Propagating two-dimensional magnetic droplets

4.3 MB·English

by  M. A. Hoefer

Propagatingtwo-dimensionalmagneticdroplets
M.A.Hoefera andM.Sommacala
aDepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,27695-8205NC,USA
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Abstract

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Propagating, solitary magnetic wave solutions of the Landau-Lifshitz equation with uniaxial, easy-axis anisotropy in thin (two-
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J dimensional) magnetic films are investigated. These localized, nontopological wave structures, parametrized by their precessional
frequencyandpropagationspeed,extendthestationary,coherentlyprecessing“magnondroplet”tothemovingframe,anon-trivial
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1 generalization due to the lack of Galilean invariance. Propagating droplets move on a spin wave background with a nonlinear
dropletdispersionrelationthatyieldsalimitedrangeofallowabledropletspeedsandfrequencies.Aniterativenumericaltechnique

] isusedtocomputethepropagatingdroplet’sstructureandproperties.Theresultsagreewithpreviousasymptoticcalculationsin
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the weakly nonlinear regime. Furthermore, an analytical criterion for the droplet’s orbital stability is confirmed. Time-dependent
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numericalsimulationsfurtherverifythepropagatingdroplet’srobustnesstoperturbationwhenitsfrequencyandspeedliewithin
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n theallowablerange.
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[ Keywords: Landau-Lifshitzequation,ferromagneticmaterials,spinwaves,magneticdroplet,dynamicsofdomainstructures,solitarywaves
PACS:75.70.Kw,75.78.-n,05.45.Yv,02.60.Cb

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1 1. Introduction A natural question arises, then, to inquire into the
3 existence, stability, and properties of propagating,
2
Magnetic materials yield a rich variety of intrigu- nontopologicalsolitarywavesintheunderlyingcon-
.
1 ing nonlinear wave phenomena. Recent theoretical servativemodel,theLandau-Lifshitz(LL)equation.
1 and experimental developments [1–4] have enabled Previous studies of 2D propagating droplets in-
1
thecontrolledmanipulationofmagneticmomentson cludenumericalcomputationsintheisotropic[9]and
1
: the nanometer length scale, the magnetic exchange anisotropic [10] regimes as well as weakly nonlinear
v length,therebygeneratingfurtherinterestinthefield asymptotics [11]. In [10], a numerical study of topo-
i
X of nanomagnetism [5]. Dynamic, strongly nonlinear, logical and nontopological localized solutions in 2D
r localized wave structures varying on the exchange ferromagnets with easy-axis anisotropy was under-
a lengthscalehavebeenstudiedtheoreticallyforsome taken. Stationary droplets were constructed numer-
time [6] (see the exhaustive review [7]). It was re- ically which were then made to propagate in time-
cently proposed that one member of this family of dependent simulations by the imposition of an ini-
solitary waves, the two-dimensional (2D) nontopo- tial,unidirectionalphasegradient.Thefollowingap-
logicaldroplet,couldbeexperimentallyrealizedina proximate properties were observed: the excitation,
spintorquenanocontactsystemthatdeliversalocal- accompanied by spin wave radiation, moved coher-
ized torque to a ferromagnetic thin film, balancing entlywithconstantspeed,asinglefrequencyofpre-
theinherentmaterialdamping[8].Theso-calleddis- cession, and a spatially dependent phase lag. In this
sipativedropletcan,undercertainconditions,expe- work, we demonstrate that the excitations observed
rience a drift instability, propagating for some time in[10]andthemoving,coherentstructuresin[8]can
beforeeventuallysuccumbingtomagneticdamping. be identified as propagating, nontopological droplet
solitarywaveswhichweconstructdirectlybystudy-
Emailaddresses: [email protected](M.A.Hoefer), inganonlineareigenvalueproblem.
[email protected](M.Sommacal).
Preprintsubmittedto
∂M
Weundertakeacarefulnumericalandasymptotic
= γ µ M H , (1a)
0 eff
study of propagating 2D nontopological droplets, ∂t −| | ×
from here on in termed droplets. A two-parameter Heff = (1b)
(cid:18) (cid:19)
family of localized solitary wave solutions to the LL 2A 2K
∆M + H + u M e +H ,
equation with easy-axis anisotropy parametrized by µ M2 0 µ M2 3 3 d
0 s 0 s
speedV andrestfrequencyωisconstructed.Inorder
describesthedissipationlessdynamicsofthemagne-
to avoid a linear resonance, we are led to postulate
tization M : R2 [ δ/2,δ/2] R+ M S2 in an
the droplet’s existence and stability for speeds and × − × → s
unboundedferromagneticfilmofthicknessδexhibit-
frequenciesintherestrictedrange
ingperpendicular,uniaxialanisotropyandaperpen-
dicularappliedmagneticfield[13].
V2 Relevant parameters for Eq. (1) are the gyromag-
1 ω >0,
− − 4 netic ratio (γ), the free space permeability (µ ), the
0
exchange stiffness parameter (A), the perpendicular
similar to the existence conditions for moving 1D magnetic field amplitude (H0), and the crystalline
droplets[7,12].Thedroplet’sstructureisstudiedan- anisotropyconstant(Ku).
alyticallyinthelargeamplitude,smallspeedregime Themagnetization
andintheweaklynonlinearregimefollowing[11].A
3
nonlinear dispersion relation for a background spin M M m M (cid:88)m e , (2)
s s j j
wavewhichmodulatesthepropagatingdropletnatu- ≡ ≡
j=1
rallyarises.Itdegeneratestothelineardispersionre-
lationforexchangespinwavesinthesmallamplitude satisfies M(x,t) =Ms, where Ms is the saturation
| |
regimewheretheNonlinearSchr¨odingerequationap- magnetization, as can be verified by taking the dot
proximately governs the dynamics. An iterative nu- product of (1) with M. The vectors ej, j = 1,2,3,
mericalmethodisemployedtocomputethedroplet’s are the standard Cartesian basis vectors for R3 (see
structure in the comoving frame by solving an ap- Fig. 1). The magnetostatic or dipolar field Hd is,
propriateboundaryvalueproblem.Theseresultsare ingeneral,nonlocal,couplingthemagnetizationdy-
used to compute the amplitude, asymmetry, energy, namicstoMaxwell’sequationswhich,inthemagne-
momentum, and more for a wide range of propagat- tostaticregime,taketheform
ingdropletswhichagreewiththeasymptoticcalcula-
H = M,
d
tions.Dropletorbitalstabilityisconfirmedbynumer- ∇· −∇·
icalverificationofananalytical,Jacobiancondition. where M is extended to zero outside of R2
×
Time-dependentnumericalsimulationswithinitially [ δ/2,δ/2].Weareinterestedinthethinfilmregime
−
perturbedpropagatingdropletsfurtherdemonstrate where δ is sufficiently small. There are a number
therobustnessofdropletpropagation. of different thin film scalings that depend upon the
The outline of this work is as follows. In Sec. 2, length scales inherent to the problem at hand. In
severalequivalentformsoftheLLequationareintro- the static case, some thin film limits have been rig-
ducedforlateruseinasymptoticandnumericalanal- orously justified via Γ-convergence; see, e.g., [14,15]
ysis.Inaddition,propertiesofthestationarydroplet and references therein. Thin film magnetodynamics
are reviewed. The propagating droplet problem for- have also been rigorously studied [14,16]. In many
mulation and some basic properties are discussed in thin film scalings, there are two key simplifications
Sec. 3. Section 4 presents the asymptotic results. In that can be made: the magnetization is approxi-
Sec. 5, numerical results addressing droplet proper- mately uniform through the thickness of the film so
ties and stability are given. Further discussion and thattheproblemisapproximately2D
conclusions are related in Sec. 6 and 7. Finally, vali-
∂2 ∂2
dationofthenumericaltechniqueispresentedinthe ∆ + , (3)
→ ∂x2 ∂x2
Appendix. 1 2
and the dipolar field can be approximated by the
localterm
H M e . (4)
d 3 3
→−
2. Model Equations and the Stationary
Heuristically, the two-dimensionality of the problem
Droplet
is justified when the film thickness is much smaller
than the transverse magnetic excitations of interest
TheLandau-Lifshitztorqueequation becausetheexchangeenergyformagnetizationvaria-
2
tionsinthex directionwouldbeprohibitivelylarge. Thenondimensionalization(8)assumesaneasy-axis
3
Inourcase,thisamountstotherequirement anisotropy so that the quality factor Q, measuring
therelativestrengthoftheanisotropytothemagne-
δ L /(Q 1)1/2, (5)
(cid:28) ex − tostaticenergy,islargerthanunity.WhenQ<1,the
whereLex isthemagneticexchangelength anisotropy is of the easy-plane variety. Anisotropic
(cid:18) (cid:19)1/2 ferromagneticnanostructuresinthisthinfilmgeom-
2A
L = , etrycanbefabricated,e.g.,usingCobalt/Nickelmul-
ex µ M2
0 s tilayers[5].
typically on the order of several nanometers, and Q Transformingtotherotatingframe
isthedimensionlessqualityfactor
m =[m cos(h t)+m sin(h t),
(cid:48) 1 0 2 0
2K
Q= u , m1sin(h0t)+m2 cos(h0t),m3],
µ M2 −
0 s
takes Eq. (9) to the parameterless form (after drop-
assumedheretobegreaterthanunity.Assumingthat
pingtheprime)
∂ M =0,wecancomputethe2DFouriertransform
x3
ofthex averageddipolarfield(seee.g.[17]) ∂m
3 = m h , h =∆m+m e . (10)
eff eff 3 3
∂t − ×
(cid:92) k[k M(cid:99) (k)]
(Hd+Mse3)= − · k2⊥ [1−Γ(cid:98)(kδ)] (6) Therefore,withoutlossofgenerality,wewillconsider
localizedsolutionsofEq.(10).
(cid:92)
(M3 Ms)(k)Γ(cid:98)(kδ)e3, Equation (10) admits a family of localized, coher-
− −
wMhere=f(cid:98)((Mk),=M(cid:82)R)2,fa(nxd⊥)e−ik·x⊥dx⊥, x⊥ =(x1,x2), ecnortplyoprarteeceasscinomgspolleutteiornevse[6rs,7a]l.oTfhtehseomluatigonnestitzhaattioinn-,
⊥ 1 2 i.e.m(x0,t)= e3 andlimx m(x,t)= e3 for
Γ(cid:98)(kδ)= 1−e−kδ . (7) one x0 ∈R2 (m∓agnetic vort|ic|e→s)∞, are called t±opolog-
kδ ical and exhibit pinning of the soliton center, hence
TheFouriertransformsarecomputedforrapidlyde- cannotpropagate[18].Incontrast,solutionsthatdo
caying functions. In this work, we study solutions not exhibit magnetization reversal are termed non-
that rapidly decay to M M e with the decay topologicalandcanpropagate.Suchsolutionsarethe
s 3
length L /(Q 1)1/2 (see→non-dimensionalization focusofthiswork.
ex
below). Then th−e Fourier transforms in (6) are con- SufficientlylocalizedsolutionsofEq.(10)conserve
centrated on wavevectors k of order (Q 1)1/2/L themagneticenergy
ex
−
or less. Under the assumption (5), we can approxi- [m]= 1(cid:90) (cid:104) m2+(cid:0)1 m2(cid:1)(cid:105) dx, (11)
mate(6),(7)as E 2 |∇ | − 3
1 (cid:18) (Q 1)1/2(cid:19) which incorporates contributions due to exchange
Γ(cid:98)(kδ) 1 kδ =1+ δ − ,
∼ − 2 O L m2andanisotropy(1 m2).Integrationisalways
(cid:18) (Q 1)1/2e(cid:19)x |a∇ssum|edtobetakenover−the3planeR2 unlessother-
Hd = M3e3+ δ − , wise noted. The effective field in Eq. (10) is derived
− O L
ex
fromthemagneticenergyvia
hence(4)isjustified.
δ
It is convenient to nondimensionalize Eq. (1) ac- h = E .
eff −δm
cordingto(2)and
(Q 1)1/2 Otherconservedquantitiesincludethespindensity
t(cid:48) = γ µ0Ms(Q 1)t, x(cid:48) = − x,
| | − L (cid:90)
ex (8)
H [m]= (1 m )dx, (12)
h = 0 , N − 3
0
M (Q 1)
s
− andthemomentum
sothat,afterdroppingprimes,theproblemofstudy-
(cid:90)
m m m m
ing localized solutions to Eq. (1) subject to (3) and P[m]= 2∇ 1− 1∇ 2 dx. (13)
1+m
(4)becomes 3
Thequantity canbeinterpretedasthemeannum-
∂m
N
∂t =−m×[∆m+(m3+h0)e3] berofspindeviationsfromtheuniformstatee3 ina
m: R2 R+ S2, lim m(x,t)=e , t 0. localized magnetic excitation. In the quantum limit
× → x 3 ≥ it takes integer values and corresponds to the num-
| |→∞
(9) berofmagnonsinanexcitedferromagnet[7].
3
e3 agating droplets. The energy, spin density, and mo-
mentuminthesevariablestaketheform
e2 x 1(cid:90) (cid:104) (cid:16) (cid:17)(cid:105)
2 [Θ,Φ]= Θ2+sin2Θ 1+ Φ2 dx,
e1 " E 2 |∇ | |∇ |
m
Ρ
$
x (16a)
x (cid:90)
1
# [Θ]= (1 cosΘ) dx, (16b)
N −
(cid:90)
P[Θ,Φ]= (cosΘ 1) Φdx. (16c)
− ∇
Fig. 1. Schematic of the coordinate systems for a planar fer-
romagnetconsistingofthedomainofm(R2 inpolarcoordi- Notethat(15)isHamilton’sequationsintheform
nates with radius ρ, azimuthal angle ϕ) and the range of m
(unitsphereS2 withpolarangleΘ,azimuthalangleΦ). ∂Φ δ ∂cosΘ δ
= E , = E,
∂t −δcosΘ ∂t δΦ
Note that the momentum as given in (13) is am-
whereΦandcosΘarethecanonicallyconjugatevari-
biguous when the magnetization involves topologi-
ablesfortheHamiltoniansystem.
calstructures.Theambiguityhasbeenaddressedby
Anotherrepresentationof(10)canbehadbyper-
consideringmomentsofthetopologicalvorticity[18].
formingthestereographicprojection
Since we are considering nontopological structures,
m +im eiΦsinΘ
themomentumin(13)isequivalenttothemoregen- 1 2
w = = . (17)
eralformulationof[18]. 1+m3 1+cosΘ
Small amplitude exchange spin waves deviating Then(10)becomes[19]
fromthebackgroundstatee intheform
3 2w w w+w(1 w 2)
∗
iw =∆w ∇ ·∇ −| | , (18)
m1 =εcos(k x ωLt), t − 1+ w 2
· − | |
m = εsin(k x ω t)], m 1, ε 1,
2 − · − L 3 ∼ | |(cid:28) wheretheappendedasteriskdenotescomplexconju-
withwavenumberkandfrequencyω satisfythelin- gation. This form of the LL equation is amenable to
L
eardispersionrelation weaklynonlinearasymptotics(Sec.4)andnumerical
analysis(Sec.5).Theconservedquantitiesare
ω (k) ω (k)=1+k2, k = k . (14)
L ≡ L | | (cid:90) w 2+ w 2
[w]=2 |∇ | | | dx, (19a)
E (1+ w 2)2
2.1. Alternate Forms of the Landau-Lifshitz
| |
Equation (cid:90) w 2
[w]=2 | | dx, (19b)
N 1+ w 2
(cid:90) | |
Two alternative forms for Eq. (10) will be used in P[w]= 2 Im(w∗∇w)dx. (19c)
thiswork.Forthefirst,weusethesphericalbasis(see − 1+ w 2
| |
Fig.1)
InanalogywithaBose-Einsteincondensate[20],the
m=[cosΦ sinΘ,sinΦ sinΘ,cosΘ], momentum (19c) corresponds to a weighted phase
gradientora“magneticsuperfluid”momentum.
where Φ [0,2π) (mod2π) and Θ [0,π] are the For the time-dependent initial value problem dis-
∈ ∈
azimuthal and polar angles, respectively. Inserting cussed in Sec. 5.3, we will solve Eq. (10). Thus, we
thisexpressionintoEq.(10)gives will make use of all three forms (10), (15), and (17)
oftheLLequationinouranalysis.
∂Θ ∂Φ
=F[Θ,Φ], sinΘ =G[Θ,Φ], (15a)
∂t ∂t
2.2. StationaryDroplet
where
(cid:0)sin2Θ Φ(cid:1)
F[Θ,Φ]= ∇· ∇ , An azimuthally symmetric, stationary, localized
sinΘ (15b) droplet solution to the LL equation was studied in
1
G[Θ,Φ]= sin(2Θ)( Φ2+1) ∆Θ. [6,7].Wenotethatthetermmagnondropletwasused
2 |∇ | −
becauseofitsanalogywithalocalized“droplet”ofa
WewillseethatEq.(15)isinausefulformtostudy condensed, attractive one-dimensional Bose gas [7].
stationary and slowly moving droplets as well as to Inthissection,webrieflyreviewthepropertiesofthe
formulateanenergyminimizationproblemforprop- stationarydroplet.
4
Itisconvenienttowritethestationarydropletso- with decreasing frequency. As the frequency ap-
lutioninasphericalbasissatisfyingEq.(15).Ittakes proaches zero, the droplet resembles a circular do-
theform mainwallwithatransitionlayerthatmovesfurther
away from the droplet center. The nonlinear, or-
Θ(x,t)=Θ (ρ;ω), Φ(x,t)=ωt,
0
bital stability of nontopological, three-dimensional
F[Θ ,ωt]=0, G[Θ ,ωt]=ω sinΘ ,
0 0 0 solitarywavesinthecontextofanisotropic,twosub-
where ρ = (cid:112)x2+x2 is the distance from the origin lattice ferrites were studied by use of a Lyapunov
1 2
functional whose positive definiteness is guaranteed
and the polar angle function Θ , parametrized by
0
when[22]
thedropletfrequencyω,isdeterminedbysolvingthe
d
followingnonlineareigenvalueproblem
N <0, (24)
dω
(cid:18) d2 1 d (cid:19)
+ Θ +sinΘ cosΘ ωsinΘ =0, whichiscommonlyreferredtoasaslopeorVKcon-
− dρ2 ρdρ 0 0 0− 0
ditionafterVakhitov-Kolokolov’sworkonNonlinear
(20a) Schr¨odinger solitary waves [23]. Since (ω) is the
(cid:48)
N
dΘ0 Jacobian of the map from the physical variable ω to
(0;ω)=0, lim Θ (ρ;ω)=0. (20b)
dρ ρ 0 the conserved quantity , we refer to (24) as a Ja-
→∞ N
cobian condition as this will naturally generalize to
Thegroundstatesolutionthatispositiveandmono-
the stability criterion for propagating droplets. The
tonically decaying is sought. One can show that
Jacobian condition (24) has been numerically veri-
0 < Θ (ρ;ω) < π, therefore this solution is termed
0 fied for 2D stationary droplets [7], suggesting their
non-topological [7]. In contrast, there exist topolog-
orbital stability. Their robustness to perturbations
ical droplets that satisfy a related nonlinear eigen-
in 2D, time-dependent numerical simulations [8] are
value problem with the conditions Θ (0;ω) = π,
0 furtherevidenceoftheirstability.
Θ (ρ;ω) 0. Their existence and stability subject
0
→
to certain symmetric perturbations were studied
rigorouslyin[21]. 3. Propagating Droplet: Problem
Assumingtheexistenceofalocalizedsolution,the FormulationandBasicResults
followinginequalityholds
A two-parameter family of moving droplet solu-
0<ω <1, (21)
tionsof (15)isconsideredintheform
whichcanbeseenbymultiplyingEq.(20a)by dΘ0 ρ2
dρ Θ(x,t)=θ(ξ;ω,V), (25a)
andintegratingtofind
Φ(x,t)=ωt+φ(ξ;ω,V), (25b)
(cid:90) (cid:90)
2ω ∞(1 cosΘ ) ρdρ ∞sin2Θ ρdρ=0.
− 0 − 0 with
0 0
(22) ξ =x V t, (25c)
−
Weimmediatelyobservethat
where ω is the droplet’s rest frequency and V is
ω = (cid:82)(cid:82)0∞sin2Θ0 ρdρ >0. tEhqe.(d1r5o)plreets’usltvselioncity. Insertion of this ansatz into
2 0∞(1−cosΘ0)ρdρ
Using the fact that 2(1 cosΘ) > sin2Θ for 0 < sinθ (ω V φ)=
− ·∇
Θ<π,wealsohaveω <−1. ∆θ+sinθ cosθ (cid:0)1+ φ2(cid:1) , (26a)
− |∇ |
ThedecayrateofΘ canbeobtainedbyexpanding
0
(20a) for large values of ρ and neglecting nonlinear sinθV θ = (cid:0)sin2θ φ(cid:1). (26b)
termswhichgives − ·∇ ∇· ∇
Thepropagatingnontopologicaldropletthatweseek
d2Θ 1dΘ
0 0 +(1 ω)Θ 0, satisfiestheboundaryconditions
− dρ2 − ρ dρ − 0 ∼
lim θ(ξ;ω,V)=0, lim φ(ξ;ω,V)=const,
giving ξ ξ ∇
| |→∞ | |→∞
(cid:32) (cid:33) (27)
e−ρ√1−ω with θ(ξ;ω,V) < π for all ξ R2. Assuming decay
Θ (ρ)= , ρ 1. (23)
0 O √ρ (cid:29) for ξ sufficiently large, θ ∈1 and φ U a con-
| | (cid:28) ∇ ∼
stantvector,Eq.(26b)isasymptotically
From numerical computations, see, e.g., [10], it is
known that the amplitude of the droplet increases V θ =2U θ.
− ·∇ ·∇
5
Therefore, A necessary condition for θ (ξ), φ (ξ) to be a mini-
∗ ∗
mumof (31)istherefore
V
lim φ(ξ;ω,V)= . (28) (cid:90)
ξ ∇ − 2 d(1)= sin2θ dξ V 2ω =0, (35)
| |→∞ (cid:48) 1
∗ − P − N
Due to the SO(2) invariance of the LL equation
whichcanbederivedmoregenerallybydirectmanip-
(10),itissufficienttoconsider
ulation of conservation laws for the Landau-Lifshitz
V =Ve1, V >0. (29) equation[27].SimilaridentitieswerestudiedbyDer-
rick[28]andPohozaev[29].Wewillmakeuseofthis
Fortherestofthiswork,expression(29)isassumed
relationinordertoverifythevalidityofournumeri-
tohold.
calcomputationsintheAppendix.
By use of (16), we observe that (26) is equivalent
Another formulation of the boundary value prob-
tothefollowing
lem (26), (27), (28) can be derived by considering
δ δ Eq.(18)withatravelingwavesolutionintheform
(ω +V )= E , (30a)
1
δθ N P δθ w(x ,x ,t)=f(x V t,x )eiωt
δ δ 1 2 1− 2
(ω +V 1)= E , (30b) =f(ξ ,ξ )eiωt. (36)
δφ N P δφ 1 2
where isthenonvanishingcomponentofP when Inserting the ansatz (36) into Eq. (18) leads to the
1
P followingnonlineareigenvalueproblem
condition (29) holds. Equations (30) are the neces-
saryconditionsforaminimizerofthefunctional
ωf +iVf +f +f
ξ1 ξ1ξ1 ξ2ξ2
D[θ,φ]=E[θ,φ]−ωN[θ,φ]−VP1[θ,φ], (31) 2f∗(fξ21 +fξ22) f1−|f|2 =0, (37)
− 1+ f 2 − 1+ f 2
whereωandV areLagrangemultipliersensuringthe
| | | |
constraints of fixed spin density and momentum withf 0as ξ .Letting
N → | |→∞
.Propagating,localizedstructuresin2Disotropic
P[91]and3Danisotropic[24]ferromagnetswerestudied f(ξ1,ξ2)=u(ξ1,ξ2)e−iV2 ξ1,
numericallybyminimizingasimilarfunctional.Note then(37)becomes
that, V2
∂ ∂ ωu+ u+u +u
= D , = D . (32) 4 ξ1ξ1 ξ2ξ2
1
N −∂ω P −∂V
Whilethevariationalformulationofthedropletleads 2u∗[(uξ1 −iV2u)2+u2ξ2] u1−|u|2 =0.
− 1+ u2 − 1+ u2
toaparameterizationintermsof and 1,forphys- | | | |
N P (38)
ical applications, it is natural to parameterize the
droplet in terms of the physical variables ω and V. We observe that for V = 0, u is generally complex-
(cid:54)
SolvingEqs.(26)forfixedωandV providesthemap valued. Assuming localization and linearizing
(ω,V) ( , ). Invertibility of this map requires Eq.(38)for ξ ,weobtain
1
(cid:55)→ N P | |→∞
thattheJacobian (cid:18) V2 (cid:19)
u +u + ω+ 1 u=0.
J = ∂P1∂N ∂N ∂P1 , (33) ξ1ξ1 ξ2ξ2 4 −
∂V ∂ω − ∂V ∂ω
Therefore,anecessaryconditionforlocalizationis
be nonzero. In the context of other magnetic and
nonlinear wave models, the orbital stability of two- ν2(V,ω)>0, ν(V,ω)=(cid:18)1 ω V2(cid:19)1/2 , (39)
parametersolitarywavesisdeterminedbytheJaco- − − 4
biancondition[25,26]
and the exponential decay rate of u is ν(V,ω). We
discuss the consequences of this observation in the
J <0. (34)
nextsection.
NotethattheJacobiancondition(34)degeneratesto
the Jacobian condition (24) for stationary droplets
3.1. Thenonlineardispersionrelation
when V = 0 because ∂ = 0 and ∂ > 0.
ω 1 V 1
P P
DropletstabilitywillbediscussedfurtherinSecs.4.1
From the boundary condition in (28), we observe
and5.3.
thatthepropagatingdropletismodulatedbyaspin
Ifθ (ξ),φ (ξ)areaminimizerof(31),thendilation
ofξ b∗ythef∗actorλgives wavebackgroundwithwavenumber
V
d(λ)= [θ (λ),φ (λ)]. k = , (40)
2
D ∗ · ∗ ·
6
andfrequencyinthelaboratoryframe
11
V2
ωNL(k,ω)=ω+ =ω+2k2. (41) Ω(cid:2)Ω(cid:2) Ν(cid:4)0
2 (cid:1)(cid:1)
22(cid:1)(cid:1)ΩΩ yy
Icnhacnognetrsapsitntwoavtheseilnin(e1a4r),dtihspeenrsoinolninreealratdiroonplfeotrdeixs-- quencyquency 11 ΩL(cid:3)k(cid:4) equencequenc
pqueresnicoynωrelaantidonthine s(4p1e)eddeVp.enTdhseocnobnodtithiotnhe(3re9s)tcforer-- otalFreotalFre ΩΩ ΩNL(cid:3)k,Ω(cid:4) RestFrRestFr
respondstoanegativenonlinearfrequencyshift TT 00 opletoplet00
rr
ω (k) ω (k,ω)=1 ω k2 =ν2(V,ω)>0. DD
L NL
− − −
(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)
This yields an upper bound on the droplet’s speed 00 WWaavveennuumm11bb(cid:1)(cid:1)eerrΩΩ(cid:1)(cid:1)kk(cid:2)(cid:2) 00 (cid:1)(cid:1)SSpp(cid:2)(cid:2)11(cid:2)(cid:2)(cid:2)(cid:2)ee(cid:2)(cid:2)(cid:1)(cid:1)(cid:2)(cid:2)ee(cid:2)(cid:2)(cid:2)(cid:2)dd(cid:2)(cid:2)ΩΩ(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:1)(cid:1)VV(cid:2)(cid:2) 22
foragivenrestfrequency
0<V <V (ω)=2√1 ω,
max
− Fig. 2. Dispersion relations in the laboratory frame (left)
or, equivalently, an upper bound on the rest fre- and the comoving droplet frame (right) with wavenumber
quencyforfixedspeed k = V/2. The shaded regions represent accessible droplet
solutions.
V2
ω <ω (V)=1 , V >0. (42)
max − 4 andadmitssolitonsolutions.Thesingledropletsoli-
tonsolutioncanbewrittenasfollows(see[12])
In the limit ω 1, condition (42), via (40), re-
→
quiresthatk 0.Thenonlineardispersionrelation V2 2+3ω+√V2+ω2cosh[2νξ]
→
(41)thencoincideswiththelineardispersionrelation cosΘ= − (43a)
2 ω+√V2+ω2cosh[2νξ]
(14) for a spatially uniform, small amplitude excita- −
Vξ
tion at the bottom of the spin wave band. Alterna- Φ=ωt
tively, in the limit k k √1 ω, we observe − 2
max (cid:20) (cid:21)
→ ≡ − 4νtanh(νξ/2)
tthoaatpωrNoLp(akg,aωti)ng→ex2c−haωng=e sωpLin(kwmaaxv)e.cTorhreessppoeneddinVg +tan−1 V2+2(ω+√V2+ω2) , (43b)
ofthemovingdropletisthegroupvelocityV ofthe
g where ξ = x V t and ν is given in (39). Thus, the
background spin wave, V = dω /dk = 2k = V. −
g L 1D soliton solution exists for all V > 0, ω satisfying
Thephasespeedofthebackgroundspinwaveω /k =
L (39).WhenV =0,therestfrequencyislimitedtothe
ω/k+2k can be greater than (when ω > 0) or less
unitinterval(21).Forω <0,V =0,thesoliton(43)
than (when ω <0) the droplet traveling wave speed
exhibitsaphasesingularityattheorigincorrespond-
V =2k.Inthiswork,wefocusonthecaseω >0.
ing to topological behavior in this one dimensional
Thedispersionrelationsareillustratedinthelab-
setting.However,forV >0,thenecessarycondition
oratory frame and the comoving droplet frame in
ω <ω (V)in(42)hasnolowerbound.Therefore,
max
Fig. 2. The shaded regions correspond to the local-
nontopologicalsolitonswitharbitrarilynegativerest
ization condition (39) hence localized, propagating
frequencies exist even with vanishingly small prop-
dropletscanbeinterpretedasnonlinearboundstates
agation speeds. The phase gradient (43b) has the
lyingbelowthelinearspinwaveband[7].
asymptotics
V V
3.2. 1DDroplet Φ (0) − , 0< 1,
ξ ∼ ω+ ω +V2/(2ω ) ω (cid:28)
| | | | | |
The nonlinear dispersion relation in Eq. (41) for hencecanbequitelargewhenω <0.Ournumerical
a 2D propagating droplet is the same for a propa- methodforcomputing2Dpropagatingdropletsfails
gating 1D droplet [7]. As will be shown, a number for modes with large momentum due to large phase
of 2D droplet properties are qualitatively similar to gradients(seeSec.5.2)sowearelimitedinexploring
the 1D droplet properties. Therefore, in this section thisregimeofparameterspaceinthetwodimensional
we briefly review some basic results about moving case.
dropletsolutionsoftheLLequationin(1+1)dimen- Explicitexpressionsfortheenergy (1D),thespin
E
sions. density (1D)andthemomentum (1D),asfunctions
N P
The Landau-Lifshitz Eq. (10) in one dimension, ofV andω,forasinglemovingdropletsolution,can
∆ ∂ ,issolvablebymeansoftheinversescatter- beobtainedfromsolution(43)viaEqs.(11),(12)and
xx
→
ing transform (see, e.g., [30] and references therein) (13):
7
(cid:112)
(1D) =2 4 4ω V2, Combining (45) with the relations (32), the propa-
E − −
(cid:32) (cid:33) gatingdroplet’sspindensity,momentum,andenergy
√4 4ω V2
(1D) =2 tanh−1 − − , intheweaklynonlinearregimeareapproximately
N 2 ω
− (cid:20) b(1 V2/4)(cid:21)
(cid:18) V2+2ω (cid:19) NT 1+ν2 − (46a)
(1D) =π 2 tan 1 . N ∼ 1+V2/4 (1+V2/4)2
−
P − V √4 4ω V2 V (cid:20) (cid:18) 1 V2/4
We observe the notable fact tha−t the−momentum P1 ∼ 2(1N+TV2/4) 1+ν2 b(1+−V2/4)2 (46b)
(1D) is bounded, 0 (1D) < 2π, but the energy (cid:19)(cid:21)
P ≤ P 1
andspindensityarenot. + ,
1+V2/4
(cid:20) (cid:18)2b(1 V2/4)
4. PropagatingDroplet:AsymptoticResults 1+ν2 − (46c)
E ∼NT (1 V2/4)3
1−+3V2/4 1 (cid:19)(cid:21)
In this section, asymptotic methods are employed + .
to study the behavior of the propagating droplet in (1+V2/4)2 − 1+V2/4
thesmallamplitudeandsmallspeedregimes. TheseresultsarenumericallyverifiedinSec.5.2.
For the initial value problem in the weakly non-
4.1. Weaklynonlinearregime linear regime, one can derive at leading order the
time-dependent cubic Nonlinear Schr¨odinger equa-
tion (NLS) from Eq. (18) for the slowly varying en-
Small amplitude, propagating droplets were stud-
velope of a carrier wave. In two spatial dimensions,
ied asymptotically in [11]. In this section, we briefly
the NLS equation exhibits critical blow up with the
reviewtheresultswithaviewtowardtheirnumerical
Townesmodeplayinganimportantrole[32].Forini-
verificationinSec.5.
tial data with L2(R2) norm less than u , the so-
Using a slowly varying envelope, small amplitude T 2
(cid:107) (cid:107)
lution disperses to zero as t . In other cases,
approximationtheauthorsin[11]constructapropa-
e.g. for data close to u in L→2(R∞2) but with larger
gatingdropletbifurcatingfromthesetoflinearwaves T
norm,thesolutionblowsupinfinitetime.Thus,the
wherethesmallparameteristhedeviationfromthe
Townesmodeisunstabletosmallperturbations.Itis
linearbandedge(39)
naturalthentoinquireintothestabilityoftheprop-
0<ν(V,ω)2 1.
agating droplet for the LL Eq. (18). The Jacobian
(cid:28)
The droplet solution for Eq. (18) takes the approxi- condition(34)wasverifiedin[11].Insertionof(46a)
mateform and(46b)intoEq.(33),leadsto
ν(V,ω) bω
w(x1,x2,t) (cid:112) e−i[V(x1−V t)/2−ωt] J = NT + (ν2), (47)
∼ 2(2 ω) (44) −2(2 ω)4 O
u (cid:16)ν(V,ω)(cid:2)−(x V t)2+x2(cid:3)1/2(cid:17) , suggesting that the h−igher order corrections to
× T 1− 2 the NLS approximation stabilize the propagating
with ν > 0 having the meaning of the inverse half droplet. Further discussion of stability in the large
width. As the band edge is approached, ν 0, amplitudecaseispresentedinSec.5.3.
→
the propagating droplet is approximately a wide,
smallamplitude,propagatingTownesmode[31],the
4.2. Smallpropagationspeed
unique,positive,monotonicallydecayingsolutionof
1
u u u3 +u =0, WenowconsiderEqs.(26a)and(26b)withbound-
− (cid:48)T(cid:48)−R (cid:48)T − T T ary conditions (27) and (28) for small values of the
u (0)=0, lim u (R)=0.
(cid:48)T T dropletspeed0<V 1.Polarcoordinatesareused
R→∞ (seeFig.1) (cid:28)
Using(44)anditsnextordercorrection,thefunc-
tional(31)canbeexpandedas[11] ξ =x Vt=ρ cosϕ, ξ =x =ρ sinϕ. (48)
1 1 2 2
−
(cid:20) b(1 V2/4) (cid:21) Asolutionissoughtintheform
NT ν2+ν4 − , (45)
D ∼ 1+V2/4 2(1+V2/4)2 θ =Θ (ρ;ω)+ (V2),
0
O
where (cid:18) 1 (cid:19) (49)
φ=V Φ (ρ,ϕ) ρ cosϕ + (V2),
NT ≡(cid:107)uT(cid:107)2L2(R2) ≈11.7009, 1 − 2 O
b 2π(cid:82)0∞u2Tu(cid:48)T2RdR 1.35825. whereΘ0isthesolutionofEq.(20)forthestationary
≡ ≈ droplet with 0 < ω < 1 and θ and φ are given in
T
N
8
the ansatz (25). The azimuthal angle φ in Eq. (49) verified by the analytical criterion J > 0 and time
shouldnotbeconfusedwiththeangleϕofEq.(48); dependentnumericalsimulations.
seeFig.1.
AtfirstorderinV,thefunctionΦ1 satisfies 5.1. Iterative Method to Compute Propagating
∂Φ Droplets
1
sinΘ ∆Φ +2Θ cosΘ =(cosΘ 1)Θ cosϕ,
0 1 (cid:48)0 ∂ρ 0 0− (cid:48)0
(50) Asolutiontothenonlineareigenvalueproblem(37)
subjecttotheboundarycondition iscomputedbythespectralrenormalizationmethod
lim Φ =0. [33],ageneralizationofthePetviashvilimethod[34].
1
ρ ∇ WebegintheprocedurebyrephrasingEq.(37)as
→∞
Here and hereafter the prime indicates derivation follows
with respect to ρ. Equation (50) admits separation ωf +iVf +∆f =N[f], (52a)
x1
ofvariablesintheform
with
Φ1(ρ,ϕ)= Φ˜1(ρ) cosϕ, N[f]= 2f∗(fx21 +fx22)+f(1−|f|2), (52b)
sinΘ0(ρ) 1+ f 2
| |
where Φ˜1 is the solution of the ordinary differential and,upontakingitsFouriertransform,resultsin
equation(ODE)
L Φ˜ =(1 cosΘ )Θ , (51a) (cid:0)ω−V κ1−κ2(cid:1) f(cid:98)=N(cid:91)[f],
Φ 1 − 0 (cid:48)0
d2 1 d 1 wheref(cid:98)= [f](κ)istheFouriertransformoff and
F
LΦ =−dρ2 − ρdρ + ρ2 +VΦ(ρ), (51b) κ = (κ1,κ2) is the Fourier wavenumber, with |κ| =
κ.Theiteration
2
V (ρ)= Θ +(cosΘ ω) cosΘ , (51c)
Φ − (cid:48)0 0− 0 (ω V κ1 κ2 c)F(cid:98)n+1 =N(cid:92)[fn] cf(cid:98)n, (53)
subjectto − − − −
isintroducedsothat
d (cid:104) (cid:105)
ρlim dρ √reρ√1−ωΦ˜1(ρ) =0, Φ˜1(0)=0, N(cid:92)[fn] cf(cid:98)n
→∞ (51d) F(cid:98)n+1 = ω c V−κ κ2 ,
1
− − −
fromtheasymptoticsofthestationarydroplet(23).
wherecisaconstantrealparameter,choseninorder
Equation (51a) is solvable for any choice of the
to avoid divergence of the iteration – its value is ex-
parameter 0 < ω < 1. This can be seen as follows.
perimentallyobtained,see[33].Thevalueofcaffects
Standardarguments(e.g.usingvariationalmethods)
the convergence speed of the iteration and must be
showthatL isapositivedefiniteoperator.Theso-
Φ takenlargerasthedropletamplitudeisincreased.In
lutionto(51a),
mostofourcomputations,wefindc=10tosuffice.
Φ˜1 =L−Φ1{(1−cosΘ0)Θ(cid:48)0} , Inordertoiteratethesequence{fn},wescaleFn+1
accordingto
satisfies the boundary condition (51d) as shown by
theasymptotics, f =η F , (54)
n+1 n+1 n+1
(cid:32) (cid:33)
Φ˜ = e−ρ√1−ω ρ 1, and choose the complex parameter ηn+1 so that the
1 O √ρ (cid:29) projection of Eq. (52) onto fn+1 is satisfied. Insert-
ing (54) into Eq. (52) and taking the L2(R2) inner
Φ˜ = (ρ) 0<ρ 1.
1 productwithf resultsin
O (cid:28) n+1
Therefore,fromtheansatz(25)and(49),wefindthat (cid:90) (cid:20) ∂F
anapproximatedropletsolutionforsmallV consists ω|Fn+1|2+iV Fn∗+1 ∂nx+1 (55)
ofthestationarysolutionmodulatedbythespatially (cid:21)
varyingphase F 2 dx dx = η 2
n+1 1 2 n+1
−|∇ | | | ×
Φ=ωt− V2x1+V Φ1(ρ,ϕ)+O(cid:0)V2(cid:1) . (cid:40) (cid:90) (cid:0)Fn∗+1(cid:1)2 (cid:20)(cid:16)∂F∂nx+1(cid:17)2+(cid:16)∂F∂ny+1(cid:17)2(cid:21)
2 dx dx
5. PropagatingDroplet:NumericalResults 1+ ηn+1 2 Fn+1 2 1 2
| | | |
(cid:90) F 2 (cid:0)1 η 2 F 2(cid:1) (cid:41)
n+1 n+1 n+1
In this section, a numerical method is introduced + | 1|+ η−| 2 F| | 2 | dx1dx2 .
n+1 n+1
to solve Eq. (37) for propagating droplets. The so- | | | |
lutions’ properties are investigated and its stability Equation(55)determines η .
n+1
| |
9
The numerical computations are implemented on withω <0werecomputedandexhibitsimilarprop-
thefinite,discretegrid erties to those in S. We were unable to compute
large momentum droplets for particular V with ω
x = L+(j 1)∆, j =1,2,...,N,
1,2j − − sufficiently small or negative due to inherently large
with phase gradients. Illustrative example droplet solu-
2L
∆= . tionsareshowninFig.3.Asω andV areincreased,
N
the droplet amplitude, defined as 1 min(m ) =
Due to the localization of F , derivatives are esti- − 3
n+1 1 m (0,0), decreases. The spin wave background
matedusingthefastFouriertransform.WetakeL= − 3
is evident, exhibiting a decreasing wavelength as V
50andN =1536,implying∆ 0.065,unlessother-
≈ is increased or as ω is decreased. This is in accord
wise stated. To obtain droplets with large speed, we
with the nonlinear dispersion relation discussed in
use continuation in V. For small V, the initial guess
Sec. 3.1. The precessional motion of a propagating
f isthestationarydropletdiscussedinSec.2.2with
0 dropletisdepictedinFig.4inthecomovingframe.
thesameω.ForlargeV,f isapreviouslycomputed
0 Using Eq. (19), the energy , momentum , and
propagatingdropletwiththesameω butsmallerV. E P1
spindensity werecomputedoverS.Contourplots
The integrals in (55) are computed using the spec- N
of these quantities are shown in Figs. 5(a)-5(c). Ad-
trallyaccuratetrapezoidalrule.
ditional values of the conserved quantities outside
Equation(55)issolvedfor η 2 usingNewton’s
| n+1| S were obtained by interpolation using the weakly
method and the phase of η is determined by the
n+1 nonlinearasymptotics(46).Boththeenergyandthe
condition that f have a vanishing phase at the
n+1 spindensityaremonotonedecreasingfunctionsofV
originsothat
andω.Onthecontrary,Fig.5(b)showsthatthemo-
fn+1 = ηn+1 Fn+1e−i[argFn+1(0,0)], (56) mentumdependenceisnotmonotonic.Forfixedval-
| |
uesofω,themomentum exhibitsamaximumfor
Thisimpliesthattheprojectionofthemagnetization P1
V >0.This“ridgeline”isshownasthedashedcurve
attheoriginontotheplanepointsalonge .
1
in Fig. 5(b). The droplets with the largest momen-
Itishelpfultointroducetheresidual
tum exhibit the largest phase gradient near the ori-
r[fn]= −1(cid:104)N(cid:100)fn (cid:0)ω Vκ1 κ2(cid:1)f(cid:98)(cid:105), gin.See,forexample,thecaseV =0.212,ω =0.3in
F − − −
Figs. 3 and 4 which is on the ridgeline of large mo-
which is zero at a droplet solution. We iterate from
mentumdroplets.
f until
0 Figure 6 depicts the numerical verification of the
f f <tol , weakly nonlinear results (46) for fixed ω = 0.4. The
n n 1 f
(cid:107) − − (cid:107)∞ resultsagreenearthebandedgewhereV V
r[f ] <tol , max
n r − (cid:28)
(cid:107) (cid:107)∞ 1 and we have verified that the error exhibits the
η˜ η˜ <tol ,
| n− n−1| η expectedν4 scaling.
where we set tolf = tolη = tolr = 10−9, unless oth- The droplet’s amplitude dependence is shown
erwisestated. in Fig. 5(d). As observed earlier, larger amplitude
Further details and validation of this numerical droplets correspond to smaller speeds and rest fre-
schemearepresentedintheAppendix. quencies.
ThedropletsshowninFigs.3and4shownegligible
5.2. DropletProperties signs of asymmetry in their localization extent. To
furtherinvestigatethis,wedefinethedropletwidths
inthex andx directionsaccordingtothestandard
Usingthenumericalschemepresentedintheprevi- 1 2
deviations
oussection,wecomputedalargenumberofsolutions
to the boundary value problem (37) for a propagat- (cid:18) 1 (cid:90) (cid:19)12
width = x2 (1 cosΘ) dx , (57a)
ingdropletwithparameterslyingintheset 1 1 −
S =(cid:8)(V,ω) 0 V 2√1 ω 0.25, (cid:18)N1 (cid:90) (cid:19)12
|0.2≤75 ≤ω 0−.95(cid:9)−. width2 = x22 (1−cosΘ) dx . (57b)
≤ ≤ N
This representative set of droplet parameters was Acontourplotofthequantitywidth for(V,ω) S
1
∈
chosen because accurate solutions were obtained is shown in Fig. 5(e), exhibiting non monotonic be-
without having to change the numerical parameters havior for increasing V and ω, especially for large
ofourscheme.Outsideofthisset,eitherhigherreso- amplitude modes. The droplet’s width diverges ac-
lutionorlargerdomainswereoftenrequiredtoaccu- cordingto (1/ν(V,ω))asthelinearspinwaveband
O
rately resolve the droplets (see Appendix). Droplets isapproached0<ν 1(seeEq.(44)).
(cid:28)
10