1 兩 4 ⫺1 兲 cosh ⌽ 1 sin ⌽ 2 sin 0 cos 0 ⫺2k ⬙1 2 关 4k ⬘1 2 ⫹ 兩 k 21 ⫺1 兩 2 sin2 ⌽ 2 兴 , , for an easy plane, 共47兲 and FIG. 4. Some graphical illustrations of the motion of the center and the change of shape of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共43兲 in a ferromagnet with an easy axis, where ⫽0.3, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. 12 904 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU cos ⫽1⫺ tan ⫽ PRB 60 2k 1⬙ 2 共 兩 k 1 兩 2 ⫹1 兲 2 , 共 兩 k 1 兩 4 ⫺1 兲 2 cosh2 ⌽ 1 ⫹16k 1⬘ 2 k 1⬙ 2 sinh2 ⌽ 1 ⫹4k 1⬙ 2 共 兩 k 1 兩 2 ⫹1 兲 2 16k ⬘1 2 k 1⬙ 2 sinh ⌽ 1 cos ⌽ 2 ⫺ 共 兩 k 1 兩 4 ⫺1 兲 2 cosh ⌽ 1 sin ⌽ 2 16k 1⬘ 2 k ⬙1 2 sinh ⌽ 1 sin ⌽ 2 ⫹ 共 兩 k 1 兩 4 ⫺1 兲 2 cosh ⌽ 1 cos ⌽ 2 共48兲 , for an easy axis. We can find the following property: cos共 ⫺x,⫺t 兲 ⫽ cos共 x,t 兲 . 共49兲 It means that under the action of an external magnetic field the z component of nonlinear magnetization is a symmetric function of space and time, while the orientation of the nonlinear magnetization in the plane orthogonal to the anisotropic axis changes with an external field, and it will be constant when an external field vanishes. In order to analyze the feature of the previous soliton solutions, setting the preliminary values as zero in the moving coordinates of the soliton, for the classical ferromagnet with two single-ion anisotropies in an external magnetic field, we can obtain 2cs 共 k 1 兲 ⬙ 2 cos ⫽1⫺ tan ⫽ 4ns 共 k 1 兲 ⬘ 2 cosh2 关 2 cs 共 k 1 兲 ⬙ x 兴 ⫹4ds 共 k 1 兲 ⬙ 2 sinh2 关 2 cs 共 k 1 兲 ⬙ x 兴 ⫹cs 共 k 1 兲 ⬙ 2 , ds 共 k 1 兲 ⬙ 2 sinh关 2 cs 共 k 1 兲 ⬙ x 兴 cos关 2 cs 共 k 1 兲 ⬘ 共 x⫺V 2 t 兲兴 ⫺ds 共 k 1 兲 ⬘ 2 cosh关 2 cs 共 k 1 兲 ⬙ x 兴 sin关 2 cs 共 k 1 兲 ⬘ 共 x⫺V 2 t 兲兴 ns 共 k 1 兲 ⬙ 2 sinh关 2 cs 共 k 1 兲 ⬙ x 兴 sin关 2 cs 共 k 1 兲 ⬘ 共 x⫺V 2 t 兲兴 ⫹ns 共 k 1 兲 ⬘ 2 cosh关 2 cs 共 k 1 兲 ⬙ x 兴 cos关 2 cs 共 k 1 兲 ⬘ 共 x⫺V 2 t 兲兴 , 共50兲 and cos ⫽ cos 0 2k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 1 兩 4 ⫺1 ⫺ 冋 sinh 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 兩 k 21 ⫺1 兩 2 sin 0 sin 0 ⫺ tan ⫽ 4k 1⬘ k 1⬙ 兩 k 1 兩 4 ⫺1 再 册冋 x tan 冋 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 册 共 x⫺V 2 t 兲 ⫹ 册 冋 冋 册冎 2k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 cosh 兩 k 1 兩 4 ⫺1 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 8 k ⬙1 共 兩 k 1 兩 2 ⫹1 兲 8 k ⬘1 共 兩 k 1 兩 2 ⫺1 兲 2 2 2 2 k 1⬘ cosh x ⫹k 1⬙ sin 共 x⫺V 2 t 兲 兩 k 21 ⫺1 兩 2 兩 k 21 ⫺1 兩 2 冋 sinh 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 sin 0 cos 0 ⫺2k ⬙1 2 册 冋 x ⫺ cosh 册冋 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 x tan 兩 k 21 ⫺1 兩 2 再 冏 冏 冋 4k ⬘1 ⫹ k 21 ⫺1 2 sin2 2 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 共 x⫺V 2 t 兲 共 x⫺V 2 t 兲 册冎 册 x 册 , , 共51兲 for an easy plane, and 2 cos ⫽1⫺ 共 兩 k 1 兩 4 ⫺1 兲 2 册 冋 册 册冋 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 16k 1⬘ 2 k 1⬙ 2 4k 1⬙ 2 共 兩 k 1 兩 2 ⫹1 兲 2 2 cosh2 x ⫹ sinh x ⫹ 兩 k 21 ⫺1 兩 2 兩 k 21 ⫺1 兩 2 共 兩 k 1 兩 4 ⫺1 兲 2 共 兩 k 1 兩 4 ⫺1 兲 2 冋 冋 16k ⬘1 2 k ⬙1 2 sinh tan ⫽ 冋 4k 1⬙ 2 共 兩 k 1 兩 2 ⫹1 兲 2 16k 1⬘ 2 k 1⬙ 2 sinh 8 k ⬙1 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 册 册冋 冋 x ⫺ 共 兩 k 1 兩 ⫺1 兲 cosh x tan 4 2 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 8 k ⬙1 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 册 x tan 8 k 1⬘ 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 冋 共 x⫺V 2 t 兲 ⫹ 共 兩 k 1 兩 4 ⫺1 兲 2 cosh for an easy axis. , 共 x⫺V 2 t 兲 8 k 1⬙ 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 x 册 册 , 共52兲 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . PRB 60 12 905 FIG. 5. Some graphical illustrations of the change of amplitude and width of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共51兲 in a ferromagnet with an easy plane, where 0 ⫽300 , ⫽0.2, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. We can also find that under the action of an external magnetic field the amplitudes and widths of the nonlinear magnetization are not constants but vary periodically with time. According to Eqs. 共51兲 and 共52兲, Fig. 5 shows that the amplitude and shape of the z component of the nonlinear magnetization (M1 ) z in a ferromagnet with an easy plane also changes with a velocity V 2 and it is not symmetrical with respect to the center. Its shape in a ferromagnet with an easy axis is symmetrical with respect to the center by means of Fig. 6, where ⫽0.2, k 1⬘ ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, x 20⫽0, and 0 ⫽300 . Obviously, when an anisotropic parameter →0, these soliton solutions in an uniaxial anisotropic ferromagnet reduce to those in an isotropic ferromagnet, for example, the single-soliton solutions 共42兲 and 共43兲 are transformed to 共 M1 兲 x ⫽ 2k 1⬙ 兩 k 1兩 2 再 冋冉 冉 冊 冊 册冎 sech 关 k ⬙1 共 x⫺4k ⬘1 t⫺x 10兲兴 k 1⬙ sinh关 k ⬙1 共 x⫺4k ⬘1 t⫺x 10兲兴 sin k ⬘1 x⫺2 k 1⬘ ⫺ 2 冋冉 冉 ⫹k 1⬘ cosh关 k ⬙1 共 x⫺4k ⬘1 t⫺x 10兲兴 cos k ⬘1 x⫺2 k 1⬘ ⫺ k ⬙1 2 k 1⬘ t⫺x 20 k ⬙1 2 k ⬘1 冊 冊册 t⫺x 20 , FIG. 6. Some graphical illustrations of the change of amplitude and width of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共52兲 in a ferromagnet with an easy axis, where ⫽0.2, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU 12 906 PRB 60 FIG. 7. Some graphical illustrations of the motion of the center and the change of shape of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共53兲 in an isotropic ferromagnet, where ⫽0, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10 ⫽0, and x 20⫽0. 共 M1 兲 y ⫽ 2k 1⬙ 兩 k 1兩 2 再 冋冉 冉 ⫺k 1⬘ cosh关 k 1⬙ 共 x⫺4k 1⬘ t⫺x 10兲兴 sin k 1⬘ x⫺2 k 1⬘ ⫺ 共 M1 兲 z ⫽M 0 ⫺ 冋冉 冉 冊 冊 册冎 sech 关 k 1⬙ 共 x⫺4k ⬘1 t⫺x 10兲兴 k ⬙1 sinh关 k 1⬙ 共 x⫺4k 1⬘ t⫺x 10兲兴 cos k 1⬘ x⫺2 k ⬘1 ⫺ 2 2k 1⬙ 2 兩 k 1兩 2 sech 关 k 1⬙ 共 x⫺4k 1⬘ t⫺x 10兲兴 . 2 共53兲 These results are equal to Eq. 共27a兲 obtained by the method of an inverse scattering transformation in Ref. 26. We also find that under the action of an external magnetic field the center and shape of the z component of nonlinear magnetization do not move with the two velocities V 1 and V 2 as showed by Fig. 7. While taking the z axis as the polar axis in the polar coordinates, we can obtain k ⬙1 2 k 1⬘ t⫺x 20 k 1⬙ 2 k 1⬘ 冊 冊册 t⫺x 20 , 2k 1⬙ 2 cos ⫽1⫺ 兩 k 1兩 2 sech2 关 k 1⬙ 共 x⫺4k ⬘1 t⫺x 10兲兴 , 冋 冉 冊 册 再 冎 ⫽ 0 ⫹k 1⬘ x⫺2k ⬘1 1⫺ ⫹tan⫺1 k ⬙1 k 1⬘ k ⬙1 2 k 1⬘ 2 t⫺x 20 tanh关 k ⬙1 共 x⫺4k ⬘1 t⫺x 10兲兴 . 共54兲 FIG. 8. Some graphical illustrations of the amplitude and width of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共54兲 in an isotropic ferromagnet, which do not change periodically with time, where ⫽0, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . PRB 60 12 907 FIG. 9. Some graphical illustrations of the change of the z component of the total magnetic momentum P z expressed by Eq. 共56兲 in a ferromagnet with an easy plane, where 0 ⫽300 , ⫽0.1, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. It means that the amplitudes and widths of the z component of the nonlinear magnetization do not also vary periodically with time. Figure 8 give some graphical illustrations of the amplitudes and width of the z component of the nonlinear magnetization (M1 ) z expressed by Eq. 共54兲 in an isotropic ferromagnet, where ⫽0, k 1⬘ ⫽0.1, k 1⬙ ⫽0.2, x 10⫽0, and x 20⫽0. When t→0, these results are equivalent to Eq. 共22兲 obtained by means of the method of separating variables in moving coordinates shown in Ref. 4. The total magnetic momentum P⫽M 0 冕 dx 共 1⫺ cos 兲 䉮 共55兲 depends on time and it is not a constant under the action of an external magnetic field. The integral of the motion coincident with the z component of the total magnetic momentum P z ⫽M 0 冕 dx 共 1⫺ cos 兲 共56兲 is also not a constant. Figures 9 and 10 have given some graphical illustrations of the z component of the total magnetic momentum P z expressed by Eq. 共56兲 varying periodically with time in an anisotropic ferromagnet with an easy plane and with an easy axis, respectively. In the two figures, we took the following parameters: k 1⬘ ⫽0.1, k 1⬙ ⫽0.2, x 10 ⫽0, x 20⫽0, ⫽0.10, and 0 ⫽300 for an easy plane, respectively. We find that under the action of an external magnetic field, P z depends periodically on time for a ferromagnet with an easy plane, while P z in a ferromagnet with an easy axis will decrease as time increases, where P z has the sense of the mean number of spins deviated from the ground state in localized magnetic excitations. This feature did not appear in the study of all other nonlinear problems in magnetism. When an anisotropic parameter vanishes, the ground state of the isotropic ferromagnet has a constant spin pointing in the z direction and the fixed boundary condition M→(0,0,M 0 ) when x→⫾⬁. When an external magnetic field vanishes, the Hamiltonian H, the total magnetic momentum P, and the z component of the total magnetic momentum P z , i.e., the three constants of motion associated with the global symmetries of the time translation, space translation, and spin rotation, respectively, are in the action angle representation given by the diagonal expressions. In terms of soliton solutions 共56兲, we find that only in the case of an isotropic ferromagnet are the Hamiltonian, the total magnetic momentum P, and the z component of the total magnetic momentum P z constants of motion, E⫽4JM 20 k ⬙1 ⫹4JM 0 B(k ⬙1 / 兩 k 1 兩 2 ), P ⫽4M 0 sin⫺1(k⬙1/兩k1兩), and P z ⫽4M 0 (k 1⬙ / 兩 k 1 兩 2 ). Tjon and FIG. 10. Some graphical illustrations of the change of the z component of the total magnetic momentum P z expressed by Eq. 共56兲 in a ferromagnet with an easy axis, where ⫽0.1, k ⬘1 ⫽0.1, k ⬙1 ⫽0.2, x 10⫽0, and x 20⫽0. 12 908 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU Wright4 took advantage of this feature in solving the equation of motion. These properties are important for the classical ferromagnet with an anisotropy in an external magnetic field, but they have never been obtained by all the other methods. ⌫ 2 ⬃ ␥ 2 共 1,2, . . . , j;0,1,2, . . . , j⫺1 兲 . Substituting the explicit expressions into Eqs. 共19兲, 共24兲, and 共30兲, for the classical ferromagnet with two single-ion anisotropies in an external magnetic field, we can obtain the following relations: IV. THE ASYMPTOTIC BEHAVIOR OF MULTISOLITON SOLUTIONS ⌫ 0 ⬃1⫹ Supposing all k n⬙ ⬎0 and k 1⬘ ⬎k 2⬘ ⬎•••⬎k N⬘ , the vicinity of x⫽x in⫹V int(i⫽1,2) is denoted by ⌰ n . In the extreme by large t, these vicinities are separated from left to right as ⌰ N ,⌰ N⫺1 , . . . ,⌰ 1 . In the vicinity ⌰ j , there are the following limits: (x⫺V int⫺x in0 )→⫺⬁, 兩 f n 兩 →⬁, if n⬍ j; (x ⫺V im t⫺x im0 )→⬁, 兩 f m 兩 →⬁, if m⬎ j, while j⫺1 F j⫽ ⌫ 1 ⬃ ␥ 1 共 1,2, . . . , j⫺1;1,2, . . . , j⫺1 兲 ⌫ 0⬃ ⌫ 1⬃ ⌫ 2⬃ F j⫽ 冦 兿 兿 fj 冦 冦 冦 1⫹ 兩 F j 兩 2 N 兿 兿 N 兿 兿 n⫽1 m⫽ j⫹1 1⫹ 兩 F j 兩 2 1⫹ 兩 F j 兩 2 1⫹ 兩 F j 兩 2 Fj Fj k j 共 k j 2 ⫹1 兲 k j 共 k 2j ⫹1 兲 ns 共 k j 兲 ns 共 k j 兲 兩 F j兩 2, F j, 关 cs 共 k j 兲 ⫺cs 共 k n 兲兴关 cs 共 k j 兲 ⫺cs 共 k m 兲兴 关 cs 共 k j 兲 ⫺cs 共 k n 兲兴关 cs 共 k j 兲 ⫺cs 共 k m 兲兴 fj. k j 共 k 2j ⫹1 兲 k j 共 k j 2 ⫹1 兲 ; ; 共 k 2j ⫺1 兲共 k j 2 ⫹1 兲 共 k 2j ⫹1 兲共 k j 2 ⫺1 兲 共 k 2j ⫹1 兲共 k j 2 ⫺1 兲 共 k 2j ⫺1 兲共 k j 2 ⫹1 兲 4k ⬘j k ⬙j 共 k j 2 ⫺1 兲 k j 兩 k 2j ⫺1 兩 2 共 k j 2 ⫹1 兲 兩 k 2j ⫺1 兩 2 2 ⫺1 兲共 k n 2 ⫺1 兲共 k 2n ⫺k 2j 兲共 k m 2 ⫺k 2j 兲 共km ; ; ; 4k ⬙j 共 k j 2 ⫺1 兲共 兩 k j 兩 2 ⫹1 兲 2 2 2 2 n⫽1 m⫽ j⫹1 共 k n ⫺1 兲共 k m 2 ⫺1 兲共 k n 2 ⫺k j 兲共 k m ⫺k j 兲 j⫺1 2cs 共 k j 兲 ⬙ ns 共 k j 兲 Similarly, for an uniaxial anisotropic ferromagnet in an external magnetic field, we can also find ⫹ ␥ 1 共 1,2, . . . , j;1,2, . . . , j 兲 , fj cs 共 k j 兲 ns 共 k j 兲 ⌫ 1 ⬃1⫹ 兩 F j兩 2, where ⫹ ␥ 0 共 1,2, . . . , j;1,2, . . . , j 兲 , N cs 共 k j 兲 ns 共 k j 兲 ⌫ 2⬃ ⌫ 0 ⬃ ␥ 0 共 1,2, . . . , j⫺1;1,2, . . . , j⫺1 兲 j⫺1 PRB 60 ; , for an easy axis, 2 ⫺1 兲共 k n 2 ⫺1 兲关 k j 共 k 2n ⫺1 兲 ⫺k n 共 k 2j ⫺1 兲兴关 k j 共 k m 2 ⫺1 兲 ⫺k m 共 k 2j ⫺1 兲兴 共km 2 2 2 2 n⫽1 m⫽ j⫹1 共 k n ⫺1 兲共 k m 2 ⫺1 兲关 k j 共 k n 2 ⫺1 兲 ⫺k n 共 k j ⫺1 兲兴关 k j 共 k m ⫺1 兲 ⫺k m 共 k j ⫺1 兲兴 共57兲 , for an easy axis. It can be concluded from the results given above that the classical ferromagnet with two single-ion anisotropies in an external magnetic field has multisoliton solutions in a strict sense. When t→⫾⬁, nonlinear magnetization appear to be the trains of N separating single solitons. The trains at t→⫺⬁ turn out to be trains at t→⬁ after the collision in the duration of time with the number and shape of solitons unchanged, and the position of center the of mass displaced in the traveling coordinates. The total displacement of the center of the jth peak in the course from t→⫺⬁ to t→⬁ is determined by X j⫽ 1 cs 共 k j 兲 ⬙ 再 冏 j⫺1 兿 n⫽1 ln 冏 cs 共 k j 兲 ⫺cs 共 k n 兲 cs 共 k j 兲 ⫺cs 共 k n 兲 N 兿 m⫽ j⫹1 ⫺ln 冏 冏冎 cs 共 k j 兲 ⫺cs 共 k m 兲 cs 共 k j 兲 ⫺cs 共 k m 兲 . 共58兲 PRB 60 NONLINEAR MAGNETIZATION DYNAMICS OF THE . . . 12 909 However, even in the traveling coordinates the angle tan ⫽arctan(M y /M x) contains a linear term in time t. This shows that M x and M y manifest themselves as solitons. The total phase shift of the jth peak can be written as 再 冋 ⌽ j ⫽2 arg j⫺1 cs 共 k j 兲 ⫺cs 共 k n 兲 兿 n⫽1 cs 共 k j 兲 ⫺cs 共 k n 兲 册 冋 ⫺arg N 兿 m⫽ j⫹1 cs 共 k j 兲 ⫺cs 共 k m 兲 cs 共 k j 兲 ⫺cs 共 k m 兲 册冎 共59兲 . For a uniaxial anisotropic ferromagnet in an external magnetic field, the total displacement of the center and the total phase shift of the jth peak in the course from t→⫺⬁ to t→⬁ are X j⫽ ⌽ j⫽ 冦 冦 兩 k 2j ⫺1 兩 2 2 k ⬘j k ⬙j 再 兿冏 冋 兿冏 j⫺1 ln n⫽1 2 k ⬙j 共 兩 k j 兩 2 ⫹1 兲 再 冋兿 冋 冉兿 j⫺1 2 arg ln n⫽1 N 兿 ⫺ln m⫽ j⫹1 冏 2 ⫺k 2j 兲共 k m 2 ⫺1 兲 共km 共 k 2n ⫺1 兲关 k j 共 k n 2 ⫺1 兲 ⫺k n 共 k 2j ⫺1 兲兴 共 k 2n ⫺k 2j 兲共 k n 2 ⫺1 兲 册 冋兿 N ⫺arg 冏冎 2 ⫺1 兲 共 k m 2 ⫺k 2j 兲共 k m 共 k n 2 ⫺1 兲关 k j 共 k 2n ⫺1 兲 ⫺k n 共 k 2j ⫺1 兲兴 2 2 n⫽1 共 k n ⫺1 兲共 k n 2 ⫺k j 兲 j⫺1 2 arg 共 k 2n ⫺1 兲共 k n 2 ⫺k 2j 兲 j⫺1 兩 k 2j ⫺1 兩 2 冏 共 k 2n ⫺k 2j 兲共 k n 2 ⫺1 兲 冏 N 2 ⫺1 兲 共 k m 2 ⫺k 2j 兲共 k m 2 2 m⫽ j⫹1 共 k m ⫺k j 兲共 k m 2 ⫺1 兲 共 k n 2 ⫺1 兲关 k j 共 k 2n ⫺1 兲 ⫺k n 共 k 2j ⫺1 兲兴 2 2 n⫽1 共 k n ⫺1 兲关 k j 共 k n 2 ⫺1 兲 ⫺k n 共 k j ⫺1 兲兴 冊 冉兿 N ⫺arg 兿 ⫺ln m⫽ j⫹1 册冎 ; 冏 2 ⫺1 兲 ⫺k m 共 k 2j ⫺1 兲兴 共 k m 2 ⫺1 兲关 k j 共 k m 2 ⫺1 兲关 k j 共 k m 2 ⫺1 兲 ⫺k m 共 k 2j ⫺1 兲兴 共km 冏册 . , for an easy plane; 2 ⫺1 兲 ⫺k m 共 k 2j ⫺1 兲兴 共 k m 2 ⫺1 兲关 k j 共 k m 2 2 m⫽ j⫹1 共 k m ⫺1 兲关 k j 共 k m 2 ⫺1 兲 ⫺k m 共 k j ⫺1 兲兴 冊册 , for an easy axis. 共60兲 When an anisotropic parameter →0, the displacement of the center and the phase shift of the jth peak of an isotropic ferromagnet in an external magnetic field are X j⫽ k ⬙j 冋冉 ⌽ j ⫽2 arg 冉 兿 冏 冏 兿 冏 冏冊 j⫺1 1 k n ⫺k j ln j⫺1 兿 n⫽1 n⫽1 k n ⫺k j k n 共 k n ⫺k j 兲 k n ⫺k j N k m ⫺k j m⫽ j⫹1 k m ⫺k j ⫺ln 冊 冉 ⫺arg N 兿 m⫽ j⫹1 , k m 共 k m ⫺k j 兲 k m ⫺k j 冊册 共61兲 , These results are equal to Eqs. 共28a兲 and 共28b兲 obtained by the method of an inverse scattering transformation in Ref. 26. V. CONCLUSION In this section we will compare the present results with those obtained by other methods, then give some concluding remarks. According to Eqs. 共39兲, 共42兲, and 共43兲, we can find that under the action of an external magnetic field nonlinear magnetization in a ferromagnet with an anisotropy depends essentially on two parameters V 1 and V 2 in Eqs. 共41兲 and 共45兲. The center of the nonlinear magnetization moves with a constant velocity V 1 , while its shape also changes with another velocity V 2 ; the depths and widths of a surface of nonlinear magnetization vary periodically with time, and its shape is unsymmetrical with respect to the center. By means of these features, we find that the soliton solutions in a ferromagnet with an anisotropy in the external magnetic field are not expressed in the form of the product of separated variables in moving coordinates.4 Only when an anisotropic parameter →0, these soliton solutions in an anisotropic ferromagnet reduce to those in an isotropic ferromagnet, for example, the single-soliton solutions 共47兲 in the polar coordinates are equivalent to Eq. 共22兲 obtained by means of the method of separating variables in the moving coordinates in Ref. 4. Therefore, it is very difficult to investigate the exact soliton solutions in a ferromagnet with an anisotropy in an external magnetic field by means of the method of separating variables. Reducing the Landau-Lifschitz equations to an appropriate form, Kosevich, Ivanov, and Kovalev5 found a solution. In terms of Eq. 共47兲 in the polar coordinates, there exist 再 冉冊 再 ⫽ tan2 2 k 1⬙ 2 k 1⬘ 2 兩 k 21 ⫺1 兩 2 sin2 冋 冋 8 k ⬙1 共 兩 k 1 兩 2 ⫹1 兲 兩 k 21 ⫺1 兩 2 册 冎 册 冎 共 x⫺V 1 t⫺x 10兲 ⫹4k 1⬘ 2 8 k ⬘1 共 兩 k 1 兩 2 ⫺1 兲 兩 k 21 ⫺1 兩 2 cosh2 共 x⫺V 2 t⫺x 10兲 ⫺4k ⬙1 2 2 2 兩 k 1 ⫺1 兩 . 共62兲 WU-MING LIU, WU-SHOU ZHANG, FU-CHO PU, AND XIN ZHOU 12 910 If we compared Eq. 共62兲 with an approximate solution given by Ref. 5, we find that the previous properties of the soliton solutions remain even in the approximation of the order of 2 . The solutions of Ref. 5 did not satisfy the LandauLifschitz equation for a ferromagnet with an anisotropy even in the first order of anisotropy, and there is no reason to consider it as an approximate solution, since all attempts in this approximation were not successful. Using the Hirota method, Bogdan and Kovalev10 sought the soliton solutions of the Landau-Lifschitz equation in a ferromagnet with an anisotropy in the form Mx ⫹iMy ⫽ 2fg 兩 f 兩2⫹兩g兩 , Mz ⫽ 2 兩 f 兩2⫺兩g兩2 兩 f 兩2⫹兩g兩2 共63兲 , where [N/2] f⫽ 兺 兺 a 共 i 1 , . . . ,i 2n 兲 exp共 i ⫹•••⫹ i 兲 , 1 n⫽0 C 2 n [(N⫺1)/2] g *⫽ 兺 m⫽0 2n 兺 a 共 j 1 , . . . , j 2m⫹1 兲 C 2m⫹1 ⫻exp共 j 1 ⫹•••⫹ j 2m⫹1 兲 , a 共 i 1 , . . . ,i n 兲 ⫽ 再 共64兲 (n) 兺 a 共 i k ,i l 兲 , for n⭓2; k⬍l 1, for n⫽0,1. where 关 N/2兴 is the maximum integer in addition to N/2, C n represents the summation over all combinations of N elements in n, and i ⫽(k i ⫹ i t⫹ 0i ). According to the expression of the single-soliton solutions 共42兲 and 共43兲 in this paper, we find that soliton solutions are difficult to express in the form of the Hirota factorization. Obviously, Bogdan and Kovalev10 did not obtain the desired results. We have introduced some transformations in Eq. 共13兲, while k⫽⬁ and 0 correspond to ⫽⫾2 共or ⫽0) and ⫽⫾ 共or ⫽0). In the complex plane, these two points are the edges of the cuts. This is important to ensure that the Jost solution generated satisfies the corresponding Lax equations. It indicates that the edges of the cuts in the complex plane in an inverse scattering transformation must give a contribution even in the case of nonreflection. Unfortunately, Borovik and Kulinich24,25 did not apparently consider these effects. Evidently, they did not obtain any expression of the solution. In the present paper we have used the stereographic projection of the unit sphere of the magnetization vector onto a complex plane for the equations of motion in the classical ferromagnet with two single-ion anisotropies in an external magnetic field, and the effect of a magnetic field for integrability of the system is discussed. Then, introducing some transformations instead of the Riemann surface in order to avoid the double-valued function of the usual spectral parameter, the properties of the Jost solutions and the scattering data in detail are obtained. The Gel’fand-Levitan-Marchenko equation is derived. In the case of no reflection the exact multisoliton solutions are investigated. This method is more effective than the Darboux transformation. The asymptotic behavior of multisoliton solutions in the long-time limit as well as the total displacement of the center and the phase shift of the jth peak are also given. The total magnetic momentum and its z component are obtained. The present inverse scattering transformation method includes the contributions due to the continuous spectrum of the spectral parameter. They may be useful for further theoretical research and practical application. ACKNOWLEDGMENTS W.M. 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