Propagation of electromagnetic soliton in antiferromagnetic medium

Physics Letters A 302 (2002) 77–86

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Propagation of electromagnetic soliton in antiferromagneticmedium

M. Daniel∗, V. Veerakumar

Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli 620 024, India

Received 8 May 2002; received in revised form 20 May 2002; accepted 8 August 2002

Communicated by A.R. Bishop

Abstract

We study propagation of electromagnetic waves in an antiferromagnetic medium by solving Maxwell equations coupledwith a nonlinear spin equation for magnetization of the medium. In the case of isotropic medium, the propagation is governedby perturbed electromagnetic soliton of the perturbed modified Korteweg–de Vries equation. When anisotropy is added as asmall correction to the exchange force, we do not observe any qualitative change in the propagation of EMW. However, whenanisotropy is treated as a dominant force the propagation is governed by electromagnetic soliton of the derivative nonlinearSchrödinger equation. In both isotropic and anisotropic cases the magnetization of the antiferromagnetic medium is also foundto exhibit solitonic behaviour (electromagnetic spin soliton) similar to the propagating electromagnetic soliton. 2002 Published by Elsevier Science B.V.

1. Introduction

There has been an increased interest in the re-cent times in the study of electromagnetic wave(EMW) propagation in different material media. No-table among them is the propagation of optical pulsesin the form of solitons through dielectric medium[1–3]. This brings about a natural question as to seewhat will happen when the magnetic field compo-nent of the propagating high intensity EM field in-teracts with a magnetic medium. In this context, re-cently, there was a systematic study on the prop-agation of electromagnetic soliton in ferromagnetic

* Corresponding author.E-mail address: [email protected] (M. Daniel).

medium which is inherently nonlinear. Nakata [4] andLeblond [5], using a reductive perturbation methodshowed that the EMW propagates in the form ofsoliton in a ferromagnetic medium, however by ne-glecting the spin–spin exchange energy. Recently, thepresent authors began a rigorous study in this directionby taking into account the basic magnetic interactionnamely the spin–spin exchange interaction in isotropicand anisotropic ferromagnetic media [6–8]. The re-sults show that when the EMW propagates in charge-free isotropic and anisotropic ferromagnetic media,the plane EMW is modulated in the form of electro-magnetic (EM) soliton and the magnetization of themedium also exhibits soliton excitations [6,8]. How-ever, when free charges are present in the medium,the EM soliton decelerates and the amplitude of itdecreases and gets damped [7]. Also, the shape of

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78 M. Daniel, V. Veerakumar / Physics Letters A 302 (2002) 77–86

the EM soliton is getting distorted at only one end,showing certain structural stability [7]. Very recently,the present authors have also shown the simultane-ous propagation of many signals without loss in ananisotropic ferromagnetic medium [9]. Antiferromag-net is also an equally important medium and the studyof propagation of EMW in this medium is also inter-esting. This type of problem is found to have potentialapplications in the areas of magneto-optical recording,switching etc. [10]. Though nonlinear spin dynam-ics of ferromagnetic systems have been studied ex-tensively [11–18], the study of antiferromagnetic spinsystems is still in the infant stage. Interesting finite en-ergy solutions including multi-solitons and twists havebeen obtained in these systems only in certain limitingcases [19–23].

2. Two sublattice antiferromagnetic model andelectromagnetic wave equation

The dynamics of spins in an anisotropic antiferro-magnetic system represented by the Hamiltonian

H = −N∑i=1

[JSi · Si+1 −A(Sx

i )2 + γ̂Si · H

]in the classical limit is governed by the equation ofmotion

dSj

dt= Sj ∧ [J (Sj+1 + Sj−1)− 2ASx

j n + γ̂H],

(1)n = (1,0,0).

HereSj represents the spin angular momentum opera-tor and the external magnetic fieldH(r, t) ≡(Hx,Hy,Hz) in our problem is in fact the magneticfield component of the propagating EMW. In Eq. (1),J represents the exchange integral corresponding tonearest neighbour spin–spin interaction which takesonly values less than zero,A is the anisotropy parame-ter with easy axis of magnetization alongx-directionand γ̂ = gµB , whereg is the gyromagnetic ratio andµB is the Bohr magneton.

In antiferromagnets as the neighbouring spins arealigned antiparallel to each other it is natural to treatthe problem as a two sublattice model by putting theup (Sa) and down(Sb) spins corresponding to spins atodd and even sites separately in two different lattices

saya andb respectively. Hence, the discrete equationof motion for the spin vectorsSa,j (up) andSb,j−1(down) can be written as

(2a)

dSa,j

dt= Sa,j ∧ [J (Sb,j−1 + Sb,j+1)

− 2ASxa,jn + γ̂H

],

(2b)

dSb,j−1

dt= Sb,j−1 ∧ [J (Sa,j + Sa,j−2)

+ 2ASxb,j−1n + γ̂H

].

We now make continuum approximation for neigh-bouring spin vectors of each sublattice by replacingSa,j (t) by Sa(r, t) and Sb,j−1(t) by Sb([r − λ], t),wherer = (r1, r2, . . . , rn), r = jλ, whereλ is the lat-tice constant vector and introducing the series expan-sions

Sa,j + Sa,j−2

= 2[Sa(r, t)− λ∇Sa + λ2∇2Sa − · · ·],

Sb,j−1 + Sb,j+1

= 2[Sb[(r − λ), t] + λ∇Sb + λ2∇2Sb + · · ·].

Using them in Eq. (2), the equations of motion in thecontinuum limit at O(λ) is written as

(3a)

∂Sa(r, t)∂t

= Sa ∧ [2J (Sb + λ∇Sb

− 2ASxan + γ̂H

],

(3b)

∂Sb([r − λ], t)∂t

= Sb ∧ [2J (Sa − λ∇Sa

+ 2ASxbn + γ̂H

],

Sa = (Sxa , S

ya , S

za), Sb = (Sx

b , Syb , S

zb),

(3c)Sa2 = 1= Sb

2.

Now, defining two unit vectors [20,21]

M = 1

2S√(1− ρ2)

(Sa(r, t) − Sb([r − λ], t)),

M′ = 1

2Sρ(Sa(r, t)+ Sb([r − λ], t)),

M · M′ = 0 and ρ2 = 12(1 + Sa·Sb

S2 ) and assumingthat the low energy configurations correspond to|Sa(r, t)−Sb([r−λ], t)| ≈ 2S and|Sa(r, t)+Sb([r−λ], t)| ≈ 0, whenρ � 1, after suitable rescaling andredefinition of parameters, we find that the dynamics

M. Daniel, V. Veerakumar / Physics Letters A 302 (2002) 77–86 79

is dominated by the equation [20]

(4)∂M∂t

= M ∧ {H − Jλ∇M − 2AMxn}.Eq. (4) is analogous to the Landau–Lifshitz equationfor ferromagnets [24]. While in the case of ferromag-nets the field contribution due to exchange energy ap-pears as∇2M, in the case of antiferromagnets thereis a nonvanishing contribution from the first derivativeof the staggered magnetization variableM (i.e.,∇M).This is because in the case of antiferromagnets wehave taken continuum limit for the two sublattices in-dividually. Unlike antiferromagnets the nonlinear spindynamics of ferromagnets with different magnetic in-teractions has been extensively studied by solving theassociated Landau–Lifshitz equations especially forsoliton-like excitations (see for, e.g., Refs. [11–18]).

The dynamics of electromagnetic field in a materialmedium is described by Maxwell equations which inthe absence of stationary and moving charges can bewritten as [25]

(5)∂2

∂t2[H + M] = c2[∇2H − ∇(∇.H)

],

wherec = 1√µ0ε0

is the phase velocity of the EMW

in a magnetic medium andµ0, ε0 are respectively themagnetic permeability and dielectric constant of themedium. The magnetic fieldH in our problem is con-nected to the magnetic inductionB by the relation [25]H = B

µ0− M, whereM represents the effective stag-

gered magnetization of the antiferromagnetic mediumwhich is nonzero. The set of coupled Eqs. (4) and (5)completely describe the propagation of EMW in ananisotropic antiferromagnetic medium when the ad-jacent antiparallel spins are locked under low energyconfigurations.

3. Electromagnetic wave propagation inantiferromagnetic medium

In order to find the nature of propagation of EMWin the antiferromagnetic medium, we now have tosolve the set of coupled Eqs. (4) and (5). However, thenonlinear character of Eq. (4) makes the task of solv-ing them difficult and therefore to make the calcula-tions simpler, we restrict the problem to one dimen-sion. This is an admissible approximation, because

quasi-one dimensional antiferromagnets are not un-common in nature. We are interested in finding solitonsolutions to the above coupled equations so that onecan have lossless propagation of EMW in the mediumin the form of electromagnetic solitons. For this westudy the nonlinear modulation of the slowly varyingEM plane wave of small but finite amplitude in theantiferromagnetic medium into solitons using a reduc-tive perturbation method developed by Taniuti and Ya-jima [26]. Very recently, the above reductive perturba-tion method has been successfully used to study theelectromagnetic soliton propagation in different ferro-magnetic media by the present authors and also by fewothers [4–9]. Experience suggests that for carrying outthe perturbation the magnetization of the medium andthe magnetic field components of the EM field haveto be expanded uniformly and nonuniformly respec-tively in the case of isotropic and anisotropic media.Also, an examination of the dispersion relation for theplane EMW propagation suggests us to introduce dif-ferent coordinate stretchings in the case of isotropicand anisotropic media. This suggests us to treat thepropagation of EMW in isotropic and anisotropic me-dia separately.

3.1. Oblique propagation in an isotropic medium

We first consider the propagation of EMW in anisotropic antiferromagnetic medium and therefore tryto solve the set of coupled Eqs. (4) and (5) in onedimension in the isotropic limit by settingA = 0.We expand the magnetization and the magnetic fieldabout an undisturbed uniform state specified byM0(saturated magnetization) andH0 as

(6a)M = M0 + εM1 + ε2M2 + · · · ,(6b)H = H0 + εH1 + ε2H2 + · · · .

Without loss of generality we assume that the one-dimensional plane EMW propagates alongx-directionand therefore assume thatH andM can be treated asfunctions of(x − vt) alone wherev is the group ve-locity of the wave. We also introduce slow variablesto separate the system into rapidly varying and slowlyvarying parts by stretching the time and the newly in-troduced wave variable asτ = ε3t andξ = ε(x − vt),whereε is a small parameter. Also we make the lat-tice spacing to be very small by rescalingλ asλ = ελ


andλx = λy = λz = λ. We now substitute the expan-sions (6a) and (6b) in the one-dimensional (say-x)-component equations of Eqs. (4) and (5) and use thenewly introduced slow variables and collect the co-efficients of different powers ofε, and solve the re-sultant equations. On solving the equations at O(ε0),we obtainHx

0 = −Mx0 = 0, Hy

0 = kMy

0 , Hz0 = kMz

0,

wherek = v2

c2−v2 . Similarly, on solving the equations

at O(ε1), after using the solutions at O(ε0) we obtainHx

1 = −Mx1 , Hy

1 = kMy

1 , Hz1 = kMz

1, and

(7a)∂M

y

0

∂ξ= σMz

0Mx1 ,

(7b)∂Mz

0

∂ξ= −σM

y

0Mx1 ,

whereσ = (1+k)v

. Finally, at O(ε2) we obtain

(8a)Hx2 = −Mx

2 ,

(8b)∂

∂ξ

[H

y

2 − kMy

2

]= −∂My

0

∂τ,

(8c)∂

∂ξ

[Hz

2 − kMz2

]= −∂Mz0

∂τ,

and

(9a)

−v∂Mx

1

∂ξ= β

[Mz

0

∂My

0

∂ξ−M

y

0

∂Mz0

∂ξ

]+M

y

0

(Hz

2 − kMz2

)−Mz0

(H

y

2 − kMy

2

),

(9b)∂M

y

1

∂ξ= σ

[Mz

0Mx2 +Mz

1Mx1

],

(9c)−∂Mz1

∂ξ= σ

[M

y0M

x2 +M

y1M

x1

].

While writing Eqs. (9), we have used the results fromthe previous orders of perturbation and in Eqs. (8b)and (8c),τ is rescaled asτ → σ

2k τ andβ = Jλ.To proceed further we assume that the EMW is

propagating obliquely with reference to the uniformfield in the medium. Hence we represent the unper-turbed uniform magnetizationM0 in terms of polarcoordinates in a unit sphere. As the magnetizationM0is restricted to the(y − z)-plane in the lowest orderof perturbation (i.e.,Mx

0 = 0), we can choose the az-imuthal angleφ as π

2 so thatM0 is represented by

(10)M0 = (0,sinθ(ξ, τ ),cosθ(ξ, τ )

).

In view of this, Eq. (7a) can be rewritten as

(11)Mx1 = −1

σ

∂θ

∂ξ,

which on using in Eq. (9a) gives

α∂2θ

∂ξ2 + β∂θ

∂ξ+ cosθ

∂

∂τ

ξ∫−∞

sinθ dξ ′

(12)− sinθ∂

∂τ

ξ∫−∞

cosθ dξ ′ = 0,

whereα = vσ

. After differentiating Eq. (12) twice andusing the results of the previous steps and integratingonce with respect toξ and after very lengthy calcula-tions, we finally obtain the following perturbed modi-fied Korteweg–de Vries (PMKDV) equation

∂f

∂τ+ 6f 2∂f

∂ξ+ ∂3f

∂ξ3

(13)= −β ′[∂2f

∂ξ2 + 4∂

∂ξ

(f

ξ∫−∞

f 2 dξ ′)]

,

wheref = ∂θ∂ξ

andβ ′ = βα

. It may be noted that whenJ = 0 (thenβ ′ = 0) the right hand side of Eq. (13) van-ishes thus reducing to the well known completely in-tegrable modified Korteweg–de Vries (MKDV) equa-tion for which theN -soliton solutions have been foundusing the Inverse Scattering Transform (IST) method[27]. For instance, the one-soliton solution of theMKDV equation can be written as

(14)f = 2η sech[2η(ξ − 4η2τ )+ ξ0

],

whereη is the imaginary part of the spectral parameterand ξ0 is the phase constant. Knowingf , θ can bestraightaway calculated and further the components ofmagnetizationM can be obtained using Eqs. (10) and(11) and thereupon the magnetic field component ofthe EMW. The one soliton for thex-component of themagnetizationM1 or the negative of the magnetic fieldcomponent of the EM fieldH1 at O(ε) is thus given by

(15)Mx1 = −Hx

1 = −2η

σsech

[2η(ξ − 4η2τ )+ ξ0

].

Fig. 1 shows the soliton behaviour ofHx1 or −Mx

1as given in Eq. (15) forη = 0.1, σ = 2.0 andξ0 =


Fig. 1. x-component of magnetic fieldH1 or negative of themagnetizationM1 (Eq. (15)) exhibiting soliton behaviour forη = 0.1, σ = 2.0 andξ0 = 0.09.

0.09. This implies that when the Zeeman energy dom-inates over the exchange energy in the antiferromag-netic medium that normally happens at microwave fre-quencies the magnetic field component of the EM fieldon interacting with the magnetization of the mediumgenerates excitation of magnetization in the form ofsoliton (EM-spin soliton) in the medium and alsothe plane EMW is modulated in the form of soli-ton (EM-soliton). In this case the corresponding spinequation (4) assumes the form∂M

∂t= M∧H and shows

that even in the absence of the nonlinear spin–spin ex-change interaction, the torque produced by the exter-nal EM-field is sufficient to provide enough nonlin-earity to support the generation of EM-soliton in themedium.

In the general case, whenJ �= 0 (β ′ �= 0) we havethe full PMKDV equation (13) which on solving willbring out the effect of perturbation due to spin–spinexchange interaction on the soliton during evolution.This can be done using a soliton perturbation [7,28]based on the IST theory. The results show that a smallstructural difference will lead to slow variation ofsoliton parameters and distortion of the soliton shape.The one soliton solution of Eq. (13) under perturbationcan be written in the form

(16)f = 2η(τ)[sech(u)+W(u, τ)

],

whereu = 2η(τ)[ξ − φ(τ)]. The unknown functionW(u, τ) is assumed to be zero whent = 0. Thefunctionsη(τ) andφ(τ) are found from the relations[28,29]

(17)dη

dτ= 8β ′η3,

Fig. 2. Deceleration of thex-component ofH1 or −M1 solitonunder perturbation (Eq. (20)) forv′(0)= 1.0 andβ′ = −0.008.

and

(18)dφ

dτ≡ v′(τ ) = 4η2,

wherev′(τ ) represents the velocity of the soliton. Onsolving Eq. (17), we obtainη(τ) as

(19)η(τ) = 1√(η(0))−2 − 16β ′τ

.

Using Eq. (19) in Eq. (18), we find the velocity of thesolitonv′(τ ) = dφ

dτas

(20)v′(τ ) = 4

[4

v′(0)− 16β ′τ

]−1

,

wherev′(0) is the initial velocity of the soliton. InFig. 2 we have plotted the variation of the velocityof the soliton under perturbation by choosing theinitial velocity asv′(0) = 1.0 andβ ′ = −0.008 whichshows that the velocity of the soliton decreases as timeprogresses after the perturbation is switched on.

The correction to soliton namelyW(u, τ) whichmanifests the change in shape of the soliton is deter-mined from a cumbersome expression which has theasymptotic forms

(21a)2ηW = β ′u2 exp(−u), u → ∞,

(21b)2ηW = 0, u → −∞.


Fig. 3. Damping ofx-component ofH1 or −M1 during evolutionwhenβ′ = −0.008 andσ = 2.0.

From Eq. (21), we further observe that in a referenceframe connected with a moving soliton the distortionin the shape of the soliton does not change with respectto time. Thus the soliton displays a certain stabilitywith respect to the structural excitation. Besides,Eq. (21a) shows that only the front part of the solitonundergoes a distortion.

Again, knowingf from Eq. (16), the value ofθcan be found using the relationf = ∂θ

∂ξand then the

magnetization of the medium and consequently themagnetic induction and the magnetic field componentof the EMW can also be computed as done in theunperturbed case. For example,Mx

1 or −Hx1 when

(J �= 0) is given by

(22)Mx1 = −Hx

1 = −2η(τ)

σ

[sech(u)+W(u, τ)

].

Eq. (22), is plotted forHx1 or −Mx

1 in Fig. 3 withβ ′ = −0.008,σ = 2.0 and exhibits soliton damping inthe asymptotic limitu → −∞. Knowing the EM-spinsoliton (M) and EM-soliton (H) the magnetic induc-tion soliton (B) can be obtained straight away fromthe relationH = B/µ0 − M.

Now, we study the effect of anisotropy as a smallcorrection to the exchange force. For this we firstrescale the anisotropy parameterA in Eq. (4) using thesmall perturbation parameterε by writing A → ε2A.Then we solve Eqs. (4) and (5) as done in isotropiccase. The results are similar to the previous case at allorders except an additional term proportional to theanisotropy parameter parameterA (−2Acosθ sinθ) inEq. (12) at O(ε2). By repeating the same procedurecarried out in the case of isotropic medium we finally

Fig. 4. Decrease in the amplitudeη of the EM-spin and EM-solitonsfor β′ = −0.008 andA′′ = 0.1.

arrive at the following new PMKDV equation.

∂f

∂τ+ 6f 2∂f

∂ξ+ ∂3f

∂ξ3

= −β ′[∂2f

∂ξ2 + 4∂

∂ξ

(f

ξ∫−∞

f 2 dξ ′)]

(23)

+ 6A′[

4f 2 sin 2

ξ∫−∞

f dξ ′ − ∂f

∂ξcos2

ξ∫−∞

f dξ ′],

whereA′ = Aα

. The effect of perturbation in Eq. (23)including that due to the anisotropic force (i.e., theterm proportional toA′) can be understood by con-structing equations similar to Eqs. (17), (18) and (21)as before representing the variations in the amplitude,velocity and shape of the soliton respectively. As a re-sult, the change in the soliton amplitude and solitonvelocityv′′(τ ) are found to be

(24a)dη

dτ= 8η2(β ′η +A′′),

(24b)v′′(τ ) ≡ dφ

dτ= 4η2 + 2A′ = v′(τ )+ 2A′,

whereA′′ = 165 A′. On solving Eq. (24a) we obtain


(1+G(τ)

)exp(−G(τ))

= (1+G(0)

)exp

(8A′′2

β ′ τ −G(0)

),

whereG(τ) = A′′β ′η(τ) and the results are plotted in

Fig. 4 forβ ′ = −0.008 andA′′ = 0.1. From the figurewe observe that the amplitude of the soliton decreasesas the rescaled time progresses as in the isotropic case.Further Eq. (24b) shows that the addition of weakanisotropy does not alter the nature of deceleration ofthe soliton qualitatively. Also, as done in the isotropiccase we can study the structural stability by evaluatingW(u, τ). The asymptotic forms ofW(u, τ) in this caseread

(25a)2ηW =[β ′ + 2A′

5η

]u2 exp(−u), u → ∞,

(25b)2ηW = 3πA′

ηexp(σ ′u), u → −∞,

whereσ ′ = 8∫η2dτ . Thus we observe that unlike the

isotropic case here the weak anisotropy has induceddistortion in both the ends of the soliton.

3.2. Parallel propagation in an anisotropic medium

Now, we study the propagation of EMW in theanisotropic antiferromagnetic medium by assumingthat the anisotropic energy is equally dominant bysolving the full (A �= 0) one-dimensional version ofthe set of coupled Eqs. (4) and (5). The anisotropiccharacter of the medium suggests us to make anonuniform expansion of the magnetizationM and themagnetic fieldH. Since, the easy axis of magnetizationof the anisotropic medium lies parallel to the directionof propagation (x-direction), we assume that at thelowest order of expansion the magnetization of themedium and the magnetic field lie parallel to theanisotropic axis and turn around to the(y − z) planeat higher orders.

(26a)Mx = M0 + εMx1 + ε2Mx

2 + · · · ,(26b)My = ε

12[M

y1 + εM

y2 + · · · ],

(26c)Mz = ε12[Mz

1 + εMz2 + · · · ],

and

(26d)Hx = H0 + εHx1 + ε2Hx

2 + · · · ,

(26e)Hy = ε12[H

y

1 + εHy

2 + · · · ],(26f)Hz = ε

12[Hz

1 + εHz2 + · · · ].

The magnetization and the magnetic field have beenexpanded about the uniform valuesM0 andH0 respec-tively along the direction of propagation of the EMW.

We now substitute the expansions ofM and Has given in Eqs. (26) in the component form ofthe one-dimensional version of Eqs. (4) and (5) andcollect terms proportional to different powers ofε. Onsolving the resultant equations at O(ε0), we obtainthe relationsHy

1 = kMy

1 , Hz1 = kMz

1, where k ≡(H0/M0) = v2

c2−v2 and from the constraint on the

conservation of length of magnetization vectorM2 =1, we find thatM2

0 = 1. At O(ε1), after using theresults at O(ε0) we finally obtain Mx

1 = −Hx1 =

12M0

[(My1 )

2 + (Mz1)

2] and also

(27a)∂

∂ξ

[H

y2 − kM

y2

]= −(∂H

y

1

∂τ

),

(27b)∂

∂ξ

[Hz

2 − kMz2

]= −(∂Hz

1

∂τ

),

(27c)

∂Mx1

∂ξ= β

v

[M

y

1

∂Mz1

∂ξ−Mz

1

∂My

1

∂ξ

]

+ 1

v

[M

y

1

ξ∫−∞

∂Mz1

∂τdξ ′ −Mz

1

ξ∫−∞

∂My

1

∂τdξ ′],

(27d)

∂My

1

∂ξ= σMx

1Mz1 − βM0

v

∂Mz1

∂ξ+ AM0

vMz

1

− M0

v

ξ∫−∞

∂Mz1

∂τdξ ′,

(27e)

∂Mz1

∂ξ= βM0

v

∂My

1

∂ξ− σMx

1My

1 − AM0

vM

y

1

+ M0

v

ξ∫−∞

∂My

1

∂τdξ ′.

While writing Eqs. (27) we have rescaledτ → τ2k

and also Eqs. (27a) and (27b) have been used inEqs. (27c)–(27e). Unlike the isotropic case, here weassume parallel propagation by choosing the directionof propagation of the EMW along the easy axis ofmagnetization. In order to identify Eqs. (27) with more


standard nonlinear evolution equations, we define

(28a)ψ = (M

y1 − iMz

1

).

Usingψ in the relation for conservation of length ofthe magnetization vectorMx

1 = −Hx1 = 1

2M0[(My

1 )2+

(Mz1)

2], we obtain

(28b)|ψ|2 = −2M0Mx1 .

After a single differentiation of Eqs. (27d) and (27e)and using the new field as given in Eqs. (28) and thetransformationX = ξ + Aτ , the resultant equationsafter some algebra can be finally written as

(29)i∂ψ

∂τ+ ∂2ψ

∂X2 + iα′ ∂[|ψ|2ψ]∂X

= 0,

whereα′ = 2M20σv[ v

M0+ iβ]−1/2 andX is rescaled

asX → [− vM0

+ iβ]1/2X. It may be verified that onusing Eqs. (28), Eq. (27c) can also be rewritten in theform of Eq. (29).

Eq. (29) is the well known completely integrablederivative nonlinear Schrödinger (DNLS) equationwhich has been solved for soliton solutions by Kaupand Newell [30] using IST method. Even though themulti-soliton solutions to Eq. (29) has been calculatedusing Hirota’s bilinear method in [31] it appears thatthe form of the solution seems to be incorrect andthey do not satisfy Eq. (29) in its present form. Anexact one-soliton solution of Eq. (29) is also obtainedin the context of nonlinear asymmetric self-phasemodulation and self-steepening of pulses in opticalfibers [32]. The one-soliton solution is obtained aftertransforming Eq. (29) using the transformationsζ =X − Qτ , τ = τ and ψ = ρ(ζ )exp[i(κτ + φ(ζ ))],where Q and κ are real constants. The resultantequations read

(30a)d2ρ

dζ 2−{(1−Q)

dφ

dζ+ κ

}ρ − α′ dφ

dζρ3 = 0,

(30b)ρd2φ

dζ 2−Q

dφ

dζ+ (

2+ 3α′ρ2)dρdζ

= 0.

Solving Eq. (30b) fordφdζ

and substituting in Eq. (30a),and integrating the resultant equation once, we get

(31)

(dρ

dζ

)2

= −α′2

16ρ6 + α′Q

4ρ4 +

(κ − Q2

4

)ρ2.

Here we have chosen the integration constant to beequal to zero by using the condition thatρ and dρ

dζ

Fig. 5. x-component ofH1 or −M1 exhibiting soliton behaviourwhenM0 = 1.0, ρ0 = √

2.0, ν = 1.0, α′ = 2.0 andQ = 1.0.

vanish atζ → ±∞. The formal solution of Eq. (31),can be given in terms of elliptic integrals, which inappropriate limit becomes

(32)ρ = ρ0/√

2− ν√cosh2Dζ +C

,

whereν = −α′Qρ20

2D2 , (κ − Q2

4 ) = D2 = α′ρ20

4 [3α′ρ20

4 −Q]andC = ν−1

2−ν. Here the constantQ corresponds to in-

verse soliton velocity shift,ρ0 is the peak amplitudeof the soliton andκ is the wave number shift. UsingEq. (32) in the transformationψ = ρ(ζ )exp[i(κτ +φ(ζ ))] andζ = X − Qτ , we can find the componentsof magnetization of the medium at O(ε1). For exam-ple, the one soliton for thex-component of magneti-zationM1 or the negative of the magnetic fieldH1 canbe written as

(33)Mx1 = −Hx

1 = −ρ20/(2− ν)

2M0(cosh2D(X −Qτ) +C).

In Fig. 5 we have plotted thex-component of the mag-netic field(Hx

1 ) or the negative of the magnetization(Mx

1 ) by choosingM0 = 1.0, ρ0 = √2.0, ν = 1.0,

α′ = 2.0 andQ = 1.0. From the figure it is interest-ing to observe that, the EM-spin soliton as well as theEM-soliton travel without change in shape. The othercomponents of magnetization and magnetic field canbe written down straight away.

4. Conclusions

In this Letter, we studied propagation of EMWin isotropic and anisotropic charge-free antiferromag-


netic media by treating the antiferromagnetic systemas a two-sublattice model. The propagation in theantiferromagnetic medium is governed by Maxwellequations coupled with a nonlinear spin equation forthe magnetization of the medium. The problem isanalysed using a reductive perturbation method bystretching the space and time variables and expand-ing the magnetization of the medium and the magneticfield component of the EM field suitably. In the caseof isotropic medium, we considered oblique propaga-tion under uniform expansion of the fields and arrivedat a perturbed MKDV equation to govern the dynam-ics of magnetization of the medium and the magneticfield component of the EM field. When the Zeemanenergy is dominant over the exchange energy then thedynamics of magnetization of the medium as well asthe magnetic field component of the EMW are gov-erned by completely integrable MKDV equation. Inthis case the magnetization of the medium and themagnetic field component of the EMW are governedby EM-spin soliton and EM-soliton respectively. In thegeneral case EM-spin soliton (magnetization) and theEM-soliton (magnetic field) corresponding to the per-turbed MKDV (PMKDV) equation undergoes damp-ing as it starts to propagate. It is also found that thevelocity of the soliton decreases as time progresses.Further, the shape of the soliton is distorted and it doesnot change in time thus showing a structural stability.Further, it is found that only the front part of the soli-ton undergoes distortion. When anisotropy is added asa small correction to the exchange force, it is foundthat the magnetization and also the magnetic field dy-namics are governed by a new PMKDV with addi-tional perturbation, which on solving following thesame procedure also gives rise to damping and decel-eration of the soliton, however with distortion at bothends of the soliton.

In the case of anisotropic medium when the aniso-tropic energy is treated as equally dominant we con-sidered the EMW to propagate in a direction parallelto the anisotropic axis. In this case, unlike the isotropicmedium, we made a nonuniform expansion of fieldsand found that the dynamics of magnetization of themedium and the magnetic field component of the EMfield are governed by completely integrable DNLSequation. This shows that the magnetic field compo-nent of the EMW is modulated in the form of EM-soliton and also the magnetization of the antiferromag-

netic medium is excited in the form of EM-spin soli-ton. While formulating the equation of motion for thedynamics of magnetization of the antiferromagneticmedium we arrived at Eq. (4), that corresponds to stag-gered magnetization in the medium due to locking ofadjacent spins. Studying the more general model with-out locking, that is by considering Eq. (3) instead ofEq. (4) is an open problem and work is under progress.

Acknowledgements

The work of M.D. forms part of a major DSTproject. V.V. wishes to thank the CSIR for financialsupport in the form of Senior Research Fellowship.

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Propagation of electromagnetic soliton in antiferromagnetic medium