PHYSICAL REVIEW B VOLUME 58, NUMBER 17 1 NOVEMBER 1998-I Intrinsic localized spin waves in classical one-dimensional spin systems: Studies of their interactions S. Rakhmanova and D. L. Mills Department of Physics and Astronomy, University of California, Irvine, California 92697 Received 24 March 1998; revised manuscript received 6 July 1998 One-dimensional classical spin systems can support nonlinear excitations referred to as intrinsic localized spin-wave modes. These entities have internal frequencies which lie outside the spin-wave bands of linear theory, and are localized by virtue of the intrinsic nonlinearity present in spin systems. By numerical methods, we have explored collisions between these objects. We explore the influence of a defect on the spectrum of intrinsic nonlinear spin waves, to find a new class of modes localized at the defect. We also examine the interaction of a propagating nonlinear mode with the defect, to find rich behavior. The mode may be trans- mitted with no reflected component, trapped or reflected depending on the strength of the perturbation asso- ciated with the defect spin. S0163-1829 98 01841-4 I. INTRODUCTION tions and their properties is provided by Kosevich, Ivanov, and Kovalev,7 though the ILSM's just described are not dis- For many decades, since the work of Bloch in the early cussed. In the present paper, we address interactions experi- 1930s,1 it has been known that in Heisenberg magnets, the enced by intrinsic nonlinear spin-wave modes. We first ex- elementary excitations are spin waves. These extended, amine collisions between two such modes. The behavior we plane-wave modes control the thermodynamics of such sys- find here is complex. There are circumstances where the tems at low temperatures. In the late 1970s and early 1980s,2 modes emerge from a collision unchanged in shape or form, attention was directed toward unique aspects of one- and thus behave as solitons. However, more generally, we dimensional Heisenberg magnets. If the spins are viewed as see spin waves emitted as a consequence of such a collision, classical objects and suitable anisotropy is present, the equa- so, in fact, these localized entities interact in a complex man- tions of motion admit static domain-wall solutions, and so- ner. We provide several examples. In what follows, we use lutions in which such walls move with finite velocity. That the term soliton to describe these modes on occasion, but the this is so quite generally had been known much earlier.3 The reader should appreciate their interactions are complex. unique aspect of the one-dimensional spin chain is that the We also place a defect in our one-dimensional line of excitation energy is on the microscopic scale. Thus, moving spins and find a class of ILSM's localized on the defect. We domain walls may be excited thermally, and contribute to the then study the interaction of the ILSM solitons with the de- thermodynamics of the system. At low temperatures, the fect spin, to find very rich behavior. If the perturbation pre- thermal excitations may be viewed as a dilute soliton gas, sented by the defect is weak, the soliton passes over it with a with spin waves present as well. transmissivity of unity, though its center-of-mass velocity is In the recent literature, considerable theoretical attention altered. Upon increasing the strength of the perturbation, we has been devoted to objects referred to as intrinsic localized reach a regime where the entity is self-trapped. Then a fur- spin modes ILSM states .4­6 These are localized entities, ther increase in strength of the perturbation takes us into a stabilized by the intrinsic anharmonicity inherent in the equations of motion of the spin system; so far primary atten- domain where the soliton is reflected completely. In this lat- tion has been directed toward the one-dimensional line of ter regime, the soliton may be trapped between two defects. classical spins. These entities differ in one important, quali- One may inquire if one may realize a physical system tative regard, when compared to the domain-wall structures described by a model Hamiltonian such as that employed discussed some years ago.2 When the ILSM is at rest, all here and in our earlier studies. Magnetic superlattices can be spins in the system are engaged in circular precession, at synthesized which meet this requirement. An ultrathin mag- some frequency which lies outside the frequency bands netic film in such a structure is characterized by its total associated with the spin waves of linear theory. In contrast to magnetization M(t), which may be viewed as a classical this, when the domain walls described in the previous para- spin. Two neighboring such ``spins'' can experience ferro- graph are at rest, the spins are static. The ILSM states are magnetic couplings such as contained in our model, by fab- found to exist for any internal frequency above the linear ricating a structure with appropriate nonmagnetic films sand- spin-wave bands, in a model we have studied. For a one- wiched between the ferromagnetic films. It is possible to dimensional line of a finite number N of spins, we have synthesize easy-plane ferromagnetic films, with very small demonstrated previously that for any such frequency , the in-plane anisotropy. Thus, the ground state is ferromagnetic, equations of motion also admit solutions with two, three, with spins in the plane normal to the growth axis of the four,... intrinsic localized spin-wave modes.5 superlattice. We consider such a system, with spins pulled The studies that have appeared to date have explored the into ferromagnetic alignment by a magnetic field perpendicu- properties and nature of these states for various models and lar to the easy plane. Note that for a superlattice in this circumstances. A complete review of localized spin excita- configuration, when the moments precess, no intrafilm de- 0163-1829/98/58 17 /11458 7 /$15.00 PRB 58 11 458 ©1998 The American Physical Society PRB 58 INTRINSIC LOCALIZED SPIN WAVES IN CLASSICAL . . . 11 459 magnetizing fields are generated by the spin motion. Thus, there is no way of determining its form in advance of actu- such a superlattice is a physical realization of the model sys- ally solving the equations of motion. There is no analytical tem studied here. Other authors have demonstrated that solution to use as a guide, in general. When is very close ILSM's exist for other model one-dimensional spin to the top of the linear spin-wave bands, we may introduce a systems.6,8 continuum approximation, and one is led to the nonlinear This paper is organized as follows. In Sec. II, we present Schro¨dinger equation.4 We have proceeded here with the fol- the model and discuss issues related to solving the equations lowing viewpoint. If we choose a set of sn(0) which de- of motion for moving ILSM solitons. Section III presents our scribes a form close to but not a perfect representation of a studies of collisions between moving solitons. Section IV stable ILSM, then at short times this object will shed energy examines the nonlinear modes localized on a defect, and Sec. in the form of spin waves, and settle down into a stable V the interaction of ILSM solitons with magnetic defects. object, moving with some velocity v. The spin waves give rise to a slightly noisy background. We have found that the II. INITIAL CONDITIONS AND PROPAGATION following procedure works well over a wide parameter do- OF A SINGLE LOCALIZED MODE main. By this last statement, we mean the background noise THROUGH A PERFECT CHAIN has very small amplitude, after the solution settles down. The initial conditions are chosen so that the Eq. 3 will We consider a ferromagnetic chain of N spins described describe the propagation of the ILSM through the finite chain by the Hamiltonian of spins, as just discussed. It is also desirable that for the case k 0, these initial conditions generate the stationary ILSM H 2J S z z familiar from Ref. 5. The best way to proceed, in our expe- n*Sn 1 A Sn 2 H0 Sn , 1 n n n rience, is to generalize to nonzero k the equation the station- where J 0 is the exchange interaction constant, A is the ary ILSM's satisfy. Here we make use of the following ob- anisotropy constant, and H servation. Let us assume for the moment that at 0 the 0 is the magnitude of the external field applied along z axis. We chose A positive, which cor- quantity in square brackets on the right-hand side of Eq. 3 responds to the case of easy-plane anisotropy. The field H vanishes. Thus, the initial configuration satisfies 0 is assumed to be large enough so that in the ground state all spins are ferromagnetically ordered along the z axis. The sn 0 sn 0 1 sn 1 0 2 1 sn 1 0 2 equation of motion for the nth spin is found from 2Bsn 0 sn 1 0 dS i n s dt Sn ,H . 2 n 1 0 cos k 1 sn 0 2. 4 After obtaining the commutator of the spin operator with Then, given sn(0) from Eq. 4 , we find that the equation the Hamiltonian in Eq. 2 , we treat operators S ds n as classical n /d sin k(sn 1 sn 1) 1 sn 2 determines the evolu- vectors of magnitude S. To bring the equations of motion tion of the system at least during the first few time steps. For into suitable form, we introduce vectors s z small sn it describes uniform propagation of the initial profile n and sn defined by s x y z z as a whole without changing its shape. The velocity of n (Sn iSn)/S, sn Sn/S. We separate fast scale oscilla- tions by writing s x propagation is equal to 2 sin k, as one can see if one writes n in the form sn sn(t) the equation in the form ds isy n /d 2 sin k(sn 1 sn 1)/2 0. n(t) ei(kn t). Then, the complex amplitude sn(t) In numerical calculations we proceed as follows. We find sx y n(t) isn(t) obeys a solution of stationary Eq. 4 and use it as an initial con- ds figuration for the full time-dependent Eq. 3 . Note that sn is n complex in both equations. The linear spin-wave bands as- d i sn sn 1 sn 1 2 1 sn 1 2 sociated with Eq. 4 lie in the frequency region from 2B to 4 2B. For all calculations discussed here we use 2Bsn sn 1 sn 1 cos k 1 sn 2 the anisotropy constant B 4, the same value as in Ref. 5. s Then, the linear spectrum corresponds to 8 4. We n 1 sn 1 sin k 1 sn 2 3 find that for any value of k Eq. 4 also has localized solu- with ( H0)/2JS, B A/J, and (2JS/ )t. Here, tions, with frequencies above the linear spin-wave band. The szn is replaced by 1 sn 2, since the magnitude of sn is amplitude and degree of localization of these solutions are conserved, and explicit dependence of sn's on time is now determined not only by , B, but also by k. When k dropped for the sake of brevity. We use free end boundary equals zero they are the stationary ILSM's of Ref. 5. If is conditions. Therefore, the evolution of the end spins n 1 close to the linear spin-wave band region, and k is not too and n N is described by Eq. 3 without the term sn 1 for large, then the amplitude of the ILSM is small and it propa- n 1 and the term sn 1 for n N. Equation 3 together with gates without loss of its shape. We first consider a solution of the boundary conditions constitute the basis of our numerical Eq. 4 for N 501 in the form of a single localized excita- calculations. Supplemented by proper initial conditions, it is tion with 3.85 and k 0.1. The evolution of this exci- solved on a computer by fourth-order Runge-Kutta method. tation is illustrated in Fig. 1 where we plot sn( ) 2 at each To integrate Eq. 3 forward in time, we need an initial set site. The ILSM travels through the chain with a constant of values for sn(0). So far as we can see, if we assume that speed and reflects elastically from the boundaries. The veloc- a stable, moving ILSM exists for a given choice of and k, ity of the propagation is very close to the estimated value 11 460 S. RAKHMANOVA AND D. L. MILLS PRB 58 FIG. 1. The time evolution, as determined by Eq. 3 , of the FIG. 3. Collision of two ILSM solitons. Each has 3.85 initial configuration taken as the solution of Eq. 4 in the form of a and k 0.1, due to the fact that they were obtained by taking as the single soliton with 3.85 and k 0.1. The chain consists of initial configuration a two-soliton solution of Eq. 4 . 501 identical spins. the method just outlined, at all future times we are solving 2 sin k. The excitation remains stable and shows little signs the full equation of motion, Eq. 3 , for each spin in our of decay even after a very long run. system. The authors of Ref. 7 assume s This method of initialization of the traveling ILSM's n( ) to be real at all works quite well for frequencies close to the top of the times. They choose sn(0) by requiring the quantities inside linear spin-wave band and small wave numbers k. For larger the square brackets of Eq. 3 to vanish. They then project 's but k still small we observe a slowdown, stopping, and forward in time by requiring finally reversal of direction of the propagation of the ILSM, ds which is in agreement with the results of Ref. 7 for antifer- n romagnetic chains. If both and k are large in the Eq. 4 , d sn 1 sn 1 sin k 1 sn 2 5 then during the evolution the starting configuration experi- for all times. Note this procedure assumes the quantity in ences relaxation into an ILSM with lower amplitude before it square brackets in Eq. 3 vanishes at all times, and not just begins to move. This process is accompanied by energy re- 0. We have checked explicitly whether this assumption is lease in the form of low amplitude extended spin waves. The remaining localized entity moves freely through the chain as valid by using their method to find sn(0), but then solving pictured in Fig. 2. Clearly, while we have achieved a stable the full Eq. 3 for all times. We find the terms in square ILSM, our initial guess is sufficiently far removed from the brackets to vanish only at short times; one is required to final stable profile that a substantial fraction of the initial solve the full time-dependent equation, unfortunately, once energy is shed by the structure. Note the ``Cerenkov wake.'' the initial configuration is chosen. A spectral analysis of this ILSM shows that its frequency has changed and is smaller than the value that was used in III. COLLISIONS BETWEEN TWO ILSM'S Eq. 4 for obtaining the initial envelope. ON THE PERFECT CHAIN The procedure outlined above differs considerably from In this section, we present studies of collisions between that used in Ref. 7. Once we chose the initial values sn(0) by two ILSM's. In Fig. 3 we show an example of such a colli- sion. To generate this figure, we have proceeded as follows. We have a line of 501 spins. We then, for k 0.1 and the frequency 3.85 recall the top of the linear spin-wave band is at 4.0), find at 0 a two-soliton solution of Eq. 4 ; as we have demonstrated earlier,5 for the finite line, multisoliton solutions exist. As we start the integration of Eq. 3 in time, both ILSM's in the solution move to the right at the same speed. The rightmost feature reflects off the right end of the line, and subsequently the two approach each other and collide. It is evident from the figure that they re- main unchanged in shape. If we regard the two solitons in Fig. 3 as independent entities, then each has precisely the same internal frequency , and wave vector k. One may inquire if this is perhaps a special case. This does not appear to be so, as we illustrate in Fig. 4. FIG. 2. The same as in Fig. 1, but the value of the wave number At time 0, we create two distinct objects as follows. k is much larger, k 0.9. We find a single soliton solution on the line of 501 spins for PRB 58 INTRINSIC LOCALIZED SPIN WAVES IN CLASSICAL . . . 11 461 FIG. 4. Collision of two ILSM solitons, one has 3.90 and k 0.1, and another has 3.95 and k 0. The ILSM solitons are initiated as two different single-soliton state solutions of Eq. 4 with corresponding parameters and k. 3.90 and k 0.1, and then we find a second single internal frequency 3.85. We noted earlier,5 that there soliton state for 3.95 and k 0 on the same line. We are two distinct solutions of Eq. 3 for a given choice of . set up sn(0) by placing the 3.90 mode on the left side This is because the equation of motion is invariant under of the line, and the 3.95 mode on the right side of the sn sn . The second differs from the first by a 180° phase line. The initial configuration is illustrated in Fig. 4 a . Then shift. We thus have two ILSM's which, in some sense, may we integrate the full equations of motion, Eq. 2 forward in be regarded as degenerate. In Ref. 5, the states were studied time; we cannot use Eq. 3 because, of course, it assumes in the presence of an external, circularly polarized oscillating we have a single frequency and wave vector k everywhere magnetic field in the xy plane. Such a field ``splits'' these on the line. When we carry out this calculation, we use the two states, in a sense discussed in Ref. 5. The collision three equations of motion, for sx y z n , sn , and sn , respectively. illustrated in Fig. 3 is between two identical ILSM's. In Fig. We see in Fig. 4 b that the leftmost soliton moves to the 5, we show a collision between an ILSM with 3.85 right, and approaches the rightmost one, which remains sta- and a second such object with same internal frequency, but tionary. We see the two collide, and Fig. 4 f shows that in phase shifted by 180°. We now see a complex interaction the final state, each emerges unchanged in shape. Note that between these two objects. In Fig. 6, we show another ex- there is one small effect of the interaction. The k 0 mode ample. This is the interaction between an ILSM with an in- has been displaced very slightly to the left, after the collision ternal frequency of 3.90, with a very localized ILSM has been completed. at rest with 2.00. Here the ILSM at rest acts like a If one looks closely at Fig. 4 f , one sees very small- perfectly reflecting barrier. amplitude spin waves that have appeared. In our view, to the Interactions between ILSM's are, thus, complex in nature. level of accuracy of our simulation, it is not clear that these We do realize, under the circumstances outlined, that these features are significant. Keep in mind our discussion of Sec. objects can behave in a manner similar to solitons, i.e., they II, where the precise procedure for setting up the spins at pass through each other as if they are noninteracting par- 0 is not clearly defined. Such very small-amplitude spin waves may well be a reflection of the fact that at 0, the stable ILSM has not quite been depicted accurately. In Fig. 4, each ILSM has an internal frequency very close to the top of the linear spin-wave bands. As a consequence of this, the envelopes of each extends over quite a few lattice constants. As noted earlier, in this limit, one may use the continuum approximation, and Eq. 3 may be mapped into the nonlinear Schro¨dinger equation.4 In fact, the envelope function of each ILSM is reproduced nicely by this con- tinuum approximation. Since the nonlinear Schro¨dinger equation is well known to admit multisoliton solutions, and these entities are noninteractive, one may suppose the result in Fig. 4 is, thus, expected. However, the internal frequency of each ILSM in such a multisoliton solution is identical, while we have two distinctly different internal frequencies for the objects in Fig. 4. FIG. 5. Collision between two ILSM's. Each has 3.85 At this point, one is tempted to conclude that the ILSM's and k 0.1. The initial configuration is same as for Fig. 3, except have solitonic properties. Further studies show this is not the that one of the ILSM's is phase shifted by 180° with respect to the case. We return to the case examined in Fig. 3, where the other. 11 462 S. RAKHMANOVA AND D. L. MILLS PRB 58 FIG. 6. Collision of a moving ILSM with low frequency ( FIG. 8. Envelope functions of stationary ILSM on a perfect spin 3.90) with stationary ILSM with high frequency ( 1.00). chain showed by dotted line , and nonlinear localized mode trapped on a defect with B/B 0.03. The internal frequency of ticles. But this behavior is not universal; the interactions may both modes is 3.90. be complex in character. we always have a localized spin-wave mode pushed out of the top of the spin-wave bands of the linear theory. The local IV. NONLINEAR SPIN-WAVE MODES LOCALIZED spin-wave mode thus resides in the frequency region where AT DEFECTS the ILSM's are found, for this sign of B. For B 0, a local mode emerges from the bottom of the spin-wave bands. So far we have explored the properties of ILSM solitons When B 0, if on the perfect lattice. Now we place a defect spin right in the M is the maximum linear spin-wave fre- middle of the line. We model the defect as follows. All spins quency, that loc of the local mode may be written on the line continue to interact with the same nearest- neighbor exchange interaction J. For the defect spin, we loc M , 6a change the anisotropy constant from B, to B B. A real where for our model defect in one-dimensional spin system will have exchange coupling to neighbors of magnitude different than found in 2 1 B 2 2. 6b the host, and the spin s may have different magnitude as well. While such effects are easily included in our model, We find that we have stationary, nonlinear localized mode without an explicit example in mind, a meaningful choice of solutions where the localized nonlinear mode is localized on such parameters is problematical. We wish to study the non- the defect. These states can occur when the internal fre- linear properties of a system which, in linear theory, admits a quency of the localized mode exceeds loc , the frequency localized spin mode above the spin-wave bands in frequency. for localized spin-wave modes in linear theory. For B/B We achieve this by simply decreasing the anisotropy con- 0.03 we show the spatial form of the nonlinear localized stant of a selected spin, designated as the impurity. The re- modes in Fig. 7, for several frequencies. These modes have a sulting model then has just one parameter. spatial profile quite similar to the ILSM's on the perfect line, In the theory of the linearized excitations, when B 0, but have smaller amplitudes, as we see from Fig. 8, where FIG. 7. Nonlinear modes localized at a defect for which B/B FIG. 9. Dependence of the amplitude of nonlinear mode local- 0.03. We show modes for the following frequencies: a ized at a defect site on the value of the defect perturbation. The 3.60, b 3.80, c 3.85, and d 3.90. frequency of the mode is 3.85. PRB 58 INTRINSIC LOCALIZED SPIN WAVES IN CLASSICAL . . . 11 463 FIG. 10. Propagation of an ILSM soliton with 3.85 FIG. 12. Propagation of an ILSM soliton with 3.85 through the spin chain with a defect. The defect spin is at site 251, through the spin chain with a defect. The defect spin is at site 251, and has the value B/B 0.01. and has the value B/B 0.07. we make a comparison between the two. The amplitude of again have a line of 501 spins, with the defect at site 251. the nonlinear mode localized at a defect depends on the In Fig. 10, we show the ILSM soliton passing by a defect value of the perturbation associated with the defect. We il- characterized by a very small value of B/B, B/B 0.01. lustrate this dependence in Fig. 9. The soliton passes over the defect, with no reflected pulse. In Fig. 9, we show the following. For small B/B, we set Clearly, its center-of-mass velocity has decreased rather ap- up a localized ILSM, trapped on the defect. Then we exam- preciably. If one looks carefully at the figure, one sees no ine the amplitude on the impurity of the ILSM, as B/B is evidence of spin waves shed by the ILSM before collision. increased in magnitude with the internal frequency of the Afterward, however, one sees distinct small-amplitude exci- localized ILSM held fixed. We have B/B positive always, tations spread over the chain, outside the region of the ILSM so we have a defect which supports a localized spin-wave soliton. So far as we can see, we have no excitation at the mode in linear theory. We see that as B/B increases in site of the defect, after the ILSM soliton has passed by. Evi- magnitude, the amplitude of the localized ILSM decreases, dently the slowdown has its origin in the energy shed in the to vanish when form of spin waves. For B/B 0.0075, the slowdown is loc equals its internal frequency . We find no localized, nonlinear modes when less pronounced, and the amplitude of the spin waves excited loc . We now turn to our studies of the interactions of propagating ILSM's with after the collision is much smaller. the defect. As B/B increases further, we enter a regime where the ``radiative loss'' of energy by the ILSM soliton is sufficient for this entity to become trapped on the defect. We illustrate V. INTERACTION OF PROPAGATING ILSM MODES this in Fig. 11 where we have B/B 0.015. So far as we WITH THE DEFECT can tell from the simulations, the trapped entity presumably In the previous section, we saw that in the presence of a a localized ILSM such as that discussed in previous section defect in the spin chain, we may have localized ILSM exci- has an infinite lifetime. Clearly, it is perturbed periodically tations, trapped on the defect. In this section, we study the by interaction with the ``spin-wave radiation'' reflected from interaction of moving ILSM's with the same defect. We il- the ends of the chain. As B/B increases yet further, how- lustrate the various characteristic regions through a sequence ever not exceeding the limit where loc exceeds , we see of figures. In all figures, the frequency of the propagating the build up of a reflected ILSM. In Fig. 12, where B/B ILSM is set at 3.85, and its wave vector k 0.1. We will vary the ratio B/B, as we study the interactions. We FIG. 11. Propagation of an ILSM soliton with 3.85 FIG. 13. Propagation of an ILSM soliton with 3.85 through the spin chain with a defect. The defect spin is at site 251, through the spin chain with a defect. The defect spin is at site 251, and has the value B/B 0.015. and has the value B/B 0.35. 11 464 S. RAKHMANOVA AND D. L. MILLS PRB 58 0.07, we see well-defined reflected ILSM solitons, with initial condition. We conclude that in the collisions with each energy stored continuously near the defect site, in the form other, ILSM's can exhibit solitonic properties, as illustrated of a localized ILSM soliton. Notice that the propagating soli- by the examples provided. That is, they preserve their shape, ton is trapped between the defect, and the end of the chain. speed, and identity after the collision. Generally, however, Clearly, one could trap such a mode between two defects. the interactions can be complex. It is possible to have sta- As we saw in the previous section, when the internal fre- tionary nonlinear localized excitation centered at the defect quency is fixed, and loc exceeds , the defect fails to sup- site. The internal frequency of this excitation has to be port a nonlinear ILSM. In this regime, the ILSM soliton is greater than the frequency of corresponding linear localized reflected off the defect; the reflection appears elastic, in the mode. The interaction of a moving ILSM with defects has sense that in the simulations we see no evidence for radiated diverse and rich character, depending on the relation of the spin waves. We illustrate this in Fig. 13. internal frequency of the ILSM to the local mode frequency If B/B 0 in our model, in linear spin-wave theory, there is no spin-wave mode localized at the defect in the of linear spin-wave theory. If loc , so that stationary frequency region above the spin-wave bands. We find no nonlinear localization is not allowed on the defect, the trav- localized nonlinear ILSM's as well. We find for B/B 0, eling ILSM reflects elastically from the defect spin. In the the propagating ILSM is fully reflected from the defect, even range of the defect perturbations, for which nonlinear local- ization is possible, we observe a strong interaction between a when B/B is as small as 0.0075. The interaction has an traveling ILSM and a stationary nonlinear mode localized on appearance very similar to Fig. 13. In this section, we have the defect. explored the interaction of ILSM solitons with defects, to find the rich behavior outlined above. ACKNOWLEDGMENT VI. CONCLUDING REMARKS We have presented a method for simulating a moving This research was completed with support from the Army ILSM numerically, starting from a stationary ILSM as the Research Office, under Contract No. CS0001028. 1 F. Bloch, Z. Phys. 61, 206 1930 . R3828 1995 ; S. V. Rakhmanova and A. V. Shchegrov, ibid. 2 H. J. Mikeska, J. Phys. C 11, L29 1978 ; 13, 2913 1980 ; see, 57, R14 012 1998 . also K. M. Leung, D. W. Hone, D. L. Mills, P. Riseborough, and 5 S. Rakhmanova and D. L. Mills, Phys. Rev. B 54, 9225 1996 . S. E. Trullinger, Phys. Rev. B 21, 4017 1980 . A discussion of 6 R. Lai and A. J. Sievers, Phys. Rev. B 57, 3433 1998 ; 55, moving solitons, with attention to circumstances where the R11 937 1997 . simple relativistic analogy fails is given by P. S. Riseborough, S. 7 A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep. E. Trullinger, and D. L. Mills, J. Phys. C 14, 1109 1981 . 194, 117 1990 . 3 A. P. Malozemoff and J. Sloncewski, Magnetic Domain Walls in 8 R. Lai, S. A. Kiselev, and A. J. Sievers, Phys. Rev. B 54, R12 665 Bubble Materials Academic, New York, 1979 . 1996 . 4 R. F. Wallis, D. L. Mills, and A. D. Boardman, Phys. Rev. B 52,