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Darboux transformations for multidimensional quadrilateral lattices. I
21 July 1997
PHYSICS LETTERS A
physics Letters A 232 (1997) 99-105
Darboux transformations for multidimensional quadrilateral lattices. I Pa010 Maria Santini bA3 Manuel Ma&s a*l, Adam Doliwa bvc*2, a ~e~ata de Fisicu Te&ica, Universe Contpktense,E-28040 Madrid, @ain b IstitutoNa.onak di EMca Nuckare, Sezionedi Roma,P-k Aldo More 2. I-00185 Rotne,Italy c InstytutFL&i Teontycznej,UniwersytetWarszawski, rL &!a 69,lW681 Warsaw,Mand d Dijxim’mentoa?’Fisica, Universirddi Catania,Corso Italia57, I-95129 Gxtania,Italy Received 24 January 1997; acceptedfor publication 23 April 1997 Communicatedby A.P. Fordy
Abstract The vectorial Datboux transformationfor the multidimensionalquadrilaterallattice equations is constructed. Its particular reduction to the case of trivial background gives Grannn, Wronski and Casorati type ~p~~n~tions for the solutions. Some examples of the quadrilateral lattices are constructed explicitly. @ 1997 Elsevier Science B.V.
1, In&oduetion In a recent paper it was shown that rn~~~~nsion~ quadrilateral lattices (MQLs) are integrable [ 11.By a MQL we mean an N-dimensional (N 2 2) lattice; i.e., a mapping x : ZN --+ W”, M 3 N, labelled by the multi-index n := {nr,. . . , nN} E EN, .%&fying the property that each elementary quadrilater of it is planar. This condition is equivalent to the following discrete analogue of the Laplace equation, didjx f (I;:Aij)& i#j,
i,j=l,...,
=f(nl,
..,, nj+l,...,?&)
and Aj = Tj - 1 is the corresponding difference operator. For N = 2, any choice of the two functions Ar2, A21 gives rise to a possible two-dimensional qnadrilateral lattice. In order to build a MQL one h&sto repeat this ~ns~ction in every pair of dire&ions and, to make it possible, the data Aij must satisfy the following nonlinear constraints,
+ (TjAji)Ai~, N,
(1)
where x E IP is m arbitrary point of the lattice, Aij are N( N - 1) real functions of n, Tj is the translation operator in the jth variable,
AkAij = (T’Ajk)Aij + (%Akj)Aik - (TkAij)Aik+ ifj#kfi,
m
which chamctexize completely all the MQLs. It is convenient to reformnlate Bqs. ( 1) as tirst order systems [ 11. In order to do that, we introduce new functions Hi, i = 1,...,N,definedby AjHi = AijHi.
0375~9601/97/$17.00 @ 1997 Ekvier Science B.V. All rights reserved. PIISO375-9601(97)00341-l
(3)
M. Ma&s et al./Fiysics Letters A 232 (1997) 99-IO.5
100
Then, the discrete Laplace equations ( I) can be written as the first order system of equations AiXi = (Tj&)Xj,
i + j,
(4)
where the scalar functions &j and the ~-Dimensions vectors Xi are defined by the equations AiHj = &l;:Hi,
i $ j,
(3
AiX = (T;:Hi)Xi,
whose ~mpatibility Akflij =
(TkPik)Pkj+
(6)
gives the ~uations i
#
j
#
k # i.
(7)
FromnowonwerefertoEqs. (2) (or (7)) astothe multidimensional quadrilateral lattice equations. The geometric ~ns~ction just described provides, in principle, also the integrability scheme associated with these lattices. Indeed, any simultaneous solution of the linear equations ( 1) gives automatically a solution of the associated nonlinear constraint and, therefore, defines a particular MQL. The ~ntinuum limit of Eqs. (2) gives the well known Darboux equations for conjugate coordinate systems [ 41. The matrix version of the Darboux equations was discovered and solved by Zakharov and Manakov [6] using the a’ approach, while Eqs. (2) were recently derived, in the matrix case, by Bogdanov and Kono~l~henko f 71 as an integrable discrete analogue of the Darboux equations; but their geometric meaning was unknown, The Darboux transformation, which is a well known tool in the theory of integrable systems [ 9,111, is named after the work of Darboux [ 121; the classical Darboux ~sfo~on deals with a Sturm-Liouville problem, the one-dimensional stationary SchrCidinger equation, generating, at the same time, new potentials and wave functions from given ones. As it is described in Refs. [ 9, lo] the method has been successfully used for integrable systems in 1 + 1 dimensions; the idea is that the nonlinear integrable equation is a compatibility condition for an overdetermined linear system; then, one constructs a suitable Darboux transformation for this linear system. However, as is pointed out in Ref. [ lo], it was the work of Moutard [ 131 which inspired Darboux; indeed, one can already find in Ref. [ 131 the proper transformation that applies to multidimensional integrable systems. In particular,
the theory of conjugate coordinate systems provides a class of symmetry transformations of the Darboux equations [4,5], recently rediscovered in Ref. [ 151. There is an extensive literature on the applications of the method to integrable systems and one can find Wrons~~ and ~~~ rep~n~tions for essentially all known cases of integrable equations [9,10]. There has been also an intensive study on the application of this technique to discrete integrable equations (see, for instance, Refs. [ 8,9] ) ; e.g., the Darboux ~~sfo~ation for Eqs. (2) has been constructed in Ref. 17). In recent years this method has been generalized to a vectorial form, avoiding in this way the iteration of Darboux transformations and giving, from the very beginning, compact expressions for the potentials and wave functions [ 141. In this paper we construct a vectorial Darboux transformation for the MQL equations (7), and we use it to obtain classes of explicit solutions corresponding to special lattices. The paper is organized as follows. In Section 2 we present the vectorial Darboux transformation (VDT) for the MQL equations and we give a procedure to generate concrete lattices from the ingredients of the VDT. In Section 3 we show that this VDT can be expressed in terms of discrete Grammians for the zero background seed and provides, as particular case, a Wroftski and Casorati representation of the solutions. In Section 4 we consider two examples of expiicit quadrilateral lattices: the parallelogram and the “dromionic” lattice.
Consider N( N - 1) Scalar functions &j, i, j = 1, . . . , N, i + j depending on N integer variables nN, satisfying the MQL equations (7). 4...., These nonlinear equations arise as the compatibility condition for the following linear system, AjVi=(Tj&j)Vj,
i Z j,
i,j=l,...,N,
(8)
where unknown functions vi take values in a linear space V. It turns out that, in the construction of the VDT for Eqs. (7), it is also necessary to introduce the following additional linear system whose compatibility condition is again (71,
hf. h&z&s et al. /Physics LettersA 232 f1997) 9!MO5
AjLui~~j~(Z)d),
i
#
solves (7): we only know that the new linear system is satisfied for the particular functions 18,and D. However, as we will show now, the vectorial character of the Darboux transformation does imply this result. Given another arbitrary solution xi of (8), we consider the solution
(9)
j,
where unknown functions u’ take values in the dual space v” to the vector space V. The vectorial Darboux transformation, which allows one to construct a new solution of the MQL equations (7) from any seed solution, is presented in the following theorem.
i= l,,..,
N.
(
i=l,...,N,
tbi=oiJ2-‘, fiij =&j
(11)
i=l,...,
Vi
>
of (8) taking values in the doubled space V @ V, and the solution d = (0, w’) of (9). Now, we look for an operator @ E GL( V ~3 V) that satisfies ( 10) for e and w; this can be taken as
(10)
Then, the new functions 3i=K’Vf+
Xi
&=
Theonclm 1. Let &j, i, j = 1,. . . ,n, i # j, be a solution of the MQL equations (7) and Vi and o’, i = 1, . . . , N, be solutions of the associated linear systems ( 8) and (9), respectively. Let 0 be an invertible linear operator in V defined by the compatible equations diKJ=Ui@ (T*‘),
101
N,
where fiX is defined by (10) by replacing v by x. Because its inverse is
(12)
-(ClJ~hf2-']Ui),
i,j=l,...,
N,
i#j
(13)
are new solutions of Eqs. (7)-(g). In addition, any solution A of &A = bi @ (I;:&) is of the form A=c--an-‘,
the ~~sfo~ (12) are & =
&=
where C is a constant operator.
(
Xi
wave functions according to ( 11) and
-
f2x 4fl-'Vi W'Vi > ’
(O,wn-‘>,
while
Proof: From (8) and (9) it follows that
dj(Ui Q9(I;:w’)) = (Tjpij)(vj +(~~ji)(~i~~~~)
@TJjO’)
=ic*
j,
i#j.
Therefore, Eqs. ( 10) are compatible and D exists together with its inverse (at least locally). Now, applying dj to the equations q = &q.,
w’ = &i&?,
and using Eqs. (lo)-( 0( AjOi - (Tjbij)Dj)
Aid = Of @ &tj’ = fa-‘Vi @ I;(da-‘)
13) one obtains = Ajvi - (q@ij)Uj
does not depend on Xi and is the same as the one before doubling the space. NOW we CaIl Use the frtct that $ and & s&fy (8)) and conclude that Xi - &fi-‘vi and sij do also. As this holds for any arbitrary solution Xi, we conclude that the functions & satisfy (7). Finally,
= f2-‘(Ai0)(l;i2)-’
2Z0,
(Aji3’-~j~(Ti~))Tjn=AjWi-Bji(Tjoi)
=O.
of fl imply that 1131, Oj do satisfy (8) and (9) with &j IEplaced by &. This does not imply in principle that bij
cl
Given a solution &l pi, W’ and 0 of Eqs. (7)lo), we can construct a MQL in the following way. First,welet&,~~EV,aJ~V*,i~j,i=l,..., N and 0 E V ~3 V* be a solution of (7)-( lo), where dim V 2-M to avoid trivial degenerations of the lattice. (
Eqs. (8) and (9) and the (local) inv~bi~~
= -AiJz-‘.
M. Ma&s et ai./Physics Letters A 232 (1997) 99405
102
Second, we consider a constant vector wt f V, VMan M-dimensional subspace of V and Pv, a projection, Pv, : V -+ V,. Finally, we define, (i) the functions Hj : ZN 3 W r”:=(o’lw),
i=l,...,
iv,
(14)
(ii) the vector-valued functions Xi : ZN --f V, Xi:=Pv#z+,
i=l,...,
N,
(1%
(iii) the vector-valued function x : ZN -b VM x := Pv,&?w.
Gramm and Casorati representations of solutions of Eqs. (7)-(9). In this case of zero background, the linear systems (8) and (9) simply say that the vector-functions pi and 0’ depend only on the discrete variable ni. If dim V = M, an interesting way of choosing these vectors is given by the following decomposition of M, M = Ml + * f * -!-MN>N,
M~EM,
and the induced decomposition
(16)
Proposition 1. The vector x is the position vector of the N-dimensional quadrilateral lattice in VM N RM and Hi and Xi are corresponding objects defined in Eqs. (3)-(6). Proof: To prove this proposition we compare Elqs. (8)-( 10) with Eqs. (4)-(6), respectively, and •1 we make use of their linearity. In particular, inducing the basis {ei} of V, its dual basis (ei) c V* and choosing, without lack of generality, for VM the subspace spanned by {et,. . . ,e~}, Eqs. (14)-( 16) take the following form in components,
If we take vi(%) E Pi, o’(ni) E (I?“~)+, then the Darboux potential 4 which is obtained by discrete int~ration of ( lo), takes the form fi=C+diag(fi
&),
whe~Cisa~ns~tMx~~~x~d~i(ni) Mi x Mi matrix defined by
isan
We see that fi is the discrete version of a multiGrammian type matrix. The new /3% are i%j =
k
I,...,
--d-&(&+/(adj
J2)~lPi)
where (adj 0); is an Mi x Mj matrix which occupies the (i, j) place in the adjoint matrix of 52,with respect to the block decomposition induced by the splitting V = ?l4+ @ +o. @I+. The new wave functions will be
where ekl
W’
Vi
=c L&k, k
k
oiek, k
k.1
3. Gramm and Casorati nepresemtatiom In the case 0f zero background: pij = 0, OUTvat+ ~~sfo~tions allow one to obtain, for an appropriate choice of the vectors Viand w’, discrete
rial Darboux
.
rs’ =~(W’(BdjD)~,....J(adja)“,. To present a discrete Wronskian representation of solutions of (7)-(g) we proceed as follows. Given a ve&0rfunCtiOnVi=(~t,...,V~)ofn~EZwefirst define the discrete Wrofiski type m x M rectangular matrix W(Vi,m) a.5
M. hiarIaset al./Physics LettersA 232 (1997) 99-105
Ayq:1 AT-21$ *** l$l
W(Vi,rn)
:=
(
i
i
;
Ay&&f Ay-21(&f*. * l$M
Theonrm 2. Given N positive integers Ml, . . . , ikfN andNarbitraryvectorftmctionsV~(ni),i=l,...,N, taking values in RM with M = Ml + aa. + hfN, then
) .
Given N vector functions taking values in R”, V~(nhr) and the partition M = A41+ Vi(W),..., aa- + 44~ of A4 in N natural numbers, we introduce a multiple discrete Wroiiski matrix W( VI, . . . , VN) as
= (WV,,M1),...,W(VN9MN))
built up with the rectangular blocks W( Vi, Mi). In what follows we shall need a sequence of polynomials {sj}jas of a discrete independent variable, say n E Z, &fined by (-1)j
=
(“+;-
1).
These polynomials are completely characterized by dSj+r +TSj =O, Sj(0) = 0,
SO= 1,
j > 0.
CO-
J,
:=
0,
& = c(M)w--l,
i= 1,...,
det W.. /3jj=-&!,
i,j=l,...,
N
N,
N,
iZ
j
A? Vi. PmO$ Since the seed solution is taken to be flij = 0 the linear systems (8) and (9) simply say that ZIP and mi depend solely on ni. We take vi = A? Vi, with Vi(ni) in R”. The choice for the w’s relies on the splitting (P) * = (PI)* 63 .a. 6B(@)‘; in fact we choose oi = #M’) as the lift of +‘“” (ni) E (I@)* up to (W”)*. With this choice, and using property (17) to perform discrete integrations, one readily obtains that J2 can be taken as
n=w.s
We rewrite Wronskian solutions in terms of the more familiar Casorati matrices. For that aim we recall the binomial expansion
and the upper triangular m x m matrix So”), with ?I
i= 1,...,
a.
(which can be checked to satisfy ( 10) ) . This implies the stated result. q
( SO.Sl,..., Sm-I )
&?!I”’:= s._. I
t
(17)
To the first m polynomials Sj we associate the vector e(m) :=
3.I’ - s-1 . )/y’~~‘V.
are solutions of Eqs. (7)-( 9), and where W := VN) and Wij is the matrix obtained from WWl,..., W by replacing the first CO~UIINI of W( Vj, Mj) by
w(V1,..:,vNJ
Sj(tl)
103
iaj
i < j, A” =
-@-l)k(;)Tm-k,
that has the property
that implies W(Vi,m)
e(m) := ( 1, 0 ,...,
=C(Vbm)
. Bm,
where
0) E (P)‘.
. . ++ MN, we consider GiventhepanirionM=~,;t;n.) the co-vector function + pending on the jth variable only.’ We also introduce thematrixS=diag(S(M1),...,S(MN)) builtupasa block diagonal M X M III&X, with Mi X hli blocks StMiM1) in the diagonal. j
=
1,...,N
de_
23a :=
1
0
0
... 0
-(y’)
1
0
... 0
1
. ..O
(“2’)
-(“TZ)
.. (-l)m-’
(-I)“-2
(_I)“-3
.
..: ;
,1
M. M&aset al./PhysicsLettersA 232(1997)W-105
104
7yq&# 11”-21$Jf *** i&f
1
We see that the Wroiiski matrix W(V,, m) can be factor&d in terms of Casorati matrix C( Vi, m) and a lower triangular matrix with binomial coefficients. Now, we in~~uce the multiple Casorati matrix C(Vl,...,Viv>
CWI
,...,VM)
4. I. The pu~llelog ~
lattices
In the case & = 0 the vectors Viand 0’ are arbitrary vector functions of ni only. Applying Proposition 1 to this simple case, we obtain that the position vector x can be represented in the following form,
as
:=
(CCVl,1M1),...,C(V~,MN))
built UP with the m&tngnlar blocks C(Vi,Mi); we introduce also the matrix a := diag( 23~~). . . , a~,). Then, we have Proposition 2. Given N positive integers Ml, , . ., i = 1, . . ..N.~ngv~u~in~~with~=~~+...+~~, then
where xk(O) = 0 and c is a constant vector, which characterize the lattice made out of p~lelo~~s. In this case the MQL can be constructed from the initial CUIWS x(0,. . . , nip.. . (0) = Xi(%) f C{ which already contain the full information about the la&e. The simplest among the parallelogram lattices, the Cartesian one, is obtained choosing M = N and
MN and N arbitrary vector functions Ifi(
bi = S-1 . a-1 . C-lA”i i Vi, &‘==(“‘)B-’
.C-‘,
N
Xi=Ui=ei,
&i=C’,
W=CCj, j=l N
i=l,...,
izz I,...,
N,
n=C+diag(nl,...,np),
are new solutions of E&s. (7) and (8)) where C := VN) and C, is the matrix obtained from C by replacing the first column of C(Vj, Mj) by qiMiViCWl,...,
Proo$ Observe that det a = 1 and thus det W = det C. For det YV;jthe situation is more subtle in the rectangular block W(Vi, Mi), where now the first column is D”iVis But one can transform all the others columns to be of %anslational type”. Then, using the binomial formula, one can expand this special column, and add to it convenient linear combinations of the columns in the block Ci( Vi, Mi) ; these operations will not alter the value of the determinant and SO det MI, = El det Cij+
IZjej
4
C,
j=l
N, j+j
x= E
We remark that the parallelogram lattice can be localized in a finite region of R”+ To have this situation it is sufficient to choose the vector-functions xi(q) to be regular for finite Ri and to go to some constant values iii* for ai --* fCX3. 4.2. Dromionic lattices Let us apply the vectorial Darboux transformation to the zero background &j = 0, choosing V = RN and
ui
I
air\Tei,
J = &Fe’,
where Ai, pi > 1, a& > 0. Then, the operator 0 is given by
id
4. Explicit lattices In this section we consider two explicit solutions of Eqs. (7) and the corresponding MQLs.
where C is an arbitrary constant operator on V and fi(ni)
:= Qibif&(AiFi)Rie I
I
hf. Ma&s et al. /Physics Letters A 232 (1997) 99405
A more systematic investigation of the explkit solutions (of rational and exponential type) of the MQL equations is postponed to a subsequent paper.
Then, the dressed solutions are
N de. (adj&
ai = &A: C
”
det Ja
j=l
N (adjO)jej
&+p~~----
j=,
?ji
= _a.b.
det
W
’ ’
Acknowledgements
,
D
O)tj
det 0
’
where adj D is the adjoint matrix of D and therefore its elements (adj a>$ are ( - 1) i+j times the determinant of the matrix obtained by deleting the ith column and the jth row in the matrix of 0. We shall analyze in more detail the case N = 2. If C = (q) , we have det n=det
C+CII~I
+c~2f2+fl_f2
and (adjo),
=
(a4 0)1i
=~22+f2,
-Cij,
105
i # j (a4ifB;?;?=ctt+.fi.
Therefore, ,
@ 2=
a2Ai detfz -(
-CZI cll+fl >’
In the generic case (det C, cr 1, ~22$0) the vectors & and the scalars & go to zero “exponentially” when nf + 4 + 00, while the matrix J2 goes to a constant matrix. Therefore, in the (ni , n2) plane, this solution represents the discrete analogue of the onedromion solution [ 161; the corresponding lattice x(nr , n2) is asymptotically a parallelogram lattice, but localized in a finite region of R2. If det C = 0 the lattice is not localized when ~1 and nz are negative and large. If, in addition, ~22= 0 then the lattice is not bounded also for u: + nf --+ co and ni > 0 and n2 < 0. Similar considerations can be made in the case N=3.
M.M. acknowledges partial support from CXCYT proyecto PB95-0401 and from the exchange agreement between Universita I.a Sapienza of Rome and Universidad Complutense of Madrid. References [ 1] A. Doliwa and P.M. Santini, Multid~~ion~
Quadrihtteral Lattices am Integrable, Preprint JNFN ( 1996). [2] R. Sauer, Differenzengeometrie (Springer, Berlin, 1970). [3] A. Doliwa, Geometric Discretisation of the Toda System, INFN preprint (19%). [4] C. Darhoux, Le9ons sur la thdorie g&&ate des surfaces Iv, Liv. VIII, Chap. XII (Gauthier-VKlars. Paris, 18%). [ 5 ] L.R Bisenhatt, A Treatise on the Di&umial Geometry of Curves and Surfaces (Gkm, Boston). [6] V.E. Zakharov and SE. Manakov, Zap. N. S. LGMI 133 (1984) 77; Funk. Anal. Ego FriJozh. 19 (1985) 11. [?‘I L.V. Bogdanov and B.G. Konopelchenko, J. Phys. A 28 (1995) L173. [8] K. Kajiwarn and J. Satsuma, J. Phys. Sot. Japan 36 (1991) 3987; Y. Ohta, K. Kajiwara, J. Matsukidaira and J. Satsuma, J. Math. Phys. 34 (1993) 5190: K. Kajiwara, Y. Ohta and J. Satsuma, J. Math. Phys. 36 f 1995) 4162. [9] V.B. Matveev and MA. Salle, Darboux Transformations and Solitons (Sprhrger, Berlin, 1991). [lo] C. Athorne and J.J.C. Nimmo. Jnv. Rob. 7 (1991) 809. [ 1l] W. Gevel, Physica A 195 (1993) 533. 1121 G. Darboux, CR. Acad. !ki. Paris 94 (1882) 1456, 120, 158, 1290. [ 131 Th.F. Moutard. C.R. Acad. Sci. Paris 80 (1875) 729; J. de 1’Jkole PoIytechnique 45 (1878) 1. [141 l? Guil and M. Ma&s, Phys. Lett. A 217 (1996) 1; M. Maitas, J. Phys. A 29 (19%) 7721. f 151 B.G. Konopelchenko and WK. Schiif, Lam6 and Z&hmvManakovsystems:Co~urc. Dadx~x and B8cklund Transformations, Applied Mathematics Pmprhtt AM93/9, The University of New South Waies. 1161 A.S. Fokas and RM. Saotini, Physica D 44 (1990) 99.
PHYSICS LETTERS A
physics Letters A 232 (1997) 99-105
Darboux transformations for multidimensional quadrilateral lattices. I Pa010 Maria Santini bA3 Manuel Ma&s a*l, Adam Doliwa bvc*2, a ~e~ata de Fisicu Te&ica, Universe Contpktense,E-28040 Madrid, @ain b IstitutoNa.onak di EMca Nuckare, Sezionedi Roma,P-k Aldo More 2. I-00185 Rotne,Italy c InstytutFL&i Teontycznej,UniwersytetWarszawski, rL &!a 69,lW681 Warsaw,Mand d Dijxim’mentoa?’Fisica, Universirddi Catania,Corso Italia57, I-95129 Gxtania,Italy Received 24 January 1997; acceptedfor publication 23 April 1997 Communicatedby A.P. Fordy
Abstract The vectorial Datboux transformationfor the multidimensionalquadrilaterallattice equations is constructed. Its particular reduction to the case of trivial background gives Grannn, Wronski and Casorati type ~p~~n~tions for the solutions. Some examples of the quadrilateral lattices are constructed explicitly. @ 1997 Elsevier Science B.V.
1, In&oduetion In a recent paper it was shown that rn~~~~nsion~ quadrilateral lattices (MQLs) are integrable [ 11.By a MQL we mean an N-dimensional (N 2 2) lattice; i.e., a mapping x : ZN --+ W”, M 3 N, labelled by the multi-index n := {nr,. . . , nN} E EN, .%&fying the property that each elementary quadrilater of it is planar. This condition is equivalent to the following discrete analogue of the Laplace equation, didjx f (I;:Aij)& i#j,
i,j=l,...,
=f(nl,
..,, nj+l,...,?&)
and Aj = Tj - 1 is the corresponding difference operator. For N = 2, any choice of the two functions Ar2, A21 gives rise to a possible two-dimensional qnadrilateral lattice. In order to build a MQL one h&sto repeat this ~ns~ction in every pair of dire&ions and, to make it possible, the data Aij must satisfy the following nonlinear constraints,
+ (TjAji)Ai~, N,
(1)
where x E IP is m arbitrary point of the lattice, Aij are N( N - 1) real functions of n, Tj is the translation operator in the jth variable,
AkAij = (T’Ajk)Aij + (%Akj)Aik - (TkAij)Aik+ ifj#kfi,
m
which chamctexize completely all the MQLs. It is convenient to reformnlate Bqs. ( 1) as tirst order systems [ 11. In order to do that, we introduce new functions Hi, i = 1,...,N,definedby AjHi = AijHi.
0375~9601/97/$17.00 @ 1997 Ekvier Science B.V. All rights reserved. PIISO375-9601(97)00341-l
(3)
M. Ma&s et al./Fiysics Letters A 232 (1997) 99-IO.5
100
Then, the discrete Laplace equations ( I) can be written as the first order system of equations AiXi = (Tj&)Xj,
i + j,
(4)
where the scalar functions &j and the ~-Dimensions vectors Xi are defined by the equations AiHj = &l;:Hi,
i $ j,
(3
AiX = (T;:Hi)Xi,
whose ~mpatibility Akflij =
(TkPik)Pkj+
(6)
gives the ~uations i
#
j
#
k # i.
(7)
FromnowonwerefertoEqs. (2) (or (7)) astothe multidimensional quadrilateral lattice equations. The geometric ~ns~ction just described provides, in principle, also the integrability scheme associated with these lattices. Indeed, any simultaneous solution of the linear equations ( 1) gives automatically a solution of the associated nonlinear constraint and, therefore, defines a particular MQL. The ~ntinuum limit of Eqs. (2) gives the well known Darboux equations for conjugate coordinate systems [ 41. The matrix version of the Darboux equations was discovered and solved by Zakharov and Manakov [6] using the a’ approach, while Eqs. (2) were recently derived, in the matrix case, by Bogdanov and Kono~l~henko f 71 as an integrable discrete analogue of the Darboux equations; but their geometric meaning was unknown, The Darboux transformation, which is a well known tool in the theory of integrable systems [ 9,111, is named after the work of Darboux [ 121; the classical Darboux ~sfo~on deals with a Sturm-Liouville problem, the one-dimensional stationary SchrCidinger equation, generating, at the same time, new potentials and wave functions from given ones. As it is described in Refs. [ 9, lo] the method has been successfully used for integrable systems in 1 + 1 dimensions; the idea is that the nonlinear integrable equation is a compatibility condition for an overdetermined linear system; then, one constructs a suitable Darboux transformation for this linear system. However, as is pointed out in Ref. [ lo], it was the work of Moutard [ 131 which inspired Darboux; indeed, one can already find in Ref. [ 131 the proper transformation that applies to multidimensional integrable systems. In particular,
the theory of conjugate coordinate systems provides a class of symmetry transformations of the Darboux equations [4,5], recently rediscovered in Ref. [ 151. There is an extensive literature on the applications of the method to integrable systems and one can find Wrons~~ and ~~~ rep~n~tions for essentially all known cases of integrable equations [9,10]. There has been also an intensive study on the application of this technique to discrete integrable equations (see, for instance, Refs. [ 8,9] ) ; e.g., the Darboux ~~sfo~ation for Eqs. (2) has been constructed in Ref. 17). In recent years this method has been generalized to a vectorial form, avoiding in this way the iteration of Darboux transformations and giving, from the very beginning, compact expressions for the potentials and wave functions [ 141. In this paper we construct a vectorial Darboux transformation for the MQL equations (7), and we use it to obtain classes of explicit solutions corresponding to special lattices. The paper is organized as follows. In Section 2 we present the vectorial Darboux transformation (VDT) for the MQL equations and we give a procedure to generate concrete lattices from the ingredients of the VDT. In Section 3 we show that this VDT can be expressed in terms of discrete Grammians for the zero background seed and provides, as particular case, a Wroftski and Casorati representation of the solutions. In Section 4 we consider two examples of expiicit quadrilateral lattices: the parallelogram and the “dromionic” lattice.
Consider N( N - 1) Scalar functions &j, i, j = 1, . . . , N, i + j depending on N integer variables nN, satisfying the MQL equations (7). 4...., These nonlinear equations arise as the compatibility condition for the following linear system, AjVi=(Tj&j)Vj,
i Z j,
i,j=l,...,N,
(8)
where unknown functions vi take values in a linear space V. It turns out that, in the construction of the VDT for Eqs. (7), it is also necessary to introduce the following additional linear system whose compatibility condition is again (71,
hf. h&z&s et al. /Physics LettersA 232 f1997) 9!MO5
AjLui~~j~(Z)d),
i
#
solves (7): we only know that the new linear system is satisfied for the particular functions 18,and D. However, as we will show now, the vectorial character of the Darboux transformation does imply this result. Given another arbitrary solution xi of (8), we consider the solution
(9)
j,
where unknown functions u’ take values in the dual space v” to the vector space V. The vectorial Darboux transformation, which allows one to construct a new solution of the MQL equations (7) from any seed solution, is presented in the following theorem.
i= l,,..,
N.
(
i=l,...,N,
tbi=oiJ2-‘, fiij =&j
(11)
i=l,...,
Vi
>
of (8) taking values in the doubled space V @ V, and the solution d = (0, w’) of (9). Now, we look for an operator @ E GL( V ~3 V) that satisfies ( 10) for e and w; this can be taken as
(10)
Then, the new functions 3i=K’Vf+
Xi
&=
Theonclm 1. Let &j, i, j = 1,. . . ,n, i # j, be a solution of the MQL equations (7) and Vi and o’, i = 1, . . . , N, be solutions of the associated linear systems ( 8) and (9), respectively. Let 0 be an invertible linear operator in V defined by the compatible equations diKJ=Ui@ (T*‘),
101
N,
where fiX is defined by (10) by replacing v by x. Because its inverse is
(12)
-(ClJ~hf2-']Ui),
i,j=l,...,
N,
i#j
(13)
are new solutions of Eqs. (7)-(g). In addition, any solution A of &A = bi @ (I;:&) is of the form A=c--an-‘,
the ~~sfo~ (12) are & =
&=
where C is a constant operator.
(
Xi
wave functions according to ( 11) and
-
f2x 4fl-'Vi W'Vi > ’
(O,wn-‘>,
while
Proof: From (8) and (9) it follows that
dj(Ui Q9(I;:w’)) = (Tjpij)(vj +(~~ji)(~i~~~~)
@TJjO’)
=ic*
j,
i#j.
Therefore, Eqs. ( 10) are compatible and D exists together with its inverse (at least locally). Now, applying dj to the equations q = &q.,
w’ = &i&?,
and using Eqs. (lo)-( 0( AjOi - (Tjbij)Dj)
Aid = Of @ &tj’ = fa-‘Vi @ I;(da-‘)
13) one obtains = Ajvi - (q@ij)Uj
does not depend on Xi and is the same as the one before doubling the space. NOW we CaIl Use the frtct that $ and & s&fy (8)) and conclude that Xi - &fi-‘vi and sij do also. As this holds for any arbitrary solution Xi, we conclude that the functions & satisfy (7). Finally,
= f2-‘(Ai0)(l;i2)-’
2Z0,
(Aji3’-~j~(Ti~))Tjn=AjWi-Bji(Tjoi)
=O.
of fl imply that 1131, Oj do satisfy (8) and (9) with &j IEplaced by &. This does not imply in principle that bij
cl
Given a solution &l pi, W’ and 0 of Eqs. (7)lo), we can construct a MQL in the following way. First,welet&,~~EV,aJ~V*,i~j,i=l,..., N and 0 E V ~3 V* be a solution of (7)-( lo), where dim V 2-M to avoid trivial degenerations of the lattice. (
Eqs. (8) and (9) and the (local) inv~bi~~
= -AiJz-‘.
M. Ma&s et ai./Physics Letters A 232 (1997) 99405
102
Second, we consider a constant vector wt f V, VMan M-dimensional subspace of V and Pv, a projection, Pv, : V -+ V,. Finally, we define, (i) the functions Hj : ZN 3 W r”:=(o’lw),
i=l,...,
iv,
(14)
(ii) the vector-valued functions Xi : ZN --f V, Xi:=Pv#z+,
i=l,...,
N,
(1%
(iii) the vector-valued function x : ZN -b VM x := Pv,&?w.
Gramm and Casorati representations of solutions of Eqs. (7)-(9). In this case of zero background, the linear systems (8) and (9) simply say that the vector-functions pi and 0’ depend only on the discrete variable ni. If dim V = M, an interesting way of choosing these vectors is given by the following decomposition of M, M = Ml + * f * -!-MN>N,
M~EM,
and the induced decomposition
(16)
Proposition 1. The vector x is the position vector of the N-dimensional quadrilateral lattice in VM N RM and Hi and Xi are corresponding objects defined in Eqs. (3)-(6). Proof: To prove this proposition we compare Elqs. (8)-( 10) with Eqs. (4)-(6), respectively, and •1 we make use of their linearity. In particular, inducing the basis {ei} of V, its dual basis (ei) c V* and choosing, without lack of generality, for VM the subspace spanned by {et,. . . ,e~}, Eqs. (14)-( 16) take the following form in components,
If we take vi(%) E Pi, o’(ni) E (I?“~)+, then the Darboux potential 4 which is obtained by discrete int~ration of ( lo), takes the form fi=C+diag(fi
&),
whe~Cisa~ns~tMx~~~x~d~i(ni) Mi x Mi matrix defined by
isan
We see that fi is the discrete version of a multiGrammian type matrix. The new /3% are i%j =
k
I,...,
--d-&(&+/(adj
J2)~lPi)
where (adj 0); is an Mi x Mj matrix which occupies the (i, j) place in the adjoint matrix of 52,with respect to the block decomposition induced by the splitting V = ?l4+ @ +o. @I+. The new wave functions will be
where ekl
W’
Vi
=c L&k, k
k
oiek, k
k.1
3. Gramm and Casorati nepresemtatiom In the case 0f zero background: pij = 0, OUTvat+ ~~sfo~tions allow one to obtain, for an appropriate choice of the vectors Viand w’, discrete
rial Darboux
.
rs’ =~(W’(BdjD)~,....J(adja)“,. To present a discrete Wronskian representation of solutions of (7)-(g) we proceed as follows. Given a ve&0rfunCtiOnVi=(~t,...,V~)ofn~EZwefirst define the discrete Wrofiski type m x M rectangular matrix W(Vi,m) a.5
M. hiarIaset al./Physics LettersA 232 (1997) 99-105
Ayq:1 AT-21$ *** l$l
W(Vi,rn)
:=
(
i
i
;
Ay&&f Ay-21(&f*. * l$M
Theonrm 2. Given N positive integers Ml, . . . , ikfN andNarbitraryvectorftmctionsV~(ni),i=l,...,N, taking values in RM with M = Ml + aa. + hfN, then
) .
Given N vector functions taking values in R”, V~(nhr) and the partition M = A41+ Vi(W),..., aa- + 44~ of A4 in N natural numbers, we introduce a multiple discrete Wroiiski matrix W( VI, . . . , VN) as
= (WV,,M1),...,W(VN9MN))
built up with the rectangular blocks W( Vi, Mi). In what follows we shall need a sequence of polynomials {sj}jas of a discrete independent variable, say n E Z, &fined by (-1)j
=
(“+;-
1).
These polynomials are completely characterized by dSj+r +TSj =O, Sj(0) = 0,
SO= 1,
j > 0.
CO-
J,
:=
0,
& = c(M)w--l,
i= 1,...,
det W.. /3jj=-&!,
i,j=l,...,
N
N,
N,
iZ
j
A? Vi. PmO$ Since the seed solution is taken to be flij = 0 the linear systems (8) and (9) simply say that ZIP and mi depend solely on ni. We take vi = A? Vi, with Vi(ni) in R”. The choice for the w’s relies on the splitting (P) * = (PI)* 63 .a. 6B(@)‘; in fact we choose oi = #M’) as the lift of +‘“” (ni) E (I@)* up to (W”)*. With this choice, and using property (17) to perform discrete integrations, one readily obtains that J2 can be taken as
n=w.s
We rewrite Wronskian solutions in terms of the more familiar Casorati matrices. For that aim we recall the binomial expansion
and the upper triangular m x m matrix So”), with ?I
i= 1,...,
a.
(which can be checked to satisfy ( 10) ) . This implies the stated result. q
( SO.Sl,..., Sm-I )
&?!I”’:= s._. I
t
(17)
To the first m polynomials Sj we associate the vector e(m) :=
3.I’ - s-1 . )/y’~~‘V.
are solutions of Eqs. (7)-( 9), and where W := VN) and Wij is the matrix obtained from WWl,..., W by replacing the first CO~UIINI of W( Vj, Mj) by
w(V1,..:,vNJ
Sj(tl)
103
iaj
i < j, A” =
-@-l)k(;)Tm-k,
that has the property
that implies W(Vi,m)
e(m) := ( 1, 0 ,...,
=C(Vbm)
. Bm,
where
0) E (P)‘.
. . ++ MN, we consider GiventhepanirionM=~,;t;n.) the co-vector function + pending on the jth variable only.’ We also introduce thematrixS=diag(S(M1),...,S(MN)) builtupasa block diagonal M X M III&X, with Mi X hli blocks StMiM1) in the diagonal. j
=
1,...,N
de_
23a :=
1
0
0
... 0
-(y’)
1
0
... 0
1
. ..O
(“2’)
-(“TZ)
.. (-l)m-’
(-I)“-2
(_I)“-3
.
..: ;
,1
M. M&aset al./PhysicsLettersA 232(1997)W-105
104
7yq&# 11”-21$Jf *** i&f
1
We see that the Wroiiski matrix W(V,, m) can be factor&d in terms of Casorati matrix C( Vi, m) and a lower triangular matrix with binomial coefficients. Now, we in~~uce the multiple Casorati matrix C(Vl,...,Viv>
CWI
,...,VM)
4. I. The pu~llelog ~
lattices
In the case & = 0 the vectors Viand 0’ are arbitrary vector functions of ni only. Applying Proposition 1 to this simple case, we obtain that the position vector x can be represented in the following form,
as
:=
(CCVl,1M1),...,C(V~,MN))
built UP with the m&tngnlar blocks C(Vi,Mi); we introduce also the matrix a := diag( 23~~). . . , a~,). Then, we have Proposition 2. Given N positive integers Ml, , . ., i = 1, . . ..N.~ngv~u~in~~with~=~~+...+~~, then
where xk(O) = 0 and c is a constant vector, which characterize the lattice made out of p~lelo~~s. In this case the MQL can be constructed from the initial CUIWS x(0,. . . , nip.. . (0) = Xi(%) f C{ which already contain the full information about the la&e. The simplest among the parallelogram lattices, the Cartesian one, is obtained choosing M = N and
MN and N arbitrary vector functions Ifi(
bi = S-1 . a-1 . C-lA”i i Vi, &‘==(“‘)B-’
.C-‘,
N
Xi=Ui=ei,
&i=C’,
W=CCj, j=l N
i=l,...,
izz I,...,
N,
n=C+diag(nl,...,np),
are new solutions of E&s. (7) and (8)) where C := VN) and C, is the matrix obtained from C by replacing the first column of C(Vj, Mj) by qiMiViCWl,...,
Proo$ Observe that det a = 1 and thus det W = det C. For det YV;jthe situation is more subtle in the rectangular block W(Vi, Mi), where now the first column is D”iVis But one can transform all the others columns to be of %anslational type”. Then, using the binomial formula, one can expand this special column, and add to it convenient linear combinations of the columns in the block Ci( Vi, Mi) ; these operations will not alter the value of the determinant and SO det MI, = El det Cij+
IZjej
4
C,
j=l
N, j+j
x= E
We remark that the parallelogram lattice can be localized in a finite region of R”+ To have this situation it is sufficient to choose the vector-functions xi(q) to be regular for finite Ri and to go to some constant values iii* for ai --* fCX3. 4.2. Dromionic lattices Let us apply the vectorial Darboux transformation to the zero background &j = 0, choosing V = RN and
ui
I
air\Tei,
J = &Fe’,
where Ai, pi > 1, a& > 0. Then, the operator 0 is given by
id
4. Explicit lattices In this section we consider two explicit solutions of Eqs. (7) and the corresponding MQLs.
where C is an arbitrary constant operator on V and fi(ni)
:= Qibif&(AiFi)Rie I
I
hf. Ma&s et al. /Physics Letters A 232 (1997) 99405
A more systematic investigation of the explkit solutions (of rational and exponential type) of the MQL equations is postponed to a subsequent paper.
Then, the dressed solutions are
N de. (adj&
ai = &A: C
”
det Ja
j=l
N (adjO)jej
&+p~~----
j=,
?ji
= _a.b.
det
W
’ ’
Acknowledgements
,
D
O)tj
det 0
’
where adj D is the adjoint matrix of D and therefore its elements (adj a>$ are ( - 1) i+j times the determinant of the matrix obtained by deleting the ith column and the jth row in the matrix of 0. We shall analyze in more detail the case N = 2. If C = (q) , we have det n=det
C+CII~I
+c~2f2+fl_f2
and (adjo),
=
(a4 0)1i
=~22+f2,
-Cij,
105
i # j (a4ifB;?;?=ctt+.fi.
Therefore, ,
@ 2=
a2Ai detfz -(
-CZI cll+fl >’
In the generic case (det C, cr 1, ~22$0) the vectors & and the scalars & go to zero “exponentially” when nf + 4 + 00, while the matrix J2 goes to a constant matrix. Therefore, in the (ni , n2) plane, this solution represents the discrete analogue of the onedromion solution [ 161; the corresponding lattice x(nr , n2) is asymptotically a parallelogram lattice, but localized in a finite region of R2. If det C = 0 the lattice is not localized when ~1 and nz are negative and large. If, in addition, ~22= 0 then the lattice is not bounded also for u: + nf --+ co and ni > 0 and n2 < 0. Similar considerations can be made in the case N=3.
M.M. acknowledges partial support from CXCYT proyecto PB95-0401 and from the exchange agreement between Universita I.a Sapienza of Rome and Universidad Complutense of Madrid. References [ 1] A. Doliwa and P.M. Santini, Multid~~ion~
Quadrihtteral Lattices am Integrable, Preprint JNFN ( 1996). [2] R. Sauer, Differenzengeometrie (Springer, Berlin, 1970). [3] A. Doliwa, Geometric Discretisation of the Toda System, INFN preprint (19%). [4] C. Darhoux, Le9ons sur la thdorie g&&ate des surfaces Iv, Liv. VIII, Chap. XII (Gauthier-VKlars. Paris, 18%). [ 5 ] L.R Bisenhatt, A Treatise on the Di&umial Geometry of Curves and Surfaces (Gkm, Boston). [6] V.E. Zakharov and SE. Manakov, Zap. N. S. LGMI 133 (1984) 77; Funk. Anal. Ego FriJozh. 19 (1985) 11. [?‘I L.V. Bogdanov and B.G. Konopelchenko, J. Phys. A 28 (1995) L173. [8] K. Kajiwarn and J. Satsuma, J. Phys. Sot. Japan 36 (1991) 3987; Y. Ohta, K. Kajiwara, J. Matsukidaira and J. Satsuma, J. Math. Phys. 34 (1993) 5190: K. Kajiwara, Y. Ohta and J. Satsuma, J. Math. Phys. 36 f 1995) 4162. [9] V.B. Matveev and MA. Salle, Darboux Transformations and Solitons (Sprhrger, Berlin, 1991). [lo] C. Athorne and J.J.C. Nimmo. Jnv. Rob. 7 (1991) 809. [ 1l] W. Gevel, Physica A 195 (1993) 533. 1121 G. Darboux, CR. Acad. !ki. Paris 94 (1882) 1456, 120, 158, 1290. [ 131 Th.F. Moutard. C.R. Acad. Sci. Paris 80 (1875) 729; J. de 1’Jkole PoIytechnique 45 (1878) 1. [141 l? Guil and M. Ma&s, Phys. Lett. A 217 (1996) 1; M. Maitas, J. Phys. A 29 (19%) 7721. f 151 B.G. Konopelchenko and WK. Schiif, Lam6 and Z&hmvManakovsystems:Co~urc. Dadx~x and B8cklund Transformations, Applied Mathematics Pmprhtt AM93/9, The University of New South Waies. 1161 A.S. Fokas and RM. Saotini, Physica D 44 (1990) 99.