ScienceDirect
Nuclear Physics B 918 (2017) 91–114
www.elsevier.com/locate/nuclphysb
Darboux–Bäcklund
transformations,
dressing
&
impurities
in
multi-component
NLS
Panagiota Adamopoulou
a, Anastasia Doikou
a,∗, Georgios Papamikos
baDepartment of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom bDepartment of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
Received 3January2017;accepted 21February2017 Availableonline 3March2017
Editor: HubertSaleur
Abstract
Weconsiderthediscreteandcontinuousvectornon-linearSchrödinger(NLS)model.Wefocusonthe casewherespace-likelocaldiscontinuitiesarepresent,andweareprimarilyinterestedinthetimeevolution onthedefectpoint.ThisinturnyieldsthetimepartofatypicalDarboux–Bäcklundtransformation.Within thisspiritwethenexplicitlyworkoutthegenericBäcklundtransformation andthedressingassociated tobothdiscreteandcontinuousspectrum,i.e.theDarbouxtransformationisexpressedinthematrixand integralrepresentationrespectively.
©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Thenon-linear Schrödingerequation(NLS) isoneofthe fundamentalequations in
mathe-maticalphysicswithnumerousapplications,e.g.inthe theoryof non-linearopticsandocean
waves(seee.g.[1,2])tonameafew.TheNLSequationisanexactlysolvablemodel,andhas
beenintegratedusingtheInverseScatteringTransform(IST)[3,4],(seealso[5,6]).Thefirst vec-torgeneralisationofNLSwasintroducedbyManakov[7],whilefurthergeneralisationsofthe modelhavebeendiscovered(seeforinstance[8–10]andreferencestherein).Analternativeto
* Correspondingauthor.
E-mail addresses:[email protected](P. Adamopoulou), [email protected](A. Doikou),
[email protected](G. Papamikos).
http://dx.doi.org/10.1016/j.nuclphysb.2017.02.016
theISTmethodforconstructingsolutionsofintegrableequationsisthedressingmethod,which wasfirstpresentedbyZakharovandShabat(ZS)[11]andfurtherdevelopedin[12].The dress-ingformulationisbasedontheconceptof DarbouxTransformations(DTs)[13],anditisthis approachthatwefollowinSections4and5,wherewepresentthedressingmethodforthevector
NLSequation(vNLS).
Arelevantproblemwithinthisframeistheinterpretationoflocalspace(time)-likeintegrable impuritiesasDarboux–Bäcklundtransformations.Itwasfirstobservedin[14]thatclassical
de-fects inintegrable1+1 integrablefieldtheoriesmaybeseenas“frozen”Darboux–Bäcklund
transformations(seealso[15–21]).Thenalongthisspiritthenotionof quasi Bäcklund transfor-mationasdefectwasintroducedin[21]inbothdiscreteandcontinuousintegrablesystems.Inthe presentinvestigationweexplorethequasiBäcklundtransformationi.e.thedefectforthediscrete andcontinuousvectorNLSmodel.Thesl2discreteNLSmodelwasstudiedin[18],whereasthe
continuousgeneralised NLSin[20].Herewegeneralise thestudy ofthediscretevectorNLS,
andthenwealsoconsiderthecontinuousvectorNLSmodel.Inthecontinuouscasewemainly
focusonthetimeevolutionassociatedtothedefect,andinspiredbythiswegivesomegeneric expressionsontheBäcklundtransformationsanddressing.Abriefdiscussionofanovelclassof BTsthatassociatesolitonicwithanti-solitonicsolutionsisalsopresented.
More precisely,the outlineof thispaperisas follows:inSection2wepresent thediscrete
vector NLS model,after abrief review westudy the model inthe presence of alocaldefect
insection3.Theassociatedintegralsofmotionandthecorrespondingtimecomponentsofthe
Lax pairsarepresented. InSection4 wefocus onthe dressingandBäcklund transformations
(BT)forthecontinuousvectorNLSequation.Wetreatboththefocusinganddefocusingcases
simultaneouslyusinganappropriatesymmetryoftherelatedLaxpair.Suchsymmetrygroups,
knownasreduction groups,were firstintroducedin[22–24]andlaterdevelopedin,e.g.[25]. Inparticular,wepresentthedressingtransformationandgivethegeneralhigherrank1-soliton solution,as wellasthe n-solitonsolutionas aratio of determinants.Moreover,we obtainthe
BäcklundtransformationforthevectorNLSmodel,whichgeneralisestheBTforthefocusing
anddefocusingscalarNLSequationpresentedin[26,27].Wealsobrieflydiscusstheexistence of anovelclassof BTs,whichessentiallyrelateseachfieldtoits conjugateorinotherwords solitonicsolutionstoanti-solitonicones.InSection5weprovideagenericdescriptionoftheZS
dressingandthe Darboux–Bäcklundtransformsas integralrepresentations. Thenovelcaseof
differentspectralparametersassociatedtoeachfieldeveninthecaseof“one-soliton”solution isdiscussed.ThispictureismoreintunewiththequantumpictureandthenestedBetheansatz
formulation.BoththediscreteandcontinuousspectrumarediscussedforthevNLSmodel.
2. DiscretevectorNLS
LetusfirstfocusouranalysisonthediscretevectorNLSmodel,generalising essentiallythe
results presentedin[18],wherethe sl2 NLS modelwas studiedinthe presence of point-like
defects.Weshallfocusmainlyonthetimeevolutionofthedegreesoffreedomassociatedtothe defectobtainedessentiallyasequationsofmotionofthesystemevaluatedonthedefectpoint. Weconsiderthefollowinglinearsystem[5]
j+1=Ljj
d
dtj=Ajj,
foranauxiliaryfunction,with(L,A)theLaxpair.Herej denotesthelatticesiteona one-dimensionalN-siteperiodiclattice.Thecompatibilityconditionoftheabovesystemofequations reads
d
dtLj=Aj+1Lj−LjAj, (2.2)
andisequivalenttothediscrete(differential-difference)equationathand.
Inthecaseofthediscrete glN NLSmodel,theassociatedLaxoperatorL(actingonsitej)is givenby
Lj(λ)=(1+λ+ N−1
k=1
xj(k)Xj(k))e11+
N
k=2 ekk+
N
k=2
xj(k−1)e1k+X(kj−1)ek1
, (2.3)
whereλisthespectralparameterandekl areN×N matricessuchthat(ekl)pq=δkpδlq.The
Laxoperator(2.3)satisfiesthequadraticalgebra[5]
Lai(λ1),Lbj(λ2)
=rab(λ1−λ2),Lai(λ1)Lbj(λ2)
δij, (2.4)
wheretheindicesa,bdenoteauxiliaryspacesandi,jdenotesitesontheone-dimensionallattice. Thecorrespondingr-matrixis[28]
r(λ)=P
λ with P=
N
i,j=1
eij⊗ej i, (2.5)
andsatisfiestheclassicalYang–Baxterequation[29].Relation(2.4)providesthefollowing Pois-sonbracketsbetweenthedynamicalvariablesx(k),X(k)
xi(k), X(l)j
= −δijδkl. (2.6)
ThemonodromymatrixisdefinedastheproductofNLaxoperatorseachactingonasiteof
theperiodiclattice,inotherwords,
T(λ)=LN(λ)LN−1(λ) . . .L1(λ) . (2.7)
Themonodromy matrixT(λ) satisfiesthe samequadraticrelation(2.4)as L(λ).Ifwe define thetransfermatrixτ (λ)asthetraceofthemonodromymatrix,i.e.τ (λ)=trT(λ),thenonecan verifythat
τ (λ1), τ (λ2)
=0. (2.8)
Hence,expansionofthetransfermatrixτ (λ)inpowersofthespectralparameterλprovidesthe chargesininvolution.Toobtainthe localintegralsofmotion,expansionof lnτ (λ)isrequired instead.
In orderto obtain the associated integrals of motion it is convenientto utilisethe bra-ket notationfora(N−1)-dimensionalvectorandco-vector,inotherwords,
|X :=
⎛ ⎜ ⎜ ⎜ ⎝
X(1) X(2) .. . X(N−1)
⎞ ⎟ ⎟ ⎟
⎠, x| :=
x(1)x(2). . . x(N−1)
andalsodefine
Nj=1+ N−1
k=1
xj(k)Xj(k):=1+ xj|Xj. (2.10)
ThentheLaxoperator(2.3)canbewritteninblockmatrixformas
Lj=λDj+Aj=λ
1 0|
|0 0
+
Nj xj|
|Xj 1
, (2.11)
where0and1denotethe(N −1)×(N −1)zeroandidentitymatrix,respectively.
Expanding the monodromy matrix T(λ) in powers of 1λ we can thenexpress the transfer
matrixintheform
τ (λ)=1+1
λτ1+
1
λ2τ2+
1
λ3τ3+. . . , (2.12)
whereforinstance,
τ1=tr
N
i=1
DN. . . Di+1AiDi−1. . . D1
,
τ2=tr
⎛
⎝
i>j
DN. . . Di+1AiDi−1. . . Dj+1AjDj−1. . . D1
⎞
⎠. (2.13)
Theassociatedlocalintegralsofmotionareobtainedascoefficientsoftheexpansionoflnτ (λ)
inpowersof 1λ,i.e.:
lnτ (λ→ ∞)=1
λI1+
1
λ2I2+
1
λ3I3+. . . . (2.14)
ThedifferentImarefoundintermsof{τi}mi=1.Forexample,forthefirstthreelocalintegralsof
motionwehave
I1=τ1, I2= −
1 2I
2
1+τ2, I3= −
1 6I
3
1−I1I2+τ3, (2.15)
andsoon.ItturnsoutthatthefirstthreeImaregivenbythefollowingexpressions
I1=
N
i=1 Ni,
I2= −
1 2
N
i=1 N2i +
N
i=1
xi|Xi−1,
I3=
1 3
N
i=1 N3i +
N
i=1
xi|Xi−2 −
N
i=1
(Ni−1+Ni)xi|Xi−1. (2.16)
EachoftheaboveintegralsofmotionhasanassociatedLaxpair(L,A).TheoperatorAof theLaxpaircanbefoundvia[29]
Aj(λ, μ)=τ−1(λ)tra
Ta(N, j, λ) rab(λ−μ)Ta(j−1,1, λ)
, (2.17)
Ta(i, j, λ)=Lai(λ)Lai−1(λ) . . .Laj(λ) , with i > j . (2.18)
Inthepresentcase,wherether-matrixisgivenbyexpression(2.5),operatorAtakestheform:
Aj(λ, μ)= τ−1(λ)
λ−μ T(j−1,1, λ)T(N, j, λ) . (2.19)
Expanding(2.19)inpowersof 1λ resultsin
Aj(λ, μ)=
1
λA (1) j (μ)+
1
λ2A
(2) j (μ)+
1
λ3A
(3)
j (μ)+. . . , (2.20)
witheachA(i)j beingassociatedtoeachoftheintegralsofmotion(2.16).Inthepresentcasewe obtain
A(1) j (μ)=
1 0|
|0 0
, A(j2)(μ)=
μ xj|
|Xj−1 0
,
A(3) j (μ)=
μ2− xj|Xj−1 μxj| −Njxj| + xj+1|
|Xj−1μ− |Xj−1Nj−1+ |Xj−2 |Xj−1xj|
. (2.21)
Considerthepair(Lj,Aj(3)).Thentheassociatedequationsofmotionforthemulticomponent
fieldsx|and|Xareobtainedfromthecompatibilitycondition(2.2)
˙xj| =N2jxj| − xj+1|Xjxj| − xj|Xj−1xj| −(Nj+Nj+1)xj+1| + xj+2|,
| ˙Xj = −|XjN2j+ |Xjxj+1|Xj + |Xjxj|Xj−1 + |Xj−1(Nj−1+Nj)− |Xj−2.
(2.22)
3. VectorDNLSinthepresenceofdefects
WenowconsidertheDNLSmodelinthepresenceofapoint-likeintegrabledefect.We intro-ducethedefectonsitenoftheone-dimensionalN-sitelattice,withn=1,N.TheLaxoperator associatedtothedefectis
Ln(λ)=λ N
k=1 ekk+
N
k, l=1
αn(kl)ekl, (3.1)
andweassumethatitsatisfiesthesamequadraticalgebra(2.4)asthe glN Laxoperator(2.3). Hence,itfollowsthatthePoissonbracketsbetweenthedynamicalvariablesα(kl)n associatedto
thedefectaregivenby
αn(ij ), αn(kl)
=α(il)n δkj−αn(lj )δik. (3.2)
ForconveniencewewritethedefectLaxoperator(3.1)inthefollowingform
Ln(λ)=λ1N +An=λ1N +
αn βn|
|γn n
, (3.3)
αn:=αn(11), βn| :=
αn(12)α(n13). . . αn(1N)
,
|γn :=
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
α(n21) α(n31) .. . α(nN1)
⎞ ⎟ ⎟ ⎟ ⎟
⎠, n:=
⎛ ⎜ ⎜ ⎝
α(n22) . . . αn(2N) ..
. . .. ... α(nN2) . . . αn(N N)
⎞ ⎟ ⎟
⎠. (3.4)
Inthecasewhereadefectisintroducedonsiten,themonodromymatrixT(λ)reads
T(λ)=LN(λ) . . .Ln+1(λ)Ln(λ)Ln−1(λ) . . .L1(λ) . (3.5)
Toobtainthelocalintegralsofmotionwefirstexpandthemonodromymatrixinpowersof λ1, wherethistimethecontributionfromthedefectpointmustbetakenintoaccount(seealso[20]
for genericexpressions). Then, theexpansion inpowersof λ1 of thelogarithmof the transfer matrix(withthedefectincorporated)reads
lnτ (λ→ ∞)=1
λI1+
1
λ2I2+
1
λ3I3+. . . . (3.6)
Theintegralsofmotiontaketheform
I1=
i=n
Ni+αn,
I2= −
1 2
i=n N2i −1
2α
2
n+
i=n,n−1
xi+1|Xi + xn+1|Xn−1 + βn|Xn−1 + xn+1|γn,
I3=
1 3
i=n N3i +1
3α
3
n+
i=n,n±1
xi+1|Xi−1
−
i=n,n−1
(Ni+Ni+1)xi+1|Xi +Nn+1xn+1|Xn−1
+(Nn−1−αn)xn+1|Xn−1 + xn+1|n|Xn−1 + xn+2|Xn−1
+ xn+1|Xn−2 −αnxn+1|γn, (3.7)
wherewehavedefined
xn+1| = xn+1+βn|, |Xn−1 = |Xn−1+γn. (3.8)
ThecomponentsoftheLaxpairaroundthedefectpointaregivenby:
A(2) n (μ)=
μ xn+1|
|Xn−1 0
, A(n2+)1(μ)=
μ xn+1|
|Xn−1 0
. (3.9)
A(3) n−1(μ)=
μ2− xn−1|Xn−2 (μ−Nn−1)xn−1| + xn+1|
|Xn−2(μ−Nn−2)+ |Xn−3 |Xn−2xn−1|
,
A(3) n (μ)=
μ2− xn+1|Xn−1 xn+1|μ+ xn+1| + xn+2|
|Xn−1(μ−Nn−1)+ |Xn−2 |Xn−1xn+1|
,
A(3) n+1(μ)=
μ2− xn+1|Xn−1 (μ−Nn+1)xn+1| + xn+2|
|Xn−1μ+ |Xn−1 + |Xn−2 |Xn−1xn+1|
,
A(3) n+2(μ)=
μ2− xn+2|Xn+1 (μ−Nn+2)xn+2| + xn+3|
|Xn+1(μ−Nn+1)+ |Xn−1 |Xn+1xn+2|
where
xn+1| = xn+1|n−Nn+1xn+1| −αnxn+1|,
|Xn−1 =n|Xn−1 − |Xn−1Nn−1− |Xn−1αn. (3.11)
Againweconsiderthepair(Lj,A(j3)).Forj=n,n±1,n±2 thematrixA (3)
j isgivenin(2.21)
andtheequationsofmotionforthefieldsx|,|Xcoincidewithequations(2.22).However,in ordertoderivetheequationsofmotionforthefieldsintheneighbourhoodofthedefect,i.e.for
j=n±1,n±2,onemusttakeintoaccountexpressions(3.10).Hence,weobtainthefollowing differential-differenceequationsforx|,|X
˙xn−2| =N2n−2xn−2| − xn−1|Xn−2xn−2| − xn−2|Xn−3xn−2|
−(Nn−1+Nn−2)xn−1| + βn| + xn+1|,
| ˙Xn−2 = −|Xn−2N2n−2+ |Xn−2xn−1|Xn−2 + |Xn−2xn−2|Xn−3
+ |Xn−3(Nn−2+Nn−3)− |Xn−4, (3.12)
˙xn−1| =N2n−1xn−1| − xn+1|Xn−1xn−1| − xn−1|Xn−2xn−1| − βn|Xn−1xn−1|
−αnβn| −Nn−1βn| −(Nn−1+Nn+1)xn+1| + xn+1|n+ xn+2|,
| ˙Xn−1 = −|Xn−1N2n−1+ |Xn−1xn+1|Xn−1 + |Xn−1xn−1|Xn−2
+ |Xn−1βn|Xn−1 + |Xn−2(Nn−1+Nn−2)− |Xn−3, (3.13)
˙xn+1| =N2n+1xn+1| − xn+1|Xn−1xn+1| − xn+2|Xn+1xn+1| − xn+1|γnxn+1|
−(Nn+1+Nn+2)xn+2| + xn+3|,
| ˙Xn+1 = −|Xn+1N2n+1+ |Xn+1xn+1|Xn−1 + |Xn+1xn+2|Xn+1
+ |Xn+1xn+1|γn + |γnNn+1+ |γnαn+ |Xn−1αn−n|Xn−1
+ |Xn−1(Nn−1+Nn+1)− |Xn−2, (3.14)
˙xn+2| =N2n+2xn+2| − xn+2|Xn+1xn+2| − xn+3|Xn+2xn+2|
−(Nn+2+Nn+3)xn+3| + xn+4|,
| ˙Xn+2 = −|Xn+2N2n+2+ |Xn+2xn+2|Xn+1 + |Xn+2xn+3|Xn+2
+ |Xn+1(Nn+1+Nn+2)− |γn − |Xn−1. (3.15)
Moreover,onthedefectsitenthecompatibilitycondition(2.2)takestheform
d
dtLn=An+1Ln−LnAn, (3.16)
andleadstothe followingequations of motion for thedynamical variablesαn,βn|,|γn,n
associatedtothedefect
˙
αn=(αn+Nn−1)βn|Xn−1 −(αn+Nn+1)xn+1|γn + xn+2|γn − βn|Xn−2,
˙βn| = −xn+1|Xn−1βn| − xn+1|γnβn| − βn|Xn−1βn| +α2nβn|
+αn(αn+Nn+1)xn+1| −(αn+Nn+1)xn+1|n− βn|Xn−1xn+1|
| ˙γn = −|γnα2n+ |γnxn+1|γn + |γnxn+1|Xn−1 + |γnβn|Xn−1
− |Xn−1αn2− |Xn−1αnNn−1+n|Xn−1(αn+Nn−1)+ |Xn−1xn+1|γn
+ |Xn−2αn−n|Xn−2,
˙
n= −(αn+Nn−1)|Xn−1βn| +(αn+Nn+1)|γnxn+1| + |Xn−2βn| − |γnxn+2|
+ |Xn−1xn+1|n−n|Xn−1xn+1|. (3.17)
Theequationsabovegeneralise theresultspresentedin[20]intheglN case.
4. ThecontinuousvectorNLS
InthepresentsectionweconstructaDarbouxtransformationforthecontinuousvNLS
equa-tionsforboththefocusinganddefocusingcases.Theconstructionmakesuseofthereduction
group[22].ThematrixthatdefinestheDarbouxtransformation,isstructurallysimilartothe de-fectmatrixasalreadydiscussed.Thissuggestsanalternativetother-matrixconstructionofthe defectandpossibleclassificationusingthetheoryofreductiongroups.
WestartwiththeLaxoperatorofthevectorAKNShierarchynamely,
L(λ)=Dx−U(λ), U(λ)=λU1+U0 (4.1)
where
U1=α
ρ1 0
0 −1
and U0=β
0 |v
u| 0
. (4.2)
In(4.2)αandβ arecomplexnumbers,1isthe(N−1)×(N −1)identitymatrix,and|u,|v
areN −1 dimensionalvectorvaluedfields.Wechooseρ=(N −1)−1inordertoensurethat
U1∈slN.ItfollowsthatU(λ)∈slN[λ].
WeassumethattheLaxoperatorL(λ)isinvariantundertheZ2reductiongroupgeneratedby theinvolution
r:L(λ)→ −QL(λ∗)†Q (4.3)
where
Q=
1 0
0 −κ
, κ= ±1. (4.4)
TheL(λ)†in(4.3)denotestheformaladjointoperatorofL(λ),i.e.L(λ)†= −Dx−U(λ)†.
More-over,∗denotescomplexconjugationand† standsforHermitianconjugation.Theinvarianceof
L(λ)underrimpliesthat
−QU(λ∗)†Q=U(λ). (4.5)
Fromequation(4.5)followsthat
α=ia∈iR, β=√κ and |v = |u∗≡ |u∗. (4.6)
Forconveniencewechoosethenormalisationa= −(ρ+1)−1=(1−N)N−1. Itiseasytoseethattheoperator
A(λ)=Dt−V(λ), V(λ)=λ2V2+λV1+V0 (4.7)
V2= −U1, V1= −U0 and V0=
iκ|u∗u| −i√κ|u∗x
i√κu|x −iκu∗|u
(4.8)
isalsoinvariantundertheactionofrandthatthecompatibilityconditionofthetwooperators
[L,A] =0⇔Ut−Vx+ [U,V] =0 (4.9)
isequivalenttothevectorNLSequation
i|ut+ |uxx−2κ|u|2|u =0, (4.10)
where|u|2= u∗|u.Dependingonwhichκ we choosewe obtainthefocusing ordefocusing
vNLSequation(see[30,8]forgeneralfocusing/defocusingsystemsofNLSequations).Inwhat
followsnextwetreatbothvNLSequationssimultaneously.
4.1. Darboux transformation with a Z2symmetry
WearenowinthepositiontoconstructDarboux-dressingtransformationsforthevNLS(4.10)
usingthereductiongroup(4.3).Let(λ)bethefundamentalsolutionofthelinearsystem
x=U(λ), t=V(λ) (4.11)
satisfyingtheinitialcondition(λ)(x,t )=(0,0)=1.Usingtheinvariancecondition(4.5)it fol-lowsthat Q(λ∗)†−1Qisalso asolution whichsatisfies thesameinitialcondition,hence we obtainthat satisfies
Q(λ∗)†−1Q=(λ). (4.12)
ADarbouxtransformationisagaugetransformation
(λ)→(λ)=M(λ)(λ) (4.13)
that leavesthe linear system(4.11) covariant.We callthe matrix M(λ) Darboux or dressing
matrix.We are interested tofind thoseDarboux matricesso that satisfies the same initial
conditionsas .Thisimplies thatsatisfies theconstraint(4.12) too.Thenit isnothardto showthattheDarbouxmatrixM(λ)hastosatisfythesamerelation(4.12)orequivalently
M(λ)−1=QM(λ∗)†Q. (4.14)
Thetransformedfundamentalsolution(λ)satisfiesthelinearsystem
x=U(λ), t=V(λ) (4.15)
whereU(λ)=U(|u,λ)andsimilarlyforV(λ).Using(4.13)and(4.11),equations(4.15)imply thattheDarbouxmatrixsatisfiesthefollowingequations
Mx=UM−MU, Mt=VM−MV (4.16)
knownasDarboux–Laxequations.TheDarboux–Laxequationsareequivalenttothefollowing
relations
L(λ)=M(λ)L(λ)M(λ)−1, A(λ)=M(λ)A(λ)M(λ)−1, (4.17)
whereL(λ)=Dx−U(λ)andA(λ)=Dt−V(λ).Equations(4.16)arelinearinM(λ)andthus
invariantunderatransformationoftheform
wheref (λ)isanon-zeroscalarfunctionofλ.Thismeansthatwithoutanylossofgeneralitywe canassumethatM(λ)hasnopolesatλ= ∞.Thesimplestsuchmatrixwithasinglepoleisof theform
M(λ)=1+ M0
λ−μ. (4.18)
Equation(4.14)impliesthattheinverseofM(λ)hastheform
M(λ)−1=1+QM †
0Q
λ−μ∗. (4.19)
Takingtheresiduesatλ=μandμ∗ofequationM(λ)M(λ)−1=1weobtain
M0
1+QM † 0Q μ−μ∗
=0,
1+ M0
μ∗−μ
QM†0Q=0 (4.20)
respectively. Assumingthat det(M0)=0 wehave that det(QM†0Q)=0 andthus the second equationof(4.20)impliesthatM0=(μ−μ∗)1.InthiscaseM(λ)=λλ−−μμ∗1andsoMisatrivial Darbouxmatrix.HenceweassumethatM0isnotoffullrankandspecificallyweareinterested inthecasewhererank(M0)=1 andthusM(λ)willbethesimplestDarbouxmatrix(elementary
Darbouxmatrix).InthecasewheretheLaxmatricesUandVare2×2 matricestherankone
case isthe onlypossibility,howeverinourcase rank(M0)=1,. . . ,N −1. Nevertheless, we continueourinvestigationsassumingthatM0hasrank(M0)=s.
Sincerank(M0)=s,M0canbeparametrisedbytwomatricesofdimensionN ×sas
M0=pqT. (4.21)
Hereq=(q1,. . . ,qs)∈MN,s(C)withqj=(q j
1,. . . ,q
j
N)T beingN-vectorsandsimilarforp.
Thenwecansolve(4.20)withrespecttopobtaining
p=(μ−μ∗)Qq∗(qTQq∗)−1. (4.22)
Therefore,theDarbouxmatrixhastheform
M(λ)=1+μ−μ ∗
λ−μ P , μ=μ
∗, P =Qq∗(qTQ
q∗)−1qT. (4.23)
NoticethatP isaprojector(P2=P )andthatthefollowingrelations
M(μ∗)Qq∗=0, qTM(μ∗)=0, M−1(μ)Qq∗=0, qTM−1(μ)=0 (4.24)
aresatisfiedduetothefollowingidentities
(1−P )Qq∗=0 and qT(1−P )=0. (4.25)
Moreover,P andthusM(λ)isinvariantunderthetransformation
q→ qC, C:(x, t)→C(x, t )∈GL(s,C). (4.26)
It follows that M(λ) is parametrised by a non-realcomplex numberμ and apoint q inthe
complexGrassmannianGr(s,N)MN,s(C)/GL(s,C).Inthe nextsection wefocus onthe
specialcasewheres=1 andthusqisanelementoftheprojectivespacePN−1(C)Gr(1,N).
We notehere that the s=1 casehas been extensively used, see [5,13,31,32] andreferences
4.2. Dressing and Bäcklund transformations
WehaveusedthesymmetriesoftheLaxpairinordertowritethegeneralformofan
elemen-taryDarboux matrix(4.23).Moreover theDarboux matrixmustpreservethe formofthe Lax
operatorsLandA,i.e.equations(4.17)musthold.Sincethefirstpartof(4.17)musthold iden-ticallyinλweobtainthefollowingequationsfromtheregularpartatλ= ∞andtheresiduesat thesimplepolesatλ=μandλ=μ∗
U0=U0+(μ−μ∗)[P ,U1], PL(μ)(1−P )=0, (1−P )L(μ∗)P=0. (4.27) UsingthefactthatP isaprojectorandoftheform(4.23),itisnothardtoseethatthesecond andthethirdequationof(4.27)areequivalenttothefollowingeigenvalueproblems
qTx +qTU(μ)=f qT, q∗x−QU(μ∗)Qq∗=q∗f1, (4.28)
respectively, withfandf1 s×s matrixvalued functionsof x andt.Taking into accountthe
invarianceofUunderthereductiongroup(4.5)itfollowsthatequations(4.28)arecompatibleif
f1=f†.
Similarly,fromthepolesatλ=μandλ=μ∗ofthesecondequationof(4.17)weobtain
PA(μ)(1−P )=0 and (1−P )A(μ∗)P =0 whichimplythatqalsosatisfy
qTt +qTV(μ)=g qT, q∗t −QV(μ∗)Qq∗=q∗g1, (4.29)
withg,g1being s×s matrixvalued functionsofx,t.Usingagainthe invarianceof Vunder
thereduction groupimplies thatg1=g†.Wehaveprovedthat q hastosatisfythesystemof
equations
qTx +qTU(μ)=f qT, qTt +qTV(μ)=g qT. (4.30)
The compatibilityof equations (4.30)implies that f andg haveto satisfythe zero curvature
conditionft −gx+
f,g=0, andhencelocally existsamatrix valued function hsuch that
f=hxh−1andg=hth−1.Sinceq∈Gr(s,N),thetransformation
q→qhT (4.31)
preservestheformoftheDarbouxmatrixandalsomakesequations(4.30)homogeneous.
There-fore,weobtain
qT =CT(μ)−1=CTQ(μ∗)†Q (4.32)
where(μ∗)isthefundamentalsolutionofthelinearproblem
x=U(μ∗), t=V(μ∗) (4.33)
andCisaconstantmatrixofdimensionN×s.
Using (4.32) and the expression for P (4.23), we can write the projector matrix P in
terms of solutions of the linear system (4.33) that correspond to the vNLS potential |u=
(u1,. . . ,uN−1)T.Thefirstequationof(4.27)definesthetransformationforthevNLSequation
uj=uj+
i(μ∗−μ)
√
κ PNj, j=1, . . . ,N −1 (4.34)
PNj=
0 κqN1∗ · · · κqNs∗ qj1 q1TQq1∗ · · · q1TQqs∗
..
. ... . .. ...
qjs qsTQq1∗ · · · qsTQqs∗
q1TQq1∗ · · · q1TQqs∗ ..
. . .. ...
qsTQq1∗ · · · qsTQqs∗
, (4.35)
wheretheqi i=1,. . . s denotethecolumnsofmatrixq (4.32).Specific solutionsfor various differentswillbepresentedelsewhere.
Thespecialcases=rankM0=1 isthesimplestandhasadditionalinterest.InthiscaseP
takestheform
P =Qq ∗qT
qTQq∗ (4.36)
whereqisnowanN-vector.Moreover,equation(4.34)takestheform
uj=uj−i(μ∗−μ)
√
κ q
∗
Nqj
|q1|2+ · · · + |qN−1|2−κ|qN|2
, j=1, . . . ,N −1 (4.37)
whichisthedressingtransformationforboththefocusinganddefocusingvectorNLSequation
[33,34].Thetransformation(4.37)isageneralisationofthewellknowndressingtransformation forthescalarNLSequation,see[11,35,5].
In therankM0=1 case we canalso usetheDarboux matrix(4.23)inordertoderivethe
Bäcklundtransformationsfor bothfocusing anddefocusingvNLSequations andforarbitrary
N.Tothisendwefirstusetherescalingsymmetry(4.26)andwriteqinthefollowingform
q=
|q
1
. (4.38)
ThenM0takestheform
M0=
d|q∗q| d|q∗
−κdq| −κd
(4.39)
where
d= μ−μ ∗
|q|2−κ with d∗= −d . (4.40)
Thefirstequationof(4.27)impliesthat
|q = 1
i√κd(|u − |u). (4.41)
Itfollowsthat
|q|2= q∗|q = − 1 d2|u−u|
2
(4.42)
andusing(4.40)weobtainthatd=0 satisfiesthequadraticequation
Therefore,
d=μ ∗−μ
2κ ±η (4.44)
where
η=
μ∗−μ
2 2
−κ|u−u|2. (4.45)
Fromthepoleatλ=μofequations(4.16)weseethatM0hastosatisfy
M0x=U(μ)M0−M0U(μ) and M0t=V(μ)M0−M0V(μ) . (4.46)
Thefirstequationof(4.46)implies
(−κdq|)x= −idμκq| +
√
κdu|q∗q| +κ√κdu| (4.47)
whilefromthesecondweobtain
(−κdq|)t=μ(κdq|)x+i
√
κdux|q∗q| +id|u|2q| +idq|u∗u| +iκ
√
κdux|.
(4.48)
Usingrelation(4.41)and(4.44)equation(4.47)takestheform
i(|u − |u)x= −μ (|u − |u)+
μ∗−μ
2 ±η
|u
−|u|2− u|u∗ |u−u|2
μ∗−μ
2 ∓η
(|u − |u) (4.49)
andconstitutesthex-partoftheBäcklundtransformationofvNLSwhile(4.48)canberewritten as
i(|u − |u)t = −iμ(|u − |u)x−i
ux|u∗ − ux|u∗
|u−u|2
μ∗−μ
2 ∓η
(|u − |u)
+κ|u|2(|u − |u)+κ
u|u∗ − |u|2
|u +i
μ∗−μ
2 ±η
|ux. (4.50)
WhenN =2 theBäcklundtransformation(4.49),(4.50)becomestheknownBTforNLS
equa-tionwithκ= ±1 (see[26]).
4.2.1. Bäcklund transformations: solitons to anti-solitons
discussedindetailedinfutureworks.Nevertheless,weshallgiveafirstflavourofthisbehaviour here.
Letusintroducethefollowingobject:
U(X, λ)= −UT(X,−λ), (4.51)
whereT denotesusualtransposition.Thenthe correspondingtimecomponentof theLaxpair
canbederivedviathefamiliarformulabelow[29]
˜
V(x, λ, μ)=τ−1(λ)tra
˜
Ta(−L, x, λ) rabTaTb(−λ+μ)T˜a(x, L, λ)
, (4.52)
whereTa denotestranspositiononthespacecharacterised bytheindexa,and
τ (λ)=t rT˜(−L, L;λ), T˜(a, b, λ)=TT(b, a, λ), b > a
T(b, a, λ)=exp b
a
dxU(x, λ)
. (4.53)
In thecasewherer istheYangianmatrixtherTaTb
ab =rab.WorkingouttheBTforthe setting
aboveweenduptostructurallysimilarBTsastheonesdefinedearlierinthetext,butnowthe followingidentificationshold:
λ→ −λ, | ˜u → |u∗, | ˜u∗ → |u. (4.54)
InthevNLScasethesituationisquitestraightforward,howevermoreinterestingandpresumably richerscenarioscouldariseinmoreinvolvedmodels,suchasthe affineTodafieldtheoriesor higherrankLandau–Lifshitzmodels.Also,thissettingnaturallyapplies todiscreteintegrable modesassociatedtohigherrankalgebras.Allthesearesignificantissuesthatwillbediscussedin detailinfutureinvestigations,giventhatourmainpurposehereistoprovideabriefintroduction tothesolitonanti-solitontypeBTs.
4.3. Higher Darboux transformation
In this section we investigate Darboux-dressing transformationsthat correspond to
multi-soliton solutions. In principle, in order to obtain higher soliton solutions one can consider
compositionsof elementaryDarbouxtransformationsoftheform(4.18)withseveraldifferent
polesinλ,see [5,32].However,here weare interested inanon-elementaryDarboux matrix,
whichhasnpolesandisoftheform
M(λ)=1+
n
i=1 Mi λ−μi
. (4.55)
Moreover,weassumethatM(λ)hasthesamestructureasthe1-solitonDarbouxmatrix,thatis
itsatisfiesrelation(4.14).Itfollowsthattheinversematrixisoftheform
M(λ)−1=1+
n
i=1
QM†iQ
λ−μ∗i . (4.56)
ComparingtheasymptoticexpansionsofM(λ)andM(λ)−1atλ→ ∞wealsoobtainthe
n
i=1
Mi= − n
i=1
QM†iQ . (4.57)
Takingtheresidueatλ=μjandμ∗j ofequationM(λ)M(λ)−1=1wehavethat
MjM(μj)−1=0, M(μ∗j)QM
†
jQ=0, j=1, . . . n, (4.58)
respectively. The above equations imply that all Mj are not of full rank. In general we can
proceedassumingthat rank(Mj)=sj with1≤sj ≤N −1 butinsteadwe willtreatonlythe
casewhererank(Mj)=1 forallj.
Asinthe1-solitoncase,wecanexpressallMj intheformMj=pjqTj wherepj andqjare
N-vectors.Then,equations(4.58)implythat
qTjM(μj)−1=0, M(μ∗j)Qq∗j=0, j=1, . . . , n (4.59)
respectively.Astherelationsin(4.59)areequivalenttoeachother,usingoneofthemwehave that
n
i=1
qTi Qq∗j
μi−μ∗j
pi=Qq∗j, j=1, . . . , n (4.60)
withμi=μ∗jforalli,j.Wedefinethescalarquantities(qi,qj)= qTiQq∗j
μi−μ∗j andifweassumethat
theCauchytypematrix(qi,qj)isinvertible,thenusingtheCramer’srulewecansolve(4.60)
forallpi.Inthiswaythepi’scanbeexpressedintermsoftheqi’sasaratioofdeterminants
pi=
(q1,q1) · · · (q1,qi−1) Qq∗1 (q1,qi+1) · · · (q1,qn) ..
. . .. ... ... ... . .. ...
(qn,q1) · · · (qn,qi−1) Qq∗n (qn,qi+1) · · · (qn,qn)
(q1,q1) · · · (q1,qn) ..
. . .. ...
(qn,q1) · · · (qn,qn)
. (4.61)
Thesymbolicdeterminantinthenumeratorisexpandedwithrespecttothei-thcolumn. Fromtheregularpartatλ= ∞ofthedressingrelations(4.17)weobtain
U0=U0+
n
i=1
piqTi ,U1
, (4.62)
wherewehaveusedrelation(4.57),whilefromtheresidueatλ=μ∗j wehavethat
M(μ∗j)L(μj∗)QM†jQ=0, M(μ∗j)A(μ∗j)QM†jQ=0. (4.63)
Similartothesinglepolecase,usingrelations(4.59),equations(4.63)implythat
qj x−QU(μ∗j)∗Qqj=0, qj t−QV(μ∗j)∗Qqj=0, (4.64)
hencewecanwrite
qTj =CjTQ(μ∗j)†Q , (4.65)
wheretheCj areconstantN-vectorsand(μ∗j)isafundamentalsolutiontothelinearproblem
Usingequation(4.61),relation(4.62)isthedressingtransformationwhichcanbewrittenas
ui=ui−
i
√
κ τi
τ , i=1,· · ·,N−1, (4.66)
whereτ,τi standforthefollowingdeterminants
τ =
(q1,q1) · · · (q1,qn) ..
. . .. ...
(qn,q1) · · · (qn,qn)
, τi=
0 q1,i · · · qn,i κq∗1,N (q1,q1) · · · (q1,qn)
..
. ... . .. ...
κq∗n,N (qn,q1) · · · (qn,qn)
, (4.67)
withqk,m denotingthem-thcomponentoftheqkvector.Recently,brightanddarksoliton
solu-tionswereobtainedusingthedressingmethod,see[36,37].
5. IntegraloperatorsasglobalDarbouxtransformations&dressing
Weshallfocusonsituationswherethedressingisexpressedintermsofintegral representa-tions.Letusrecallthe“bare”differentialoperatorsassociatedtothevectorNLSmodel(seee.g.
[3,38]andreferencestherein):
D(1)
0 =M∂x, M=
i
αi e(iiN)
D(2)
0 =(ia ∂t−∂x2)1 (5.1)
whererecalle(ijN)areN ×N matriceswithelements(e(ijN))kl=δikδj l.
Letus brieflyrecalltheZakharov–Shabat(ZS) dressing[11,38],whichisequivalenttothe inversescatteringtransformaswellastheRiemann–Hilbertproblem,andleadstotheGelfand–
Levitan–Marchenko(GLM)equation(see[5,38,39]):
K(x, z)+F (x, z)+ ∞
x
K(x, y)F (y, z)dz=0, x < z. (5.2)
Thestartingpointoftheformulationisthefollowingfactorisation fortheoperator1+F,which holdsattheabstractoperatorlevel:
(1+K) (1+F)=(1+ ˆK), (5.3)
andwedefinetheintegralrepresentationsas:
F(f)= ∞
−∞
F (x, y)f(y)dy
K(f)= ∞
x
K(x, y)f(y)dy
ˆ K(f)=
x
−∞
ˆ
Inthematrixlanguagethefactorisation ofI+F correspondsessentiallytothedecomposition onupperandlowertriangularmatrices.NotealsothatF satisfiesthelinearequationsemanating fromthefollowinginvariantactions:
D(i)
0 F=F D
(i)
0 , i∈ {1, 2}. (5.5)
ThusviatheintegralrepresentationofFthefollowingequationsarise
ia Ft−Fxx+Fzz=0
M Fx+FzM=0 (5.6)
where
F (x, z;t )= N
j=2
fj(x, z;t ) e(1Nj )+ N
j=2
ˆ
fj(x, z;t ) e(jN1). (5.7)
Itisworthnotingthat amoregeneralchoiceof thesolutionsof thelinearproblemi.e.F
ex-pressedinthegenericGrassmannianform:
F=
0k×k Xk×N
YN×k 0N×N
(5.8)
willprovidesolutionstothematrixNLSmodel,howeverthisproblemwillbediscussedindetail elsewhere.
Weshallhenceforthfocusonsolutionsofthelinearequationsabove(5.7)thatarefactorisable i.e.,
fj(x, z;t )= n
α=1
X(α)j (x, t)Z(α)j (z)
ˆ
fj(x, z;t )= n
α=1
ˆ
X(α)j (x, t)Zˆ(α)j (z). (5.9)
It is clear that the ZS dressingmay be thought of as a global Darboux transformation; it is
essentiallyaDarbouxtransformationinanintegralrepresentation:
=B0,
=1+ ˆK, 0=1+F, B=1+K. (5.10)
Wecomenowtothemainobjective,whichisthesolutionoftheGLMequation(5.2)forthe
vectorNLS system.Given theformof the solutionF (5.7),andalsoconsideringthe generic
expressionK(x,z)=i,jKij(x,z)eij we enduptothe followingsetof equations(see also [40,38]forthesl2NLScase):
K1j(x, z)+fj(x, z)+
∞
x
K11(x, y)fj(y, z) dy=0
K11(x, z)+ ∞
x
j
Ki1(x, z)+ ˆfi(x, z)+
j
∞
x
Kij(x, y, t )fˆj(y, z) dy=0
Kij(x, z)+
∞
x
Ki1(x, y, t )fj(y, z) dy=0. j∈ {2, . . . ,N}. (5.12)
ThetwosetsofequationsabovecanbeindependentlysolvedtoprovideK1j, K11andKi1, Kij
respectively. Moreover,given the formof the dressedoperators it is clearthat K1j andKj1
providethefieldsuj−1andu∗j−1respectively,i.e. thecomponentsofu|and|u∗(seealso[41]).
Solvingthelattersystem(5.11)weobtain
K1j(x, z)+fj(x, z)−
∞
x
∞
x
dy dy˜ i
K1i(x,y)˜ fˆi(y, y)f˜ j(y, z)=0. (5.13)
Duetotheformofthelatterformulawecanconsiderthefollowingfactorisation ofthekernel
K1j
K1j(x, z;t )= n
α=1
L(α)j Zj(α)(z). (5.14)
Recallingtheformoffj, fˆj andafterintegrationweendupwiththefollowinggeneric
expres-sion:
i
β
L(β)i (x, t)Mβαij = −Xj(α)(x, t) (5.15)
wherewedefine:
Mβα
ij =δijδαβ−Piiβγ Pˆ γ α
ij (5.16)
and
Piiβγ(x, t)= ∞
x
dy Z(β)i (y)Xˆ(γ )i (y, t), Pˆijγ α(x, t)= ∞
x
dyZˆ(γ )i (y)Xj(α)(y, t). (5.17)
Noticethattheobviouschoice
X(α)j (x, t)=bjei (α) j t+iλ
(α)
j x, Z(α) j =e
iμ(α)j z
ˆ
X(α)j (x, t)= ˆbjeiˆ (α) j t+iλˆ
(α)
j x, Zˆ(α) j =e
iμˆ(α)j z
(5.18)
leadstosimpleexpressionsforP , Pˆ afterintegration(seealso[39]forrelevantexpressionsin thecontextoftheinversescatteringtransform):
Pjjβγ = − ˆb(γ )j e
iˆ(γ )j t+iλˆ(γ )j x+iμ(β)j x
i(λˆ(γ )j +μ(β)j )
ˆ
Pijγ α= −bj(α)e
i(α)j t+iλ(α)j x+iμˆ(γ )i x
i(λ(α)j + ˆμ(γ )i )
Equation(5.11)canbeexpressedinamorecompactformas:
L·M= −X⇒L= −X·M−1,
(5.20)
where
L=
α
j
L(α)j eˆj∗(N)⊗ ˆe∗α(n), M= α,β
i,j
Mαβ ij e
(N) ij ⊗e
(n) αβ,
X=
α
j
Xj(α)eˆj∗(N)⊗e∗α(n)
(5.21)
ande∗j(N) aretheN dimensionalcolumnvectorswith1atpositionj andzeroelsewhere.Our taskistoidentifyK1j,whichwillprovideinturnthefieldsuj−1;indeedK1j(x,x)∝uj−1(x) [39,41].
Similarly,wesolvethesystem(5.12)toidentifythequantitiesKi1,whichinturnprovidethe
fields u∗i−1.From the formof thesystemit iseasierfirsttosolve for Kij (i,j =1)andthen
obtainKi1,andhencetheu∗i−1fields(timedependenceisimplicitinalltheexpressionsbelow)
Kij(x, z)−
∞
x
ˆ
fi(x, y)fj(y, z)dy−
l
∞
x
∞
x
dy dy K˜ il(x,y)˜ fˆl(y, y)f˜ j(y, z). (5.22)
From the latter expression andthe chosen form of the solution of the linear system we can
considerthefollowingfactorised formforKij:
Kij(x, z)= n
α=1
L(α)ij (x, t) Z(α)j (z). (5.23)
Substitutingthelatterexpressionin(5.22)weobtainthefollowinglinearsystem:
l
β
L(β)il Mβαlj = β
ˆ
Xi(β)(x)Pˆijβα(x). (5.24)
Moreover,thequantitiesK11andKij maybealsoderivedvia(5.11),(5.12),henceweobtain:
K11(x, z)= −
j
β,γ
L(β)j (x)Pjjβγ(x)Zˆj(γ )(z)
Kij(x, z)= −
β
ˆ
X(β)i (x)Zˆ(β)i (z)− j
β,γ
L(β)j (x)Pjjβγ(x)Zˆ(γ )j (z). (5.25)
Itis easynow toextract for instancethe one-soliton solutiongiven thedescription above. Indeedexpressions(5.20),(5.24)still hold, butnow M isdefinedas (noGreekletterindices involvedanymoreasisnatural):
Mij=δij−PiiPˆij (5.26)
wherewedefine:
Pii(x, t)=
∞
x
dy Zi(y)Xˆi(y, t), Pˆij(x, t)=
∞
x
Forthesakeofsimplicityletusconsiderthecasewhereallthespectralparametersλj arethe
sameforallthefields,thenitisclearthat:
fj(x, z;t )=bjeit+iλx+iμz, fˆ= ˆbjeit+iλxˆ +iμzˆ (5.28)
bj,bˆjarethecomponentsoftheso-calledpolarisation vectors.Thenitisclearfromtheintegral
equationsK1j, Kj1:
K1j(x, z;t )=Lj(x, t )eiμz, Kj1(x, z;t )= ˆLj(x, t )eiμzˆ . (5.29)
OnecantheneasilyobtainasolutionforLˆj.Indeed,oneobtainsasimplescalarequation,which
immediatelyprovidesthesolution
ˆ
Lj(x, t )= −
ˆ
bjeitˆ +iλxˆ
1+C H(x, t ),
where C=
j
bjbˆj, H(x, t )=
ei(ˆ+)t+i(λ+μ+ˆλ+ ˆμ)x
(λ+ ˆμ)(λˆ+μ) . (5.30)
Similarly,theexpressionsforLjreduceintothesimpleformulasbelow:
L= −eit+iλxB M−1 (5.31)
wherewedefine:
L=
N
j=1
Ljeˆj∗(N), B= N
j=1
bjeˆj∗(N), M=1+P (5.32)
andPisexpressedasabi-vector
P=H(x, t )BˆTB (5.33)
andisalsoaprojector:
P2=
C H(x, t )P, (5.34)
whichleadstotheimmediateidentificationoftheinverseM−1
M−1=1− 1
1+C H(x, t )P. (5.35)
TheidentificationofLˆi isthenstraightforward
Lj(x, t )= −
bjeit+iλx
1+C H(x, t ), (5.36)
andclearlycompatiblewiththesolutionforLˆj(x,t ).
5.1. The continuous case
Itwill beinstructiveforthegeneralpurposesofstudyingsolutionsof integrablePDEs,but alsoinassociationwiththetimeevolutionofpoint-likedefectstoconsiderthecontinuumcase.
insteadofthe matrixformulationoneemploysinthiscaselinearintegralequationsas willbe evidentbelow.
Theessentialdifferencewiththediscretecasestudiedaboveisthatalldiscretesumsformally turnintointegralsi.e.
n
α=1 f(α)→
∞
−∞
dk f (k). (5.37)
Moreprecisely,consideringfactorised expressionsforthesolutionsofthelinearproblem
fj(x, z;t )=
∞
−∞
dk Xj(k;x, t) Zj(k;z)
ˆ
f (x, z;t )= ∞
−∞
dkXˆj(k;x, t)Zˆj(k;z) (5.38)
wethenobtainthecontinuumlimitforthefactorisation ofK1j:
K1j(x, z, t )=
∞
−∞
dk Lj(x, t;k)Zj(z;k). (5.39)
Thefundamentallinearequation(5.15)isthenwrittenas
i
∞
−∞
dk L˜ i(x, t; ˜k)Mij(x, t; ˜k, k)= −Xj(x, t;k), (5.40)
wherewedefine
Mij(k, k)˜ =δijδ(k, k)˜ −
dkPii(k, k˜ )Pˆij(k, k) (5.41)
andP , Pˆ arethendefinedasthecontinuumanaloguesof(5.19),i.e.
Pii(x, t;k,k)˜ =
ˆ
bj(k)e˜ i(ˆ k)t˜ +iλ(ˆk)x˜ +iμ(k)x i(λ(ˆ k)˜ +μ(k))
ˆ
Pij(x, t;k,k)˜ =
bj(k)e˜ i(k)t˜ +iλ(k)x˜ +iμ(k)xˆ
i(λ(k)˜ + ˆμ(k)) . (5.42)
Similarly,asinthediscretecasewecanobtainthefactorised form:
Kij(x, z)=
dk Lij(x, t;k)Zj(z;k) (5.43)
andtherespectivelinearequation
l
dk L˜ il(x, t; ˜k)Mlj(x, t; ˜k, k)=
theotherquantitiesarethenimmediatelydeducedvia(5.11),(5.12).Thusthefieldscanbe com-pletelyreconstructedfromtheknowledgeofthekernelKij.Similarexpressionsarethenobtained
forK11andKi1via(5.11),(5.12):
K11(x, z)= −
j
dk˜
dk Lj(x, t;k)Pjj(x, t;k,k)˜ Zˆj(z; ˜k)
Ki1(x, z)= −
dk˜Xˆi(x, t; ˜k)Zˆi(z; ˜k)
−
j
dk
dk L˜ ij(x, t;k) Pjj(x, t;k,k)˜ Zˆj(z; ˜k). (5.45)
LetusfinallydiscussinmoredetailthetimepartoftheBT.Asexplainedindetailearlierin thetext,aswellasinpreviousrelatedworkswearemostlyinterestedinthetimeevolutionof thedefect.InthepresentformulationthedefectdegreesoffreedomareencodedinK,therefore studyingthetimeevolutionofKisofgreatrelevanceinthiscontext.Thiswillnaturallyleadto BTtyperelationssimilartotheonesderivedintheprevioussectionsas willbecomeapparent
below.LetGbetheglobalDarbouxtransformationsuchthat:
ˆ
=G (5.46)
and,˜ satisfy:
ia ∂tˆ = ˆDˆ
ia ∂t=D (5.47)
where
D=D0+M, Dˆ= ˆD0+ ˆM. (5.48)
From the latter equations immediately follows the typical time part of a Darboux–Bäcklund
transformation
∂tG= ˆDG−GD. (5.49)
Takingintoaccount(5.48),(5.49)andsettingG=1+K,weobtainthefollowingglobal expres-sion:
ia ∂tK= ˆMK−KM+D0K−K D0+ ˆM−M (5.50)
M, Mˆ areN ×N matrices,andD0=1∂x2.Theintegralrepresentationoftheexpressionabove
becomes(KarealsoN ×N matrices)
ia x
−∞
dy ∂tK(x, y)f(y)= x
−∞
dy M(x)K(x, y)f(y)− x
−∞
dy K(x, y)M(y)f(y)
+
x
−∞ dy
∂x2K(x, y)−∂y2K(x, y)
f(y)
+2∂xK(x, x)f(x)+(M(x)ˆ −M(x))f(x) (5.51)
ia ∂tK(x, y)=∂x2K(x, y)−∂
2
yK(x, y)+ ˆM(x) K(x, y)−K(x, y) M(y)
2∂xK(x, x)=M(x)− ˆM(x). (5.52)
Onecanofcoursestartthe“dressing”processwithtrivialsolutionsi.e.M=0 asdescribed intheprevioussubsection.Butingeneralthetimeevolution(5.52)describestheconnection be-tweentwodifferentsolutionsofthesamenon-lineardifferentialequation.Withthisweconclude
ouranalysisontheDarbouxtransformsanddressingforthevectorNLSmodel.
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