PHYSICAL REVIEW B 1 OCTOBER 1998-IIVOLUME 58, NUMBER 14
Nonlinear excitations in ferromagnetic chains with nearest-and next-nearest-neighbor exchange interactions
Guoxiang HuangDepartment of Physics and Center for Nonlinear Studies, Hong Kong Baptist University, Hong Kong, China
and Department of Physics, East China Normal University, Shanghai 200062, China
Shudong ZhangDepartment of Physics and Center for Nonlinear Studies, Hong Kong Baptist University, Hong Kong, China
and Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China
Bambi HuDepartment of Physics and Center for Nonlinear Studies, Hong Kong Baptist University, Hong Kong, China
and Department of Physics, University of Houston, Houston, Texas 77204~Received 12 January 1998; revised manuscript received 11 May 1998!
Weakly nonlinear excitations in one-dimensional isotropic Heisenberg ferromagnetic chains with nearest-and next-nearest-neighbor exchange interactions are considered. Based on the properties of modulationalstability of corresponding linear spin waves, the existence regions of bright and dark magnetic solitons of thesystem are discussed in the whole Brillouin zone. The antidark soliton mode which is convex soliton super-imposed with a plane wave component is obtained near the zero-dispersion points of the spin wave frequencyspectrum. The analytical results are checked by numerical simulations.@S0163-1829~98!01838-4#
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I. INTRODUCTION
The linear elementary excitations out of the ground staof ferromagnetic and antiferromagnetic systems, i.e., linspin waves, have been well understood.1 In recent years, inlow-dimensional magnetism the study of the nonlinearementary excitations such as magnetic solitons has genea great deal of theoretical and experimental interest.2,3 How-ever, since in theory nearly all theoretical approachesvolved the continuum approximation, valid only for zoncenter spin wave modes~i.e., for q50, whereq is the wavenumber of the spin waves!, some important nonlinear modeof rather short wavelength have been lost. We know thatHeisenberg model for describing magnetic phenomena isherently discrete, with the lattice spacing being a fundamtal physical parameter. For such discrete systems an accmicroscopic description involves a set of nonlinedifference-differential equations and the intrinsic discreness may drastically modify the nonlinear dynamics ofsystems. The discreteness makes the systems lose contitranslational symmetry, thus producing a lower cutoff in twavelength and a finite upper bound for the frequency sptrum of the linear spin waves. Due to the interplay betwethe discreteness and nonlinearity, new types of nonlinearcitations, which have no direct analogy in continuum moels, may exist. In nonlinear atomic lattices, recent studhave shown that some novel nonlinear localized excitatiosay the intrinsic localized modes~or called the discretebreathers! and the intrinsic gap modes can be natural nonear excitations of the systems. These nonlinear localimodes in perfectly periodic lattices have shorter wavelenin carrier waves and a somewhat similar character ofpreviously studied force constant or mass defects occurin purely harmonic lattices, but they can appear at any lat
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site because of the discrete translational symmetry ofsystems.~For details see the recent review papers, Refs. 4and references therein.!
The analogy between lattices and spin waves has stilated a series of studies of intrinsic localized spin-wamodes ~ILSMs! in semiclassical and classical magnemodels.8–19 For Heisenberg ferromagnetic and antiferromanetic chains with on-site easy-axis anisotropy, an ILSM wthe vibrating frequency above spin wave bands~we call it theupper-cutoff ILSM! is impossible due to the softness of thintrinsic nonlinearity in exchange interactions, but ILSMwith the vibrating frequency below the spin wave bands~thelower-cutoff ILSMs! may exist.8–12,15–17 The lower-cutoffILSMs are related to the modulational instability of corrsponding lower-cutoff spin waves.18 If a strong magneticfield is applied which is perpendicular to the easy plane,upper-cutoff ILSM may appear in easy-plane Heisenbergromagnetic chains when the strength of the single-ion aniropy exceeds a certain value.13,14 In a recent work, the gapsoliton and kink modes have been found in the Heisenbferromagnetic chains with bond alternation20 by using a qua-sidiscreteness approach~QDA!, developed in the study of theintrinsic localized and gap modes in nonlinear monoatomand diatomic lattices.21–24 At this stage, there is a need tconsider all possible nonlinear localized excitationsHeisenberg chains in a simple and systematic way.
In this paper, we study the weakly nonlinear excitationsthe Heisenberg ferromagnetic chains with both nearneighbor ~NN! and next-nearest-neighbor~NNN! isotropicexchange interactions. Based on the QDA, our resultsvalid in the whole Brillouin zone~BZ! and all possiblesmall-amplitude nonlinear excitations are discussed. Tmodel, with the Heisenberg ferromagnetic chain with onthe NN exchange interaction as its particular case, was c
PRB 58 9195NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
sidered recently by Lai, Kiselev, and Sievers.25 Due to theintroduction of the NNN interaction, the curvature of dispesion curve of the linear spin waves is considerably changThis change results in a transition of types of the nonlinlocalized modes in the BZ, especially near the zone bouary, when the ratio of the NNN interaction to the NN onexceeds a threshold. We show that an antidark soliton mwhich is similar to the intrinsic spin-wave resonance fourecently by Laiet al.,25 exists near the zero-dispersion poi~ZDP! of the spin wave spectrum. The ZDP tends to theboundary when the ration of the NNN interaction to the None approaches its threshold.
The paper is organized as follows. In the next section,model is introduced and, based on the QDA, an asymptexpansion is made and a nonlinear amplitude equationweakly nonlinear modes is obtained. The modulational insbility analysis starting from the amplitude equation is maand the existence regions of bright and dark soliton exctions are discussed in the whole BZ. In Sec. III we consithe dark and antidark soliton solutions near the ZDPs. In SIV a numerical simulation is used to check such solutioand estimate the lifetimes of these modes. Finally, Seccontains a discussion and summary of our results.
II. THE MODEL, AMPLITUDE EQUATION, BRIGHT ANDDARK SOLITONS
A. The model
We consider a ferromagnetic Heisenberg spin chain wboth the NN and the NNN isotropic exchange interactioThe Hamiltonian of the system is given by25
H522J1(n
Sn–Sn1122J2(n
Sn–Sn12 , ~1!
whereSn5(Snx ,Sn
y ,Snz) is the spin on siten and J1 and J2
are, respectively, the NN and the NNN exchange interacconstants, both of them being taken to be positive. Sincesystem has a continuum rotational symmetry, there is noof generality in assuming that in the ground state all spalign along thez axis direction. If we definesn5Sn /(S\)and Sn
65Snx6 iSn
y , whereS is the spin length and\ is thePlanck constant, then we have the Heisenberg equatiomotion for sn
1
i1
2J1Sc
dsn1
dt5sn
1@sn21z 1sn11
z 1A~sn22z 1sn12
z !#
2snz@sn21
1 1sn111 1A~sn22
1 1sn121 !#, ~2!
whereSc5S\. A5J2 /J1 is the ratio measuring the strengof the NNN interaction relative to that of the NN interactioFollowing Ref. 25 hereafter we treat the spins as classvectors. Thus we havesn
25(sn1)* and sn
z5A12sn1sn
2
5A12usn1u2. Equation~2! becomes
-d.r
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ne
sss
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al
i1
2J1Sc
dsn1
dt5sn
1@A12usn1u21A12usn11
1 u2
1A~A12usn221 u21A12usn12
1 u2!#
2A12usn1u2@sn21
1 1sn111 1A~sn22
1 1sn121 !#.
~3!
The linear dispersion relation of Eq.~3! is
v~q!54J1Sc$12cos~qa!1A@12cos~2qa!#%, ~4!
wherev and q are the frequency and wave number of tlinear spin waves, anda is the lattice spacing between adjcent spins. Shown in Fig. 1 is the dispersion curves for dferent values ofA. We can see that the introduction of thNNN coupling changes the curvature of the dispersion cuconsiderably. WhenA,1/4, there is only a minimum at thezone center wave numberq50 and a maximum at the zonboundary wave numberq5qZB5p/a. However, ifA.1/4,q5qZB becomes a point of local minimum. A new maximuappears atq5qmax, here
qmax5p
a2
1
acos21S 1
4AD . ~5!
The corresponding maximum frequency atqmax is v5vmax, which is given by
vmax5J1Sc
2
~4A11!2
A. ~6!
Betweenqmax and qBZ an inflection point~i.e,, the pointwherev950) occurs atq5q2 , where
q25p
a2
1
acos21F11A11128A2
16A G . ~7!
FIG. 1. The dimensionless dispersion curves of linear swaves for different values ofA. WhereA5J2 /J1 with J1 (J2) isthe constant of the NN~NNN! exchange interaction. From bottomto top,A takes 0, 0.25, 0.50, and 1.0, respectively.
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9196 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
WhenA approaches 1/4,q2 approachesqZB . The inflectionpoint is just the zero-dispersion point of the dispersion curThusA5Ac51/4 is a ‘‘critical’’ value for the appearance othe inflection pointq5q2 .
B. Asymptotic expansion and nonlinear amplitude equation
We use the QDA~Refs. 20–24! to investigate the effectsof nonlinearity and discreteness of the system. In this trement one sets
sn1~ t !5esn,n
~1!1e2sn,n~2!1e3sn,n
~3!1¯, ~8!
wheree is a smallness and ordering parameter denotingrelative amplitude of the spin wave excitation andsn,n
( j )
E
n
o
.
t-
e
5s(j)(jn ,t;fn), i.e., the first~second! subscriptn of sn,n( j ) rep-
resents the variablejn (fn). jn5e(na2lt) andt5e2t aretwo multiple-scale variables~slow variables!. l is a param-eter to be determined by a solvability condition. The ‘‘fasvariable,fn5qna2v(q)t, representing the phase of carriwave, is taken to be completely discrete. Substituting Eq.~8!into Eq.~3! and equating the coefficients of the same powof e, we obtain a hierarchy of equations aboutsn,n
( j ) ( j51, 2, 3, . . . ):
L̂sn,n~ j ! 5an,n
~ j ! ~9!
with
an,n~1!50, ~10!
an,n~2!5 i
1
2J1Scl
]
]jnsn,n
~1!1a]
]jn~sn,n21
~1! 2sn,n11~1! !12aA
]
]jn~sn,n22
~1! 2sn,n12~1! !, ~11!
an,n~3!5 i
1
2J1ScS l
]
]jnsn,n
~2!2]
]tsn,n
~1! D1a]
]jn~sn,n21
~2! 2sn,n11~2! !2
a2
2!
]2
]j2~sn,n21
~1! 1an,n11~1! !
12aA]
]jn~sn,n22
~2! 2sn,n12~2! !22a2A
]2
]j2~sn,n22
~1! 1an,n11~1! !1
1
2usn,n
~1! u2~sn,n21~1! 1sn,n11
~1! !21
2sn,n
~1! ~ usn,n21~1! u21usn,n11
~1! u2!
1A
2@ usn,n
~1! u2~sn,n22~1! 1sn,n12
~1! !2sn,n~1! ~ usn,n22
~1! u21usn,n12~1! u2!#, . . . . ~12!
r
The operatorL̂ is defined by
L̂un,n~ j ! [ i
1
2J1Sc
]
]tun,n
~ j ! 1~un,n21~ j ! 1un,n11
~ j ! 22un,n~ j ! !
1A~un,n22~ j ! 1un,n12
~ j ! 22un,n~ j ! ! ~13!
with un,n( j ) ( j 51, 2, 3, . . . ) a set ofarbitrary functions. The
expressions ofan,n( j ) for j 54, 5, 6, . . . ,need not be explicitly
written down here.Using the same procedure as in Ref. 21 we can solve
~9! order by order. The leading order solution (j 51) is
sn,n~1!5F~jn ,t!exp~ ifn!, fn5qna2v~q!t, ~14!
whereF(jn ,t) is an amplitude~or envelope! function yet tobe determined andv(q) is the linear dispersion relatiogiven by Eq.~4!. At the second order (j 52) a solvabilitycondition determines
l5dv
dq54J1Sc a@sin~qa!12A sin~2qa!#. ~15!
For j 53 ~the third order!, a solvability condition yields thenonlinear amplitude equation controlling the evolutionF(jn ,t)
q.
f
i]F
]t1P
]2F
]jn2
1QuFu2F50, ~16!
where
P51
2
d2v
dq252J1Sc a2@cos~qa!14A cos~2qa!#, ~17!
Q51
2v~q!52J1Sc$12cos~qa!1A@12cos~2qa!#%.
~18!
Equation~16! is the standard nonlinear Schro¨dinger ~NLS!equation. Under the transformationF5(1/e) f , Eq. ~16! canbe written in the form
i] f
]t1P
]2f
]xn2
1Qu f u2f 50, ~19!
when returning to the original variables. In Eq.~19! xn5na2Vgt with Vg being the group velocity of the carriewave exp(ifn), i.e., Vg5dv/dq.
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PRB 58 9197NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
C. Modulational instability of extended spin waves:Bright and dark solitons
When deriving the nonlinear amplitude equation~16! @orequivalently Eq.~19!# we have not specified any given wavvector, i.e., the result is valid in the whole BZ except at tZDPsq5qj ( j 51,2), where
q151
acos21SA11128A221
16A D . ~20!
q2 has been given in Eq.~7!.26 Thus we can discuss thmodulational instability of the extended spin waves andweakly nonlinear excitations in the whole BZ in a simple aunified way.
In recent years, the use of nonlinear amplitude equatifor studying the stabilities of patterns and waves in systein and out of equilibrium is widely employed.27–29 Here bysimilar ideas we consider the stability of the linear~ex-tended! spin waves by using the amplitude equation~19!. Auniform vibrating solution of Eq.~19! is
f 5 f 0exp@ iQ~q!u f 0u2t#, ~21!
where f 0 is any complex constant, which corresponds tolinear spin wave of wave numberq with a simple frequencyshift 2Q(q)u f 0u2 and as a fixed point of the system. Nothat it is possible to eliminate the time dependence bsimple transformation, justifying our use of the term ‘‘fixepoint’’ for the uniform vibrating solution. The fixed point infact may also be written as
f 5 f 0exp$ i @Q~q! f 02t1f#%, ~22!
where f 0 in this case can be any real constant and 0<f,2p. In this sense we have a ring of fixed point charactized by the different values of the phasef. From Eq.~19! itis easy to show that the fixed point is unstable for a lowavelength small perturbation, if the sign of the product oPand Q, i.e., sgn(PQ), is positive. This kind of instability isdue to the sideband modulation of the linear wave andcalled the modulational instability~MI ! or the Benjamin-Feirinstability.27–29 It is similar to the Eckhaus instability fopatterns in extended dissipative systems outequilibrium.30,31 We believe that the MI is the basic mechnism for the formation of nonlinear localized structuresspin systems.18 By this mechanism~usually called theBenjamin-Feir resonance mechanism27!, if sgn(PQ).0 a lin-ear spin wave mode will bifurcate, grow exponentiallyfirst and then saturate due to the nonlinearity of the systAt last stage a nonlinear localized spin wave mode of stonlike structure will appear.
SinceQ(q)5v(q)/2.0,26 the regions ofq for the MI areonly determined by the sign ofP(q)5v9(q)/2. Figure 2shows in the half BZ (0,p/a# the curves ofQ(q) andv9(q)for A50.27. We have the following conclusions.
~1! In the open region (0,q1) and the half open region(q2 ,p/a#, i.e., the positive or normal dispersion regions, tlinear spin waves are unstable.
~2! In the open region (q1 ,q2), i.e., the negative oranomalous dispersion region, a linear spin wave is sta~actually it is just neutrally stable!.32
e
ss
e
a
-
g
is
f
t.
i-
le
It should be noted that the above conclusions are ovalid for A.Ac51/4. WhenA,Ac , q25p/a thus the un-stable region (q2 ,p/a# disappears. Thus the introduction othe NNN exchange interaction results in a change forstability property of the spin waves.
Next we discuss the localized solutions of the amplituequation~19!. In the positive dispersion regions, i.e.,q lo-cates in (0,q1) or (q2 ,p/a#, we have thebright soliton so-lution
f ~xn , t !5 f 0sechF S Q
2PD 1/2
f 0 ~xn2xn0!G
3expF i1
2Q f0
2t2 if0G , ~23!
where f 0 , xn0, andf0 are real integral constants. Then w
have
sn1~ t !' f 0sechH S Q
2PD 1/2
f 0 @~n2n0!a2Vgt#J3exp$ i ~qna2Vst !2 if0%, ~24!
wheren0 is an arbitrary integer and
Vs5v~q!21
2Q~q! f 0
25v~q!F121
4f 0
2G ~25!
is the vibrating frequency of the bright soliton. Sincev(q).0, we haveVs,v(q).
From Eq.~24! we see that the free parameterf 0 ~the am-plitude of the soliton! can be taken as an expansion paraeter in our QDA, i.e.,f 05O(e). Using Eq. ~25! we havef 052@12Vs /v(q)#1/2. Thus in our approach the expansioparametere, used in Eq.~8!, is proportional to the squareroot of the frequency difference between the nonlinear extation and the corresponding linear magnon mode. The Qis valid for smalle thus the frequency difference mentioneabove should be small in our approach.
FIG. 2. The curves ofv9(q)/(4J1Sca2) ~solid line! and
Q(q)/(2J1Sc) ~dash line! for A50.27. The appearance of the region (q2 ,p/a# wherev9.0 is due toA.Ac51/4. WhenA<Ac ,q2→p/a thus the length of the region (q2 ,p/a# approaches zero.
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9198 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
If we chose a special wave vectorq5p/a, Eq. ~19! inthis case is
i] f
]t12J1Sc a2~4A21!
]2f
]x214J1Scu f u2f 50, ~26!
wherex5na. Thus only whenA.Ac51/4 do we have thebright soliton solution. This particular case has been dcussed recently by Laiet al.25
In the negative dispersion region, i.e.,q is at (q1 ,q2),since sgn(PQ),0 we have generally thedark soliton ~orkink! solution33
f ~xn , t !5 f 0
~l2 in!21exp Z
11exp Zexp~ iQ f 0
2t2 if0! ~27!
with
Z5n f 0A2Q
uPu~xn2xn0
2lA2uPuQt!, ~28!
where l2512n2 and 0<n<1. When n51, Eq. ~27! be-comes a black soliton
f ~xn , t !5 f 0tanhFA Q
2uPuf 0 ~xn2xn0
!Gexp~ iQ f 02t2 if0!.
~29!
If n!1, the solution~27! can be approximated as
f ~xn , t !5 f 0 F121
2n2sech2S Z
2D Gexp@ iQ f 02t1F~Z!#
~30!
with
F~Z!522n
11exp Z, ~31!
Z5A2Q
uPun f 0Fxn2xn0
7AuPuQ2
f 0~22n2!tG . ~32!
Thusn is the parameter controlling the ‘‘blackness’’ of thdark soliton. The expression ofsn
1(t) in the negative dispersion case is given by
sn1~ t !' f 0
~l2 in!21exp Z
11exp Zexp@ i ~qna2Vkt !2 if0#
~33!
with
Vk5v~q!2Q~q! f 025v~q!F12
1
2f 0
2G ~34!
being the vibrating frequency of the dark excitation.It should be noted thatqmax, the wave number wherev
takes its maximumvmax, is in the negative dispersion regio(q1 ,q2). Thus atq5qmax we only have a dark excitationAnother point which also should be strengthened is twhen A approachesAc51/4, the length of the positive dispersion region (q2 ,p/a# tends to zero. Accordingly, ifA
-
t
,1/4, the BZ may be simply divided into two regions.(0,q1) and (q1 ,p/a) we have bright and dark excitationsrespectively.
From Eqs.~25! and ~34! we see that all the vibratingfrequency of the bright and dark excitations are lowerfrom the corresponding linear spin wave frequencyv(q).But there is a difference for the decreasing rate of thequencies between the bright and dark excitations. The phcal reason for the decrease of frequency is that the intrinnonlinearity in the exchange interaction of the system is sThis conclusion in fact can be directly obtained from tamplitude equation ~19!. Taking f (xn ,t)5 f 0exp@i(Kxn2Vt)#, whereK andV correspond, respectively, to the shifof the wave number and frequency of the carrier exp(ifn),we have
V5P~q!K22Q~q! f 02 . ~35!
BecauseQ(q).0, the frequency shiftV is always a para-bolic decreasing function of the amplitudef 0 . In fact for anytype of nonlinear excitations in the systems with a soft nolinearity, the vibrating frequency of the excitation is alwalowered.
The above analytical results are checked by numersimulations. A detailed discussion for the simulationsgiven in Sec. IV.
III. ANTIDARK SOLITONSNEAR THE ZERO-DISPERSION POINTS
A. Nonlinear amplitude equationnear the zero-dispersion points
In the last section, we studied very generally the nonlinexcitations of the system based on the amplitude equa~19!. However, as already mentioned the discussion islonger adequate near the ZDPsq5qj @ j 51, 2; for q1 andq2 , see Eqs.~7! and~20!# because near these pointsP(qj ) isvery small. It can also be seen from the expressions~23!,~24!, and~28! that the soliton and kink solutions of Eq.~19!become singular whenq→qj @thus P(q)→P(qj )50].Hence it is necessary to reconsider the amplitude equa~19! near the ZDPs. Physically, whenq→qj , the dispersionterm in Eq.~19! becomes small so that a balance betweennonlinearity and the dispersion is lost. Thus in this casnonlinear localized structure is impossible in this approac
However, the dispersive effect always exists atq5qj be-cause of the discreteness of the system. Although atq5qj ,v9(qj )50, but the third-order dispersion, representedv-(qj ), is finite. Figure 3 shows the curves ofv9(q) andv-(q) for A50.27. In this section we investigate the nolinear excitations whenq is near the ZDPs.
A natural consideration is that, near the ZDPs, the thiorder dispersion term2( i /3!)v-(q)(]3f )/(]xn
3) should beadded in Eq.~19!. Thus we have
i] f̃
]t1
1
2!v9~q!
]2 f̃
]xn2
2i
3!v-~q!
]3 f̃
]xn3
1Q~q!u f̃ u2 f̃ 50,
~36!
where we have introduced a new quantityf̃ for the amplitude~envelope! since Eq.~36! can be derived based on the QD
ua
nsth
so
r
e
e
f
PRB 58 9199NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
under the scalings jn5e(na2Vgt), t̃5e3t, sn1(t)
5e2/3(sn,n(1)1esn,n
(2)1¯) with sn,n(1)5F̃(jn , t̃)exp@i(qna
2vt)#, andv95eL with L5O(1). Let f̃ 5e3/2F̃ and returnto the original variables, we then obtain the amplitude eqtion ~36!.
B. Antidark soliton solutions
An exact analytical solution of Eq.~36! is not availableyet. In what follows we consider its approximate solutioby using the connection between the NLS equation andKorteweg-de Vries~KdV! equation34 though here we havethe third-order dispersion term. We assume thatq5q11uDqu or q22uDqu with uDqu being small. Thusv9(q) isnegative for these values ofq @i.e.,q is nearqj ( j 51, 2) butin the region (q1 ,q2)]. We can see in the following that thiis the necessary condition for making a relation betweenmodified NLS equation~36! and the KdV equation in ousystem.
Let a5uv9(q)u/2(.0), b5(1/3!)v-(q) and f̃5(2/Q)1/2u(xn ,t), then Eq.~36! becomes
i]u
]t2a
]2u
]xn2
2 ib]3u
]xn3
12uuu2u50. ~37!
Equation~37! admits the uniform vibrating solution with thform u5u0exp(2iu0
2t) ~where u0 is an arbitrary real con-stant!, which is relevant to the linear spin wave of wavnumberq with a simple frequency shift22u0
2. We predictthat the general solution of Eq.~37! takes the form
u5@u01A~xn ,t !#exp@2iu02t1 iF~xn ,t !#, ~38!
whereA andF are two real functions. Substituting Eq.~38!into Eq. ~37! we obtain
FIG. 3. The curves ofv9(q)/(4J1Sca2) ~dash line! and
v-(q)/(4J1Sca3) ~solid line! for A50.27.
-
e
ur
]A
]t2au0
]2F
]xn2
2aS A]2F
]xn2
12]A
]xn
]F
]xnD
2bS ]3A
]xn3
23u0
]F
]xn
]2F
]xn2 D 50, ~39!
u0S ]F
]t24u0AD1A
]F
]t1a
]2A
]xn2
2au0S ]F
]xnD 2
26u0A2
2bS u0
]3F
]xn3
1A]3F
]xn3
13]A
]xn
]3F
]xn3
13]F
]xn
]2A
]xn2 D 50.
~40!
Supposing thatA andF are slowly varying functions of thevariablesXn5e(xn2ct) andT5e3t, wherec is an undeter-mined parameter, and making the expansion
A5e2@A~0!1e2A~1!1e4A~2!1¯#, ~41!
F5e@F~0!1e2F~1!1e4F~2!1¯#, ~42!
then by equating the powers ofe we have a hierarchy oequations aboutA( j ) andF ( j ) ( j 50, 1, 2, . . . ).
]
]XnS cA~ j !1au0
]F~ j !
]XnD 5M ~ j !, j 51, 2, 3, . . . , ~43!
with
M ~0!50, ~44!
M ~1!5]A~0!
]T2aA~0!
]2F~0!
]Xn2
22a]A
]Xn
]F~0!
]Xn2b
]3A
]Xn3
13bu0
]F~0!
]Xn
]2F~0!
]Xn2
, . . . , ~45!
and
u0S c]F~ j !
]Xn14u0A~ j !D 5N~ j !, ~46!
with
N~0!50, ~47!
N~1!5u0
]F~0!
]T2cA~0!
]F~0!
]Xn1a
]2A~0!
]Xn2
2au0S ]F~0!
]XnD 2
26u0~A~0!!22bu0
]3F~0!
]Xn3
, . . . . ~48!
For j 50, the solution of Eqs. ~43! and~46! is c5d12Aau0 with d1561 and F (0)5
2d1(2/Aa)*2`Xn A(0)(Xn8 ,T8)dXn8 . A(0)(Xn ,T) is still a func-
tion undetermined yet. At the next order (j 51), a solvabilitycondition results in the closed equation forA(0):
lyive-
l
thethe
ht
ierce
le
nn-
am--
9200 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
]A~0!
]T1
12au0
c S 11bc
2a2D A~0!]A~0!
]Xn
21
2c~a212bc!
]3A~0!
]Xn3
50. ~49!
Equation~49! is just the KdV equation which is a completeintegrable system. It should be noted that in the positdispersion region, wherev9.0, one can only get a purimaginary wave speedc thus a reduction to the KdV equation ~49! from the modified amplitude equation~36! is im-possible.
Let G5e2A(0) and recall the definitions ofXn andT, wecan transfer Eq.~49! into the form expressed in the originavariables:
]G
]t1
12au0
c S 11bc
2a2D G]G
]yn2
1
2c~a212bc!
]3G
]yn3
50,
~50!
whereyn5xn2ct5na2(c1Vg)t. The single-soliton solu-tion of Eq. ~50! is
G~yn , t !52k2~a3/21d14bu0!
2u0~a3/21d1bu0!sech2H k
AaFyn2yn0
1d1
k2~a3/21d14bu0!
au0t G J , ~51!
wherek is an arbitrary constant andyn05n0a with n0 being
an arbitrary integer. From Eq.~51! one can obtain
F'eF~0!5d1
k~a3/21d14bu0!
u0~a3/21d1bu0!H tanhF k
AaS yn2yn0
1d1
k2~a3/21d14bu0!
au0t D G11J . ~52!
It is easy to get
sn1~ t !'F 2u0
Av~q!1Gsol~yn , t !Gexp$ iqna2 i @v~q!22u0
2#t
1 iF~yn , t !%, ~53!
e
where Gsol(yn , t)5@2/Av(q)#G(yn , t). Thus near theZDPs and whenv9(q)52uv9(q)u522a,0, sn
1(t) is asuperposition of two parts. One is a plane wave withamplitude 2u0 /Av(q) and the other one is a breather withe envelopeGsol(yn , t), which can be positive and negativdepending on the values ofa, b, u0 , andd1 . It is just thedifferent choices of the values ofa, b, u0, andd1 that makeus have the possibility of different types of dark or brigexcitations, which will be discussed in the following.
Noticing that the small amplitude approximation forAand F used in Eqs.~41! and ~42! means thatk is a smallparameter appearing in Eqs.~51! and~52!, i.e.,k;e. Hencefrom Eq. ~53! we can see that the frequency of the carrwave is slowly modulated to become time and spadependent. Sincek is small one can easily obtain a simpexpression for the vibrating frequencyV r of the carrier wavefor the spin on the siten5n0
V r5v~q!F121
2usn0
1 u2G , ~54!
whereusn0
1 u52u0 /Av(q)1Gsol(yn , 0) is the maximum spin
derivation. ThereforeV r is a parabolic decreasing functioof usn0
1 u, reflecting again the softness of the intrinsic nonli
earity of the exchange interaction of the system.Now we analyze these results for the cases whenq5q1
1uDqu andq5q22uDqu, respectively.~1! q5q22uDqu. In this case we haveb.0 ~see Fig. 3!.
The feature of the solution depends on dimensionless pareter a3/2/(bu0), i.e., the ratio of the second- to the thirdorder dispersion, and the choice ofd1 .
~i! If 1 ,a3/2/(bu0),4 andd1521, we have the solu-tion
sn1~ t !'F 2u0
Av~q!1Gsol~n, t !G
3exp$ iqna2 i @v~q!22u02#t1 iF~n, t !%,
~55!
with
G~n, t !54bu02a3/2
a3/22bu0
k2
u0Av~q!sech2H k
AaF ~n2n0!a2S Vg22Aau02
k2~4bu02a3/2!
au0D t G J , ~56!
F~n, t !54bu02a3/2
a3/22bu0
k
u0H tanhX k
AaF ~n2n0!a2S Vg22Aau02
k2~4bu02a3/2!
au0D t GC11J . ~57!
Since the functionGsol is positive, i.e., Eq.~55! is a convex soliton~i.e., the dark soliton with a amplitude of reverse sign! plusa superimposed plane wave component. We call Eq.~55! the antidark soliton. It may appear at any lattice siten0 and movewith the velocityVg22Aau02k2(4bu02a3/2)/(au0), a reflection of discretely translational symmetry of the system.
~ii ! Whena3/2/(bu0),1 or a3/2/(bu0).4 andd1521, one has
ding
PRB 58 9201NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
sn1~ t !'F 2u0
Av~q!2Gsol~n, t !Gexp$ iqna2 i @v~q!22u0
2#t2 iF~n, t !% ~58!
with
G~n, t !5U 4bu02a3/2
a3/22bu0U k2
u0Av~q!sech2H k
AaF ~n2n0!a2S Vg22Aau02
k2~4bu02a3/2!
au0D t G J , ~59!
F~n, t !5U 4bu02a3/2
a3/22bu0U k
u0H tanhX k
AaF ~n2n0!a2S Vg22Aau02
k2~4bu02a3/2!
au0D t GC11J . ~60!
Obviously, Eq.~58! is a dark soliton, consisting of a concave soliton and a superimposed plane wave.~iii ! d1511 always gives dark solitons. The expression of the solutions in this case is omitted here.~2! q5q11uDqu. In this case we haveb52ubu,0 ~see Fig. 3!. One can also have dark and antidark solitons, depen
on the values ofa3/2/(ubuu0) andd1 .~iv! If 1 ,a3/2/(ubuu0),4 andd1511, we have the antidark soliton solution
sn1~ t !'F 2u0
Av~q!1Gsol~n, t !Gexp$ iqna2 i @v~q!22u0
2#t2 iF~n, t !% ~61!
with
G~n, t !54ubuu02a3/2
a3/22ubuu0
k2
u0Av~q!sech2H k
AaF ~n2n0!a2S Vg12Aau02
k2~4ubuu02a3/2!
au0D t G J , ~62!
F~n, t !54ubuu02a3/2
a3/22ubuu0
k
u0tanhX k
AaF ~n2n0!a2S Vg12Aau02
k2~4ubuu02a3/2!
au0D t GC11. ~63!
d
nv
th
o
aree
the
~v! Whena3/2/(ubuu0) is less than 1 or larger than 4 and1511, one has dark solitons.
~vi! For any values ofa3/2/(ubuu0), d1521 always givesdark solitons.
From the results discussed above, we obtain the existecondition for the antidark solitons in the system: the wanumber should be in the negative dispersive [email protected].,v9(q),0] and
1,a3/2
ubuu0,4. ~64!
Thus the appearance of the antidark solitons is due tocompetition between the second and the third dispersionofthe system.
C. Antidark soliton for A2Ac!1
In this section we discuss the features of the antidark stons whenq5q22uDqu and A2Ac!1. Note that whenA2Ac is small we have
q25p
a2
2
aA2
3~A2Ac!
1/2138
9aA2
3~A2Ac!
3/2
1O@~A2Ac!5/2#, ~65!
cee
e
li-
v~q2!5vZBH 1120
9~A2Ac!
21O@~A2Ac!3#J , ~66!
v8~q2!5Vg~q2!5avZBH 28
3A2
3~A2Ac!
3/2
1O@~A2Ac!5/2#J , ~67!
v-~q2!5a3vZB$A6~A2Ac!1/21A6~A2Ac!
3/2
1O@~A2Ac!5/2#%, ~68!
where vZB5v(p/a)58J1Sc is the frequency of the BZboundary. We now have three small parameters. TheyA2Ac , uDqu, andk. For a given problem, they should havsome relations among each other. We assumeuDqua5(A2Ac)
3/4Q0 andk5AvZBk8(A2Ac)1/2 with Q0 andk8
being two dimensionless parameters of order unity. Fromantidark soliton solution expressed in Eqs.~55!–~57! one ob-tains the following results.
~i! The amplitude of the plane wave part in Eq.~55! is
Apw52u0
Av~q!'
2u0
Av~q2!'
u08
A11 ~20/9! ~A2Ac!2
,
~69!
ea
ha
ti-
th
xi-
itoe
ch
wrine
d
eu-
odfor
kedis
nsdes
reda-
g
a-
ime
ur
n-aryof
ingec-in.is
hes
oreterirdlu-
h
in
9202 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
whereu0852u0 /AvZB.~ii ! Noticing that a5uv9(q)u/2'uv-(q2)uuDqu/2 and b
5v-(q)/3!'v-(q2)/3!, we obtain the amplitude of thesoliton part
Asol54bu02a3/2
a3/22bu0
k2
u0Av~q!,
'2~k8!2
u0
4u082123223/2361/4Q03/2~A2Ac!
11/8
123223/2361/4Q03/2~A2Ac!
11/82u08
3~A2Ac!. ~70!
From Eqs.~69! and ~70! we can see that when the relativstrength of the NNN exchange interaction decreases, theplitude of the plane wave partApw is reduced. WhenA5Ac the soliton amplitude vanishes and the plane waveits maximum amplitudeApw5u08 . ThusA5Ac is similar to apoint of ‘‘phase transition’’ for the occurrence of the andark soliton.
~iii ! The soliton width is determined by@see Eq.~56!#
Aa/k5S 1
2uv9~q!u D 1/2
/k'S 1
2uv-~q2!uuDqu D 1/2Y k
'@221/2•61/4Q0
1/2a/k8#~A2Ac!1/8. ~71!
Therefore, the soliton width becomes narrower~more local-ized! if A2Ac decreases.
~iv! The soliton speedVsol is given by
Vsol5Vg22Aau02k2~4bu02a3/2!/~au0!
'2F2
3~vZB!3/2u08a~k8!2/Q0G~A2Ac!
1/4. ~72!
Thus it is negative and becomes zero whenA5Ac .~v! It is easy to show that in the present conditions
expression forsn1(t) has the form
sn1~ t !'~21!n@Apw1Gsol~n, t !#
3expH 2 i F2S 2
3D 1/2
~A2Ac!1/21Q0~A2Ac!
3/4Gn2 ivZBS 12
1
2~u08!2D t1 iF~n, t !J . ~73!
The vibrating frequency of the carrier wave is still appromated by Eq.~54!.
From above discussion we see that the anti-dark solsolutions nearq5q2 and their features are very similar to thintrinsic spin-wave resonances found recently by Laiet al.25
Thus our analysis provided here maybe a possible menism for their interesting findings.
IV. NUMERICAL SIMULATIONS
In order to check our analytical results, in this sectionuse molecular-dynamics simulations to test the behaviothe soliton solutions obtained in Secs. II and III. We showvarious figures the time evolution of the energy density givby
m-
s
e
n
a-
eof
n
e~n!52J1Sn•~Sn211Sn11!2J2Sn•~Sn221Sn12!2e0 ,~74!
wheree05(22J122J2)Sc2 is energy density for the groun
state.Taking the analytical solutions as initial conditions, w
investigate the evolution of the system by integrating nmerically the Heisenberg equation of motion for thexyzspincomponents, using the fourth order Runge-Kutta methwith adaptive step-size control. The allowed relative erroreach time step is set to be 1028, and the length of the spinand the conservation of total energy of the system is checevery time step. In all the figures shown below, the timeshown in the unit of 1/(2J1Sc), which is the order of theperiod of linear spin-wave mode at the BZ boundaryTZB
5p/(4J1Sc), and the energy is in the unit ofJ1Sc2 .
We have numerically checked all the analytical solutiopresented in this paper. We found that these localized moare quite long lived. They can last for at least one hundtime units. In general, the lifetime is longer for the excittions with smaller spin deviations.
As examples, we show some of them in the followinfigures. In our numerical simulation, we chose a chain ofNspin sites with proper boundary conditions.
Now let us first take a look at the regionq,q1 . Accord-ing to our analytical results obtained in Sec. II, we havebright soliton excitation in this region. For this case we impose a periodic boundary condition. Figure 4 shows the tevolution for a bright soliton withq52p(135/Na), takingEq. ~24! as an initial condition.
Similarly, in the regionq2,q<p/a, we also have abright soliton solution. Such solution is also tested by osimulations. The bright soliton solutions forq being nearp/a have been considered recently by Laiet al.25
For the regionq1,q,q2 , Eq. ~19! gives a dark solitonsolution ~27! thus sn
1(t) expressed in Eq.~33!. Since thesolution is a kink, we cannot impose periodic boundary coditions for this case. Instead, we use absorbing boundconditions by assuming two imaginary sites at each endthe chain. These imaginary sites play the role of absorbwaves from the bulk of the chain and preventing the refltion of waves from the boundaries into the bulk of the chaAs shown in Fig. 5, the dark soliton solution for this casealso long-lived, the soliton keeps it shape before it reacthe boundary.
Whenq locates in the region (q1 ,q2) but near the ZDPsof the spin wave spectrum, the excitations may be darkantidark solitons, depending on the choice of the paramu0 . Due to the competition between the second and thdispersion of the system, we have an antidark soliton sotion if the condition~64! is satisfied. In Fig. 6, we show suca solution from our simulations. Notice thata1.5/(u0ubu)51.34701, hence the condition~64! is satisfied for the pa-rameters we have chosen.
We can also get dark solitons forq nearq2 , see Fig. 7,for example. In this figurea1.5/(u0ubu)50.538806,1, hencewe have a dark soliton according to the condition givenEq. ~58!.
.
PRB 58 9203NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
FIG. 4. In the regionq,q1 , the solution is a bright soliton. This figure shows the time evolution of the energy density, taking Eq~24!with t50 as the initial condition. The parameters used for this figure areN5512, A50.4, f 050.1, n05110, f050, q52p(64/Na), andq150.966827/a'2p(79/Na).
inNNuyth
nsm-thth
su-a-are
ob--
eri-
ithofon
V. DISCUSSION AND SUMMARY
We have investigated the weakly nonlinear excitationsthe isotropic Heisenberg ferromagnetic chains with theand the NNN exchange interactions. Contrary to previostudies which are only valid for low or high frequencmodes, our results based on the QDA are adequate inwhole BZ and all possible nonlinear localized excitatioincluding the bright and dark solitonlike modes of small aplitude have been obtained. In particular, we have solvednonlinear amplitude equation near the ZDPs and obtained
s
e
ee
antidark soliton modes of the system. These modes areperposition of a plane wave and a bright soliton. Their fetures have been analyzed and we have found that theysimilar to the so-called intrinsic spin wave resonancestained recently by Laiet al.25 Thus our study here may provide a possible explanation for the findings of Laiet al. Theanalytical results obtained here have been tested by numcal simulation and good agreement is found.
The model studied here includes the particular case wonly the NN exchange interaction. Due to the introductionthe NNN interaction, the curvature of the linear dispersi
g Eq.
de-
FIG. 5. In the regionq1,q,q2 , we have a dark soliton solution. This figure shows the time evolution of the energy density, usin~33! with t50 as the initial condition. The parameters used for this figure areN5512, A50.9, f 050.2, m50.5, n05100, f050, q52p(135/Na), q150.874912/a'2p(72/Na), and q252.46539/a'2p(201/Na). The boundary condition is the absorbing one asscribed in the text.
9204 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
FIG. 6. The anti-dark soliton solution nearq2 , using Eq.~55! with t50 as the initial condition. The parameters used areN51024,A50.9, u050.02, k50.01, and n05700. We take q52.4421/a52p(398/Na), being less but close to the ZDPq252.46539/a'2p(402/Na). The boundary condition is the absorbing one as described in the text.
in
io
lithahtalyio
idd bydis-iontem.
curve of the spin waves is drastically changed. WhenA.Ac , an important feature is that a maximum of the spwave frequency appears atq5qmax, located between thecenter and the boundary of the BZ, resulting in an inflectpoint at q5q2 . Based on the amplitude equation~19! ob-tained by the QDA, we discussed the modulational stabiof the extended spin waves in the whole BZ. We found t(0,q1) and (q2 ,p/a# are unstable regions, in which a brigsoliton will form via the Benjamin-Feir resonance mechnism. In (q1 ,q2), the spin waves are stable thus we onhave dark solitons. The length of the unstable reg
n
yt
-
n
(q2 ,p/a# tends to zero whenA approachesAc . In fact, inthis limit both qmax and q2 tend to the zone boundaryq5p/a. ThusA5Ac is a ‘‘critical point’’ for the appearanceof the bright solitons in the region (q2 ,p/a#.
Near the ZDPs, i.e., whenq→qj ( j 51, 2), the amplitudeequation~19!, which is the NLS equation, is no longer valbecause the coefficient of the dispersion term, representev9(q)/2, tends to zero. But this does not mean that thepersion is not important at the ZDPs. In fact, the dispersstill plays a role because of the discreteness of the sysThus we modified Eq.~19! to a new nonlinear amplitude
FIG. 7. The dark soliton solution nearq2 , using Eq.~58! with t50 as the initial condition. The parameters areN51024,A50.9, u0
50.05,k50.015, andn05700.q52.4421/a52p(398/Na), close to the ZDPq252.46539/a'2p(402/Na). The boundary condition is theabsorbing one as described in the text.
erro-
neof the
PRB 58 9205NONLINEAR EXCITATIONS IN FERROMAGNETIC . . .
TABLE I. Possible nonlinear localized excitations of small amplitude in the isotropic Heisenberg fmagnetic chains with the NN and the NNN exchange interactions whenA (5J2 /J1).Ac51/4. The BZ isdivided into several regions, where the signs ofv9 andv-, the localized excitations related-extended plawave stability and the types of the localized excitations as well as the conditions for the appearancedark and antidark solitons near the ZDPs are shown.q1 andq2 are the ZDP wave numbers andq3 is the zeropoint of v- ~see Fig. 3!. In the last column, we have defineda5uv9u/2 andb5v-/3!.
Region ExSM-relatedof q v9 v- ExSM stability soliton type Condition
equation~36!, which includes the third-order dispersion efect. For q5q11uDqu or q5q22uDqu we solved Eq.~36!and obtained the antidark soliton solutions. The occurreof the antidark solitons is due to the competition betweensecond- and the third-order dispersion. Atq5q22uDqu thefeatures of the antidark soliton solutions have been discuin detail and we found that they are similar to the spin waresonances obtained numerically by Laiet al. recently.25 Infact, the appearance of an antidark soliton is a universal cacteristic near the ZDPs. The antidark solitons in nonlinoptics have been considered in Refs. 35 and 36.
From the results obtained in Secs. II and III, in smaamplitude approximation the types of the nonlinear localizexcitations in the isotropic Heisenberg ferromagnetic chawith the NN and the NNN exchange interactions are summrized in Table I. In Table I ExSM denotes the extended swave mode~i.e., a plane wavelike mode!. q3 is the wavenumber at whichv-(q)50. Its expression is given by
q35p
a2
1
acos21S 1
16AD . ~75!
Whenq is near the ZDPsq5qj ( j 51, 2) but within theregion (0,q1) or (q2 ,p/a#, we havev9.0. The conclusionsobtained in Sec. III are not valid since in this case there isrelation between the modified NLS equation~36! and theKdV equation. Thus an antidark soliton solution is imposible. In fact, in normal dispersion region (v9.0) Eq. ~36!with Q.0 was intensively studied in soliton communictions in optical fiber.37 For any nonzerov-, numerical inte-grations showed that an initial localized pulse evolves isolitons plus small radiations, decaying exponentially, hebasically bright solitons. Ifv950, i.e., exactly at the ZDPs
ee
ede
r-r
ds-
n
o
-
oe
one still has the bright solitons with some~small! dispersivewaves.37 Therefore in Table I we have includedq5q1 andq5q2 in the first and last lines, respectively.
The results presented in this paper show that the weanonlinear localized excitations may be mobile. But in largamplitude case these discrete nonlinear excitations canpinned by Peierls-Nabarro potentials.38 For a large-amplitudeexcitation we should extend our QDA to higher-order expasions, a future work which will be considered elsewhere. Walso noted that Lai and Sievers39 and Kiselevet al.40 havenumerically found the intrinsic resonant modes in the anferromagnetic chains with single-ion easy-plane anisotroand in the diatomic lattices with soft optical modes. Sincemain feature of the dynamics of these resonant modes sedue to the resonant coupling between the optical and acoumodes, we believe that they are different from the spin waresonances in the isotropic ferromagnetic chains with theand the NNN exchange interactions and need to look fonew mechanism for their theoretical explanation.
ACKNOWLEDGMENTS
It is a pleasure to thank Professor A. J. Sievers for kinsending a copy of Refs. 18 and 39. The discussions withmembers at the Center for Nonlinear Studies of Hong KoBaptist University were helpful and fruitful. This work wasupported by grants from the Hong Kong Research GraCouncil and the Hong Kong Baptist University. Zhangwork was also supported in part by the National Nature Sence Foundation of China, the Educational Committee ofState Council through the Foundation of Doctoral Trainiand the Youth Foundation of BNU.
cs
-n
on
pn
.
v.
s.
J.
r
ce
BZ
ett.
9206 PRB 58GUOXIANG HUANG, SHUDONG ZHANG, AND BAMBI HU
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Phys.48, 1196 ~1972!#; Also see, e.g., M. Remoissenet, Phy
t
-
.
B
B
Rev. B 33, 2386 ~1986!; K. Yoshimura and S. Watanabe,Phys. Soc. Jpn.60, 82 ~1991!; Y. S. Kivshar, Phys. Lett. A161,80 ~1991!.
25R. Lai, S. A. Kiselev, and A. J. Sievers, Phys. Rev. B56, 5345~1997!.
26Whenq approaches the zero wave number, we haveQ50. In thiscase Eq.~16! is also not valid for the description of nonlineaexcitations. This case will not be discussed here.
27J. T. Stuard and R. C. DiPrima, Proc. R. Soc. London, Ser. A362,27 ~1978!.
28M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65, 851~1993!.
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