Weakly nonassociative algebras, Riccati and KP hierarchies

0.17 MB·English

by  Aristophanes Dimakis

Weakly nonassociative algebras, Riccati and KP hierarchies
AristophanesDimakisa and FolkertMu¨ller-Hoissenb
a DepartmentofFinancial andManagement Engineering,UniversityoftheAegean
31FostiniStr., GR-82100Chios, Greece
b Max-Planck-InstituteforDynamicsand Self-Organization
Bunsenstrasse10, D-37073Go¨ttingen,Germany
E-mails: [email protected],[email protected]
Abstract
8
0
Ithasrecentlybeenobservedthatcertainnonassociativealgebras(called‘weaklynonassociative’,
0
WNA) determine, via a universal hierarchy of ordinary differential equations, solutions of the KP
2
hierarchywithdependentvariableinanassociativesubalgebra(themiddlenucleus).Werecallcentral
n
resultsandconsideraclassofWNA algebrasforwhichthehierarchyofODEsreducestoamatrix
a
J Riccati hierarchy, which can be easily solved. The resulting solutions of a matrix KP hierarchy
determine, under a ‘rank one condition’, solutions of the scalar KP hierarchy. We extend these
6
1 resultstothediscreteKPhierarchy. Moreover,webuildabridgefromtheWNA frameworktothe
Gelfand-DickeyformulationoftheKPhierarchy.

]
I
S
. 1 Introduction
n
i
l
n TheKadomtsev-Petviashvili(KP)equationisanextensionofthefamousKorteweg-deVries(KdV)equa-
[ tion to 2+1 dimensions. It first appeared in a stability analysis of KdV solitons [1,2]. In particular, it

4 describesnonlinearfluidsurfacewavesinacertainapproximationandexplainstosomeextenttheforma-
v tionofnetworkpatternsformedbylinewavesegmentsonawatersurface[2]. Itis‘integrable’inseveral
0
respects, inparticular inthe sense ofthe inverse scattering method. Various remarkable properties have
1
0 been discovered that allow to access (subsets of) its solutions in different ways, see in particular [3–5].
1
Apart from its direct relevance in physics, the KP equation and its hierarchy (see [5,6], for example)
0
7 is deeply related to the theory of Riemann surfaces (Riemann-Schottky problem, see [7] for a review).
0 Some time ago, this stimulated discussions concerning the role of KP in string theory (see [8–11], for
/
n example). Laterthe Gelfand-Dickey hierarchies, of which theKdV hierarchy is the simplest and which
i are reductions of the KP hierarchy, made their appearance in matrix models, first in a model of two-
l
n
dimensional quantum gravity (see [12,13] and references therein). This led to important developments
:
v in algebraic geometry (see [14], for example). Of course, what we mentioned here by far does not ex-
i
X haust what is known about KP and there is probably even much more in the world of mathematics and
r physicslinkedtotheKPequation anditsdescendants thatstillwaitstobeuncovered.
a
In fact, an apparently completely different appearance of the KP hierarchy has been observed in
[15]. On a freely generated ‘weakly nonassociative’ (WNA) algebra (see section 2) there is a family
of commuting derivations1 that satisfy identities which are in correspondence with the equations of the
KPhierarchy(withdependent variableinanoncommutative associative subalgebra). Asaconsequence,
there is ahierarchy of ordinary differential equations (ODEs)on this WNAalgebra that implies the KP
hierarchy. Moregenerally,thisholdsforanyWNAalgebra. InthiswayWNAalgebrasdetermineclasses
ofsolutions oftheKPhierarchy.
1Familiesofcommutingderivationsoncertainalgebrasalsoappearedin[16,17],forexample.Infact,theideasunderlying
theworkin[15]grewoutofourworkin[18]whichhassomealgebraicoverlapwith[16].
1
In section 2 we recall central results of [15] and present a new result in proposition 1. Section 3
applies theWNAapproach toderiveamatrixRiccati2 hierarchy, thesolutions ofwhicharesolutions of
thecorrespondingmatrixKPhierarchy(whichundercertainconditionsdeterminessolutionsofthescalar
KPhierarchy). Insection4weextendtheseresultstothediscreteKPhierarchy[35–39]. Furthermore,in
section5weshowhowtheGelfand-Dickeyformulation[5]oftheKPhierarchy(withdependentvariable
inanyassociative algebra) emergesintheWNAframework. Section6containssomeconclusions.
2 Nonassociativity and KP
In[15]wecalledanalgebra(A,◦)(overacommutativering)weaklynonassociative (WNA)if
(a,b◦c,d) =0 ∀a,b,c,d ∈ A, (2.1)
where(a,b,c) := (a◦b)◦c−a◦(b◦c)istheassociatorinA. ThemiddlenucleusofA(seee.g.[40]),
A′ := {b ∈ A| (a,b,c) = 0 ∀a,c ∈A}, (2.2)
isanassociative subalgebra andatwo-sided ideal. Wefixf ∈ A,f 6∈ A′,anddefinea◦ b := a◦b,
1
a◦ b := a◦(f ◦ b)−(a◦f)◦ b, n = 1,2,... . (2.3)
n+1 n n
As a consequence of (2.1), these products only depend on the equivalence class [f] of f in A/A′. The
subalgebra A(f),generated byf intheWNAalgebra A,iscalledδ-compatible if,foreachn ∈N,
δ (f):= f ◦ f (2.4)
n n
extendstoaderivation ofA(f). Inthefollowingwerecallsomeresultsfrom[15].
Theorem1 Let A(f)be δ-compatible. Thederivations δ commute on A(f)and satisfy identities that
n
are incorrespondence via δ 7→ ∂ (the partial derivative operator with respect to avariable t )with
n tn n
theequations ofthepotential Kadomtsev-Petviashvili (pKP)hierarchywithdependent variable inA′. (cid:3)
Thisisacentralobservation in[15]withthefollowingimmediateconsequence.
Theorem2 Let A be any WNA algebra over the ring of complex functions of independent variables
t ,t ,.... Iff ∈ AsolvesthehierarchyofODEs3
1 2
f := ∂ (f)= f ◦ f, n = 1,2,... , (2.5)
tn tn n
then−f liesinA′ andsolvestheKPhierarchywithdependent variable inA′. (cid:3)
t1
Corollary 1 Ifthere is a constant ν ∈ A, ν 6∈ A′, with [ν] = [f] ∈ A/A′, then, under the assumptions
oftheorem 2,
φ := ν −f ∈ A′ (2.6)
2Besides their appearance in control and systemstheory, matrixRiccati equations (see[19–21], for example) frequently
showedupinthecontextofintegrablesystems,seeinparticular[22–34].
3f hastobedifferentiable,ofcourse,whichrequiresacorresponding(e.g. Banachspace)structureonA. Theflowsgiven
by(2.5)indeedcommute[15].Furthermore,(2.5)impliesδ-compatibilityofthealgebraA(f)generatedbyfinAoverC[15].
2
solvesthepotential KP(pKP)hierarchy4
3
ε λ−1(φ −φ)+φ◦φ = 0, (2.7)
ijk i [λi] [λi] [λk]
i,Xj,k=1 (cid:0) (cid:1)
where ε is totally antisymmetric with ε = 1, λ , i = 1,2,3, are indeterminates, and φ (t) :=
ijk 123 i ±[λ]
φ(t±[λ]),wheret = (t ,t ,...)and[λ] := (λ,λ2/2,λ3/3,...). (cid:3)
1 2
Remark 1. If C ∈ A′ is constant, then f = ν′ − (φ + C) with constant ν′ := ν + C satisfying
[ν′] = [ν] = [f]. Hence,withφalsoφ+C isasolution ofthepKPhierarchy. Thiscanalsobechecked
directlyusing(2.7),ofcourse. (cid:3)
Thenextresultwillbeusedinsection 4.
Proposition 1 Supposef andf′ solve(2.5)and[f]= [f′]inaWNAalgebraA. Theequation
f′◦f = α(f′−f) (2.8)
isthenpreservedforallα ∈ C.
Proof: (f′◦f) = f′ ◦f +f′◦f = (f′◦ f′)◦f +f′◦(f ◦ f)
tn tn tn n n
= (f′◦ f′)◦f −f′◦ (f′◦f)+αf′◦ (f′−f)
n n n
+f′◦(f ◦ f)−(f′◦f)◦ f +α(f′−f)◦ f
n n n
= −f′◦ f +f′◦ f +α(f′◦ f′−f ◦ f)= α(f′−f) .
n+1 n+1 n n tn
Inthethirdstepwehaveaddedtermsthatvanishasaconsequence of(2.8). Thenweused(3.11)in[15]
(togetherwiththefactthattheproducts◦ onlydependontheequivalenceclass[f]= [f′]∈ A/A′),and
n
also(2.3),tocombinepairsoftermsintoproducts ofonedegreehigher. (cid:3)
Remark2. Infunctional form,(2.5)canbeexpressed (e.g. withthehelpofresultsin[15])as
λ−1(f −f )−f ◦f = 0. (2.9)
−[λ] −[λ]
Settingf′ = f (whichalsosolves(2.9)iff solvesit),thistakestheform(2.8)withα = −λ−1. (cid:3)
−[λ]
In order to apply the above results, we need examples of WNA algebras. For our purposes, it is
sufficienttorecallfrom[15]thatanyWNAalgebrawithdim(A/A′) =1isisomorphictoonedetermined
bythefollowingdata:
(1)anassociative algebra A(e.g. anymatrixalgebra)
(2)afixedelementg ∈ A
(3)linearmapsL,R : A → Asuchthat
[L,R] = 0, L(a◦b) =L(a)◦b, R(a◦b) = a◦R(b) . (2.10)
AugmentingAwithanelementf suchthat
f ◦f := g, f ◦a:= L(a), a◦f := R(a), (2.11)
leadstoaWNAalgebraAwithA′ = A,providedthatthefollowingcondition holds,
∃a,b∈ A : R(a)◦b 6= a◦L(b). (2.12)
Thisguarantees thattheaugmentedalgebraisnotassociative. ParticularexamplesofLandRaregiven
bymultiplication fromleft,respectively right,byfixedelementsofA(seealsothenextsection).
4ThisfunctionalrepresentationofthepotentialKPhierarchyappearedin[41,42].Seealso[15,26]forequivalentformulae.
3
3 A class of WNA algebras and a matrix Riccati hierarchy
Let M(M,N) be the vector space of complex M ×N matrices, depending smoothly on independent
realvariablest ,t ,...,andletS,L,R,QbeconstantmatricesofdimensionsM ×N,M ×M,N ×N
1 2
andN ×M,respectively. Augmenting withaconstant elementν andsetting5
ν ◦ν = −S, ν ◦A= LA, A◦ν = −AR, A◦B = AQB, (3.1)
forallA,B ∈ M(M,N),weobtainaWNAalgebra(A,◦). Thecondition (2.12)requires
RQ 6= QL . (3.2)
Fortheproducts ◦ ,n > 1,wehavethefollowingresult.
n
Proposition 2
ν ◦ ν =−S , ν ◦ A= L A, A◦ ν =−AR , A◦ B =AQ B, (3.3)
n n n n n n n n
where
R Q R Q
n n = Hn with H := . (3.4)
(cid:18) Sn Ln (cid:19) (cid:18) S L (cid:19)
Proof: Usingthedefinition(2.3),oneprovesbyinduction that
S = LS +SR , L = LL +SQ , R = QS +RR , Q = QL +RQ ,
n+1 n n n+1 n n n+1 n n n+1 n n
forn = 1,2,...,whereS = S,L = L,R = R,Q = Q. Thiscanbewrittenas
1 1 1 1
R Q R Q
n+1 n+1 = H n n ,
(cid:18) Sn+1 Ln+1 (cid:19) (cid:18) Sn Ln (cid:19)
whichimplies(3.4). (cid:3)
Using(2.6)and(3.3)in(2.5),leadstothematrixRiccatiequations6
φ = S +L φ−φR −φQ φ, n = 1,2,... . (3.5)
tn n n n n
Solutionsof(3.5)areobtained inawell-knownway(see[21,32],forexample)via
φ = YX−1 (3.6)
fromthelinearsystem
X
Z = HnZ, Z = (3.7)
tn (cid:18) Y (cid:19)
withanN ×N matrixX andanM ×N matrixY,providedX isinvertible. Thissystemissolvedby
Z(t)= eξ(H)Z where ξ(H) := t Hn . (3.8)
0 n
nX≥1
5Using(2.6),intermsoff thisyieldsrelationsoftheform(2.11).
6Thecorrespondingfunctionalformisλ−1(φ −φ)+φQφ =S+Lφ −φR,whichiseasilyseentoimply(2.7),
[λ] [λ] [λ]
seealso[43].TheappendixprovidesaFORMprogram[44,45]whichindependentlyverifiesthatanysolutionof(3.5),reduced
ton=1,2,3,indeedsolvesthematrixpKPequationin(M(M,N),◦).
4
IfQhasrank1,then
ϕ := tr(Qφ) (3.9)
defines ahomomorphism from (M(M,N),◦) into thescalars (with the ordinary product offunctions).
Hence, ifφsolves the pKPhierarchy in(M(M,N),◦), then ϕ solves the scalar pKPhierarchy.7 More
generally, if Q = VUT with V,U of dimensions N × r, respectively M × r, then UTφV solves the
r×r-matrixKPhierarchy.
GL(N+M,C)actsonthespaceofall(N+M)×(N+M)matricesH bysimilaritytransformations.
InagivenorbitthisallowstochooseforH some‘normalform’,forwhichwecanevaluate(3.8)andthen
elaborate theeffect ofGL(N +M,C)transformations (see also remark3below) onthecorresponding
solution of the pKP hierarchy, with the respective Q given by the normal form of H. By a similarity
transformation we can always achieve that Q = 0 and the problem of solving the pKP hierarchy (with
somenon-zeroQ)canthusinprinciplebereducedtosolvingitslinearpart. Alternatively,wecanalways
achievethatS = 0andthenexttwoexamplestakethisroute.
Example1. IfS = 0,wecaningeneralnotachievethatalsoQ = 0. Infact,thematrices
R Q R 0
H = and H := (3.10)
(cid:18) 0 L (cid:19) 0 (cid:18) 0 L (cid:19)
aresimilar(i.e. relatedbyasimilaritytransformation) ifandonlyifthematrixequationQ = RK−KL
hasanN ×M matrixsolution K [56–60],andthen
I −K
H = T H T−1, T = N . (3.11)
0 (cid:18) 0 IM (cid:19)
Itfollowsthat
Rn RnK −KLn
Hn =T HnT−1 = (3.12)
0 (cid:18) 0 Ln (cid:19)
andthus
eξ(R) eξ(R)K −Keξ(L)
eξ(H) = . (3.13)
(cid:18) 0 eξ(L) (cid:19)
If(3.2)holds, weobtainthefollowingsolution ofthematrixpKPhierarchyin(M(M,N),◦),
φ= eξ(L)φ (I +Kφ −e−ξ(R)Keξ(L)φ )−1e−ξ(R), (3.14)
0 N 0 0
whereφ = Y X−1. Thisinturnleadsto
0 0 0
ϕ = tr e−ξ(R)(RK −KL)eξ(L)φ (I +Kφ −e−ξ(R)Keξ(L)φ )−1
0 N 0 0
(cid:16) (cid:17)
= tr log(I +Kφ −e−ξ(R)Keξ(L)φ )
N 0 0
(cid:16) (cid:17)t1
= (logτ) , τ := det(I +Kφ −e−ξ(R)Keξ(L)φ ). (3.15)
t1 N 0 0
If rank(Q) = 1, then ϕ solves the scalar pKP hierarchy. Besides (3.2) and this rank condition, further
conditionswillhavetobeimposedonthe(otherwisearbitrary)matricesR,K,Landφ toachievethatϕ
0
7Forrelatedresultsandotherperspectivesontherankonecondition, see[46]andthereferencescitedthere. Theideato
lookfor(simple)solutionsofmatrixandmoregenerallyoperatorversionsofan‘integrable’equation,andtogeneratefromit
(complicated)solutionsofthescalarequationbyuseofasuitablemap,alreadyappearedin[47](seealso[48–55]).
5
isarealandregular solution. See[61],andreferencescitedthere,forclassesofsolutionsobtainedfrom
an equivalent formula or restrictions of it. This includes multi-solitons and soliton resonances (KP-II),
andlumpsolutions (passingtoKP-Iviat 7→ it andperforming suitablelimitsofparameters).
2n 2n
Example2. IfM = N andS = 0,letusconsider
L I I −K
H = T H T−1, H = , T = , (3.16)
0 0 (cid:18) 0 L (cid:19) (cid:18) 0 I (cid:19)
withI = I andaconstantN ×N matrixK. Asaconsequence,
N
Q = I +[L,K] . (3.17)
WenotethatH isnot similartodiag(L,L)[60]. Nowweobtain
0
Ln nLn−1+[Ln,K]
Hn = T HnT−1 = (3.18)
0 (cid:18) 0 Ln (cid:19)
andfurthermore
eξ(L) nt Ln−1eξ(L)+[eξ(L),K]
eξ(H) = n≥1 n . (3.19)
(cid:18) 0 P eξ(L) (cid:19)
If[L,[L,K]] 6= 0(whichiscondition (3.2)),weobtainthesolution
φ= eξ(L)φ (I +Kφ +F)−1e−ξ(L) (3.20)
0 0
ofthematrixpKPhierarchyin(M(N,N),◦), where
F := nt Ln−1−e−ξ(L)Keξ(L) φ . (3.21)
n 0
(cid:16)nX≥1 (cid:17)
Furthermore, usingF = e−ξ(L)(I +[L,K])eξ(L)φ ,wefind
t1 0
ϕ= tr((I +[L,K])φ) = tr(F (I +Kφ +F)−1) = (trlog(I +Kφ +F)) (3.22)
t1 0 0 t1
andthus
ϕ =(logτ) , τ := det I +Kφ +( nt Ln−1−e−ξ(L)Keξ(L))φ . (3.23)
t1 0 n 0
(cid:16) nX≥1 (cid:17)
Ifrank(I+[L,K]) = 1(seealso[46,62,63]forappearances ofthiscondition), thenϕsolvesthescalar
pKPhierarchy. Assumingthatφ isinvertible, wecanrewriteτ asfollows,
0
τ = det eξ(L)(φ−1+K)e−ξ(L)+ nt Ln−1−K (3.24)
0 n
(cid:16) nX≥1 (cid:17)
(dropping afactordet(φ )). Thissimplifiesconsiderably ifwesetφ−1 = −K.8 Choosingmoreover
0 0
L = −(q −q )−1 i 6= j, L = −p , K = diag(q ,...,q ), (3.25)
ij i j ii i 1 N
(3.24) reproduces a polynomial (in any finite number of the t ) tau function associated with Calogero-
n
Mosersystems[46,62,63]. Alternatively, wemaychoose
L = diag(q ,...,q ), K = (q −q )−1 i 6= j, K = p . (3.26)
1 N ij i j ii i
8Notethatinthiscaseφ=(Pn≥1ntnLn−1−K)−1,whichisrationalinanyfinitenumberofthevariablestn.
6
The corresponding solutions of the KP-I hierarchy (t 7→ it ) include the rational soliton solutions
2n 2n
(‘lumps’)originally obtained in[64]. Inparticular, N = 2andq =−q¯ ,p = p¯ (wherethebarmeans
2 1 2 1
complexconjugation), yieldsthesinglelumpsolution givenby
1
τ = |p +ξ′(q )|2+ where ξ′(q) := nt qn−1 . (3.27)
1 1 4ℜ(q1)2 nX≥1 n (cid:12)(cid:12){t2k7→it2k,k=1,2,...}
(cid:12)
Example 3. Let M = N and L = Sπ , R = π S, Q = π Sπ , with constant N × N matrices
− + + −
π ,π subjecttoπ +π = I. ThematrixH canthenbewrittenas
+ − + −
π
H = + S I π , (3.28)
(cid:18) I (cid:19) −
(cid:0) (cid:1)
whichletsuseasilycalculate
π Sn π Snπ
Hn = + + − . (3.29)
(cid:18) Sn Snπ− (cid:19)
Asaconsequence, weobtain
φ= (−C +eξ(S)C )(π C +π eξ(S)C )−1, (3.30)
+ − − + + −
whereC := I ∓π φ . ThissolvesthematrixpKPhierarchy inM(M,N)withtheproduct A◦B =
± ± 0
Aπ Sπ B if(3.2)holds, whichisπ S(π −π )Sπ 6=0. Iffurthermore rank(π Sπ )= 1,then
+ − + + − − + −
ϕ = tr(Qφ) = −tr(π S)+(logτ) , τ = det(π C +π eξ(S)C ) (3.31)
+ t1 − + + −
solvesthescalarpKPhierarchy(seealso[43]). Wewillmeetthebasicstructureunderlyingthisexample
againinsection5.
Remark3. AGL(N +M,C)matrix
A B
T = (3.32)
(cid:18) C D (cid:19)
canbedecomposed asfollows,
I BD−1 S 0 I 0
T = N D N , (3.33)
(cid:18) 0 IM (cid:19)(cid:18) 0 D (cid:19)(cid:18) D−1C IM (cid:19)
ifD and itsSchurcomplement S = A−BD−1C areboth invertible. Letusseewhateffect thethree
D
partsofT induceonφwhenactingonZ.
(1)WritingP =D−1C,thefirsttransformation leadstoφ 7→ φ+P,ashiftbytheconstantmatrixP.
(2) The second transformation amounts to φ 7→ DφS−1 (where φ is now the result of the previous
D
transformation).
(3)WritingK = −BD−1,thelasttransformation isφ 7→ φ(I −Kφ)−1. (cid:3)
N
4 WNA algebras and solutions of the discrete KP hierarchy
The potential discrete KP (pDKP) hierarchy in an associative algebra (A,◦) can be expressed in func-
tionalformasfollows,9
Ω(λ)+ −Ω(λ) = Ω(µ)+−Ω(µ) , (4.1)
−[µ] −[λ]
9ThisfunctionalrepresentationofthepDKPhierarchyisequivalentto(3.32)in[39].
7
whereλ,µareindeterminates,
Ω(λ) := λ−1(φ−φ )−(φ+ −φ )◦φ, (4.2)
−[λ] −[λ]
and φ = (φk)k∈Z, φ+k := φk+1. ThepDKP hierarchy implies that each component φk, k ∈ Z, satisfies
the pKP hierarchy and its remaining content is a special pKP Ba¨cklund transformation (BT) acting be-
tween neighbouring sites on the linear lattice labeled by k [35,39]. This suggests a way to extend the
method of section 3 to construct exact solutions of the pDKP hierarchy. What is needed is a suitable
extension of(2.5)thataccounts fortheBTandthisisofferedbyproposition 1.
Theorem3 LetAbeaWNAalgebrawithaconstant elementν ∈ A,ν 6∈A′. Anysolution
f = (ν −φk)k∈Z, (4.3)
ofthehierarchy(2.5)together withthecompatible constraint10
f+◦f = 0 (4.4)
yieldsasolution φ= (φk)k∈Z ofthepDKPhierarchyinA′.
Proof: Since[f+] = [f],thecompatibilityfollowsbysettingf′ = f+andα = 0inproposition1. Using
f = f ◦f,werewrite(2.9)as
t1
λ−1(f −f )+(f −f )◦f −f = 0.
−[λ] −[λ] t1
Inserting f = ν −φ,thistakestheform
λ−1(φ−φ )−φ −(φ−φ )◦φ = θ−θ
−[λ] t1 −[λ] −[λ]
withθ := −φ◦ν. Nextweuse(4.4)andf = f ◦f toobtain(f+−f)◦f +f = 0,whichis
t1 t1
φ −(φ+ −φ)◦φ= θ+−θ .
t1
Togetherwiththepreviousequation, thisleadsto
λ−1(φ−φ )−(φ+ −φ )◦φ= θ+−θ
−[λ] −[λ] −[λ]
(whichisactuallyequivalent tothelasttwoequations), sothat
Ω(λ) = θ+−θ .
−[λ]
Thisiseasilyseentosolve(4.1). (cid:3)
Let us choose the WNA algebra of section 3.11 Evaluation of (2.5) leads to the matrix Riccati
hierarchy (3.5),and(4.4)withf+ = ν +C −φ+ becomes
S +CR+(L+CQ)φ−φ+R−φ+Qφ= 0, (4.5)
whichcanberewrittenas
φ+ = (S +Lφ)(R+Qφ)−1+C = Y+(X+)−1 (4.6)
10 Note that (4.4) implies fn+ ◦n f = 0, where fkn+ := fk+n. This follows by induction from f(n+1)+ ◦n+1 f =
f(n+1)+ ◦(fn+ ◦n f)−(f(n+1)+ ◦fn+)◦n f = f(n+1)+ ◦(fn+ ◦n f)−(f+ ◦f)n+ ◦n f, whereweused(2.3) and
[fn+]=[f]inthefirststep.
11Sincethereisonlyasingleelementν,thematricesL,R,Sdonotdependonthediscretevariablek.
8
(assumingthattheinverse matricesexist),whereX+,Y+ arethecomponents of
Z+ = THZ = THeξ(H)Z(0), (4.7)
with Z,H,T taken from section 3. Deviating from the notation of section 3, we write Z(0) for the
constant vector, since Z should now denote thecomponent ofZ atthelattice site 0. Inorder that(4.7)
0
defines a pDKP solution on the whole lattice, we need H invertible. Since the matrix C, and thus also
T,maydepend onthelatticesitek,solutions of(4.1)aredetermined by
Z = T HT H···T HZ , Z = (T H)−1(T H)−1···(T H)−1Z , k ∈ N.(4.8)
k k k−1 1 0 −k −k −k+1 −1 0
Thiscorresponds toasequence oftransformations applied tothematrixpKPsolution φ determined by
0
Z ,whichgeneratenewpKPsolutions (cf.[35]). φ isthengivenby(4.6)intermsofφ ,and
0 1 0
φ = [LS +SR+LC R+(L2+SQ+LC Q)φ ]
2 1 1 0
×[R2+QS +QC R+(QL+RQ+QC Q)φ ]−1+C (4.9)
1 1 0 2
shows that the action of the T becomes considerably more involved for k > 1. In the special case
k
T = I (sothatC = 0),wehave
k N+M k
Z = eξ(H)(HkZ(0)) k ∈ Z . (4.10)
k 0
If X(0),Y(0) are the components of the vector HkZ(0), the lattice component φ of the pDKP solution
k k 0 k
determinedinthiswayisthereforejustgivenbythepKPsolutionofsection3withinitialdata(att = 0)
φ(0) = Y(0)(X(0))−1 = Lkφ(0)[Rk +(RkK −KLk)φ(0)]−1 . (4.11)
k k k 0 0
With the restrictions of example 1 in section 3, assuming that L and R are invertible (so that H is
invertible), thecorresponding solutionofthematrixpDKPhierarchy(inthematrixalgebrawithproduct
A◦B = A(RK −KL)B)is
φ = eξ(L)Lkφ(0)[Rk(I +Kφ(0))−e−ξ(R)Keξ(L)Lkφ(0)]−1e−ξ(R), k ∈ Z, (4.12)
k 0 N 0 0
whichleadsto
ϕ = (logτ ) with τ = det Rk(I +Kφ(0))−e−ξ(R)Keξ(L)Lkφ(0) k ∈ Z . (4.13)
k k t1 k N 0 0
(cid:16) (cid:17)
IfQ = RK −KLhasrank1,thisisasolution ofthescalarpDKPhierarchy.12 Asaspecialcase,letus
chooseM = N,L = diag(p ,...,p ),R = diag(q ,...,q ),andKwithentriesK = (q −p )−1.13
1 N 1 N ij i j
ThenQhasrank1andweobtainN-soliton taufunctions ofthescalardiscreteKPhierarchy. Thesecan
alsobeobtained viatheBirkhoffdecomposition methodusingappropriate initialdataasin[65,66].
With the assumptions made in example 2 of section 3, setting φ(0) = −K−1, assuming that K and
0
Lareinvertible, andchoosing forT theidentity, wefindthematrixpDKPsolution
k
−1
φ = nt Ln−1+kL−1−K , k ∈ Z . (4.14)
k n
(cid:16)nX≥1 (cid:17)
Ifrank(I +[L,K]) = 1,thisleadstothefollowingsolution ofthescalarpDKPhierarchy,
N
ϕ = (logτ ) with τ = det nt Ln−1+kL−1−K . (4.15)
k k t1 k n
(cid:16)nX≥1 (cid:17)
Inexample3ofsection3,H isnotinvertible, sothat(4.7)doesnotdetermine apDKPsolution.
12Recallthatϕ=tr(Qφ)(cf. 3.9)determinesahomomorphismifQhasrank1. Asaconsequence,ifφsolvesthematrix
pDKPhierarchy(4.1),thenϕsolvesthescalarpDKPhierarchy.
13Thecondition(3.2)requiresqi 6=pjforalli,j =1,...,N.
9
5 From WNA to Gelfand-Dickey
LetRbethecomplexalgebra ofpseudo-differential operators [5]
V = v ∂i, (5.1)
i
iX≪∞
with coefficients v ∈ A, where A is the complex differential algebra of polynomials in (in general
i
(m) (m) (m+1)
noncommuting) symbols u , m = 0,1,2,..., n = 2,3,..., where ∂(u ) = u and ∂(vw) =
n n n
∂(v)w +v∂(w) for v,w ∈ A. We demand that u(m), n = 2,3,..., m = 0,1,2,..., are algebraically
n
independent inA,andweintroduce thefollowinglinearoperators onR,
S(V) := LV, π (V) := V , π (V) := V := V −V , (5.2)
+ ≥0 − <0 ≥0
whereV istheprojection ofapseudo-differential operatorV toitsdifferential operatorpart,and
≥0
L = ∂ +u ∂−1+u ∂−2+··· . (5.3)
2 3
LetI denotetheidentityofR(whichweidentifywiththeidentityinA),andletObethesubspaceof
linear operators onRspanned byS and elements ofthe form Sπ Sπ ···π S (withany combination
± ± ±
ofsigns). Obecomesanalgebrawiththeproductgivenby
A◦B := Aπ Sπ B . (5.4)
+ −
(O,◦) isthen generated by the elements (Sπ )mS(π S)n, m,n = 0,1,.... Letus furthermore intro-
− +
duceA := {v ∈ A : v = res(A(I)), A ∈ O},whererestakes theresidue (thecoefficient of∂−1)ofa
pseudo-differential operator. Thisisasubalgebra ofA,sinceforA,B ∈ O wehave
res(A(I))res(B(I)) = res(Aπ Sπ B(I)), (5.5)
+ −
sothattheproduct ofelements ofAisagaininA. Asaconsequence ofthisrelation (read fromright to
left),Aisgeneratedbytheelementsres((Sπ )mS(π S)n(I)),m,n = 0,1,.... Basedonthefollowing
− +
preparations, wewillarguethatAand(O,◦)areactually isomorphicalgebras.
Lemma1 ForallV ∈ R,
res((Sπ )mV) = res(D V), m = 0,1,... , (5.6)
− m
whereD = Iand{D }∞ arethedifferentialoperatorsrecursivelydeterminedbyD = (D L) .
0 m m=1 m m−1 ≥0
Proof: Wedothecalculation form = 2. Thisiseasilygeneralized toarbitrary m ∈ N.
res((Sπ−)2V) = res(L(LV<0)<0) = res(L≥0LV<0)= res((L≥0L)≥0V) = res(D2V). (cid:3)
Proposition 3
m
m
res((Sπ )mS(π S)n(I)) = u(k) +termsnonlinear inu(j), m,n = 0,1,... (5.7)
− + (cid:18)k(cid:19) m+n+2−k k
Xk=0
10