- Email: [email protected]
Integrable mappings and nonlinear integrable lattice equations
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
Integrable mappings and nonlinear integrable lattice equations V.G. Papageorgiou’, F.W. Nijhoff2
Department of Mathematicsand Computer Science and Institutefor NonlinearStudies, Clarkson University, Potsdam, NY 13676, USA
and H.W. Cape! Instituut voor TheoretischeFysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, The Netherlands Received 27 February 1990; accepted for publication 25 April 1990 Communicated by A.P. Fordy
Periodic initial value problems oftime and space discretizations of integrable partial differential equations give rise to multidimensional integrable mappings. Using the associated linear spectral problems (Lax pairs), a systematic derivation is given of the corresponding sets ofpolynomial invariants. The level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.
1. Nonlinear integrable lattices (i.e. integrable partial difference equations, cf. e.g. refs. [1—4]) are of fundamental importance for the study of classically integrable systems. They are generic in the sense that their various continuous limits give rise to hierarchies of integrable PDEs [5]. Furthermore, their study opens up some new points of view on classical integrability in general, in particular in connection with Hamiltonian structures for time-discrete systems [6,71. In this Letter we report on another application of such systems. In fact, we will show how these lattices give rise to nonlinear integrable mappings. Since many interesting problems in physics are close to systems that can be solved exactly, such mappings are useful for the study of bifurcations or transition to chaos, using perturbation techniques, cf. e.g. refs. [8,9]. In contrast to continuous systems, analytic nonlinear mappings have the advantage of exhibiting many of the complex phenomena of nonlinear dynamical systems while at the same time being computationally simpler. Concrete applications of mappings include the study of the stability of beam—beam collisions in storage rings [10,11], Anderson localization [12—151and incommensurate structures [16]. We also note their importance for the numerical integration of PDEs and the study of numerically induced chaos, cf. refs. [17,181. Probably the oldest nonlinear integrable mapping is the elliptic billiard due to Jacobi [19]. More recently McMillan found a four-parameter family ofrational mappings of the plane, together with their invanants [20]. An eighteen-parameter family generalizing the one of McMillan was presented by Quispel et al. in ref. [21]. Moreover, a connection with soliton equations of differential—differencetype was established, cf. also ref. [221. However, a spectral interpretation of the integrabiity of these mappings on the basis of a Lax pair was lacking. In this Letter we take a different point of view from the one expounded in refs. [21,22] by considering integrabte lattices rather than differential—difference equations as a starting point. This is convenient, because it allows us to obtain mappings from the consideration of an initial value problem on a two-dimensional planar 2
E-mail: [email protected]. E-mail: [email protected].
106
0375-9601/90/S 03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland)
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
lattice. Furthermore, we show that these mappings do indeed carry a spectral interpretation, and that the invariants can be calculated systematically from the monodromy matrix constructed from the Lax pair. 2. We shall use as prime example for the exposition of our ideas the lattice KdV. This equation (as well as other lattice analogues ofother integrable PDEs) was obtained in ref. [l} using a discrete version of the direct linearization method introduced in ref. [23]. Similar lattice equations have been proposed in the works of Hirota [3], cf. also ref. [4]. The lattice KdV we prefer to work with reads (1) (p_q+u_u)(p+q_i~+u)=p2_q2. (1) u=u(n, m) is the dynamical variable at the lattice site (n, m), n, me7L, and the and “are shorthand notations for translation on the lattice, i.e. ü=u(n+l, m), ü=u(n, m+ 1). p and q are the lattice parameters p, qeC. Eq. (1) arises as the compatibility condition ofa pair of linear problems (Lax pair) defining the shifts (translations) of an eigenfunction ~k in the n- and rn-directions, In
—
(2)
(p—k)~k=Lk~Pk, (q~k)r~k=Mk~k,
where Lk is given by
(
p—u
1
Lk=~k22+(+)() ~ (3) and where Mk is given by a similar matrix obtained from (3) by making the replacements p—’ q and The parameter k is the spectral parameter. Let us now consider initial value problems for (1) on the lattice. One way of doing this is to give initial date u( n, 0) on a horizontal line and consider either decaying, cf. ref. [5], or periodic boundary conditions, which in either case leads to a nonlocal scheme. Another way, namely the one that we will be concerned with in this Letter, which gives rise to a local iteration scheme, is to assign initial data on a “staircase” on the lattice. By a staircase we mean a sequence of neighbouring lattice sites with rn and n nondecreasing, as e.g. illustrated in fig. 1. From the fact that eq. (1) at each site involves only the four variables situated on the four lattice sites around a simple plaquette, it follows that the information on these staircases evolves diagonally through the lattice along “parallel” staircases. Furthermore, because of the “convexity” of the staircase the initial-value problem is well-posed. Although staircases of variable length and height stairsteps can be considered, for the sake of clarity we use two standard staircases: one for an even-periodic configuration of initial data, and the other for an odd one. In fact, one can show that several families of staircases give rise to equivalent mappings. Suppose we take the standard staircase through the origin (n, m) = (0, 0), and assign initial data on the lattice along this staircase as in fig. 1, ‘
~
u(j,j)=:a 2~, u(j+1,j)=:a2~+1 (jeZ) By applying the lattice equation (1) we can calculate the data on the next staircase one vertical step down, the (multiple) primes denoting the various iterations of this procedure. Thus, performing iterations by updating the lattice variables u along a vertical shift in the rn-direction, we define u(j,j+l)=a~,,,
uU+l,j+l)=a~,+1.
We then obtain from eq. (1) the mapping a~=a2~÷1—d+
~—a2J+2 +a2J
,
a~,+~=a2~÷2,
‘
(4)
in which o=p—q, E=p+q. By introducing now the differences Y~_=a2~÷2—a2,, .X,ssa2~+i—a~j, eq. (4) can be reduced to the measure-preserving rational mapping 107
Volume 147, number 2,3
PHYSICS LETTERS A
o
~a
2 July 1990
_____
0 a~
o
:
~-~-----------~
Fig. I. Standard staircase of periodic initial data on lattice, with even period 2P.
X~=Y~,Y=xj+~+~
—4.
(5)
It seems that the reduced variables Xi,, Y~have a deeper physical meaning than the unreduced ones, namely the a2~,a2~+1,because in the former we get rid of possible constant background contributions that come in via the lattice equation after iteration. We impose now the even penodicity condition a20+p)=a~,a2u+p)+I =a~÷i. Note that P (P=2, 3, ...) can be interpreted as the period along the two diagonals of the lattice corresponding to the staircase. It is easy to see that these periodic conditions are compatible with the lattice equation, and hence will be preserved after iteration of the mapping. This means that we have to supply (5) with the periodicity constraints ~‘ P
P—I
>~X~=0, ~Y1=0. J=I
(6)
J=O
~ It is possible to relax somewhat the condition of periodicity (6) by allowing fora constant on the right-hand side. This corresponds to a shift from a center-of-mass situation to one in which total momentum is this constant. We will not treat this situation in the present note.
108
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
Eq. (5) for j= 1, 2, P together with the constraint (6) is a 2 (P— 1)-dimensional integrable mapping. The simplest case P= 2 corresponds to the well-known McMillan mapping [20]. The McMillan map was also obtained as a reduction of the discrete nonlinear Schrodinger equation and also as a special case of a 18-parameter family of two-dimensional integrable mappings [21,22]. All mappings obtained in this way exhibit P— 1 nontrivial integrals which can be found in a straightforward way by exploiting the Lax representation (2) ofthe original lattice KdV. In order to do this we need to construct a monodromy rnatrix .~, i.e. an ordered product of Lax matrices Lk and Mk along the staircase over a oneperiod distance. Let us introduce the monodromy matrix in a way that is also suitable for the odd-period case. We work with lattice parameters p~at every joint of the staircase labeled by j (i.e. the jth link on the staircase counting from the origin) to be specified later. Thus, defining ...,
n fl L,,(a~f~,a,),
N- I
.~(aN,aNl,...,aI,ao)m
(7)
v=O
where the (1 indicates that the factors in the product are arranged from right to left, and in which the translation matrices Lk(a,+l, a,) represent either the Lax matrix in the n-direction (v even, p,~=p), i.e. Lk, or the Lax matrix in the rn-direction (v odd, p,~= q), i.e. Mk. They are of the form
Lk(a,+l,a,)=(k2—p~+I)F+L~+I,,F=(? ~),
(8)
in which the matrices
L
jp,+i—a,+i *
1 p~+a, 4_1
(* stands for the product of the diagonal entries) have the nice property that any product ofa connected string of them remains of the same form up to a factor, namely /N—I LN,N_I
LN_ 1,N_2...L2.I L1,0 =(
fl
(p, +P,+i
+ a,_1
—a,~1) )LNo.
(9)
\v=I
By construction, the monodromy matrix (7) has a particularly simple behaviour under application of the mapping, namely ...,
a’s, a’0) =Lk(a~,,~
)~(aN,
...,
aO)Lk(a~,a0)~=Lk(a~,ao).~(aN, aO)Lk(a~,a0)~ (10) ...,
where in the second equality use has been made of the compatibility condition of the Lax representation and in the last equality of the periodicity requirement. From eq. (10) follows that the trace Tr(~) is invariant under the mapping (5), andtohence we can mentioned generate F— 1 invanants by developing trace in powers of calthe 2. Due the property above for the matrices La,, it the is particularly easy to spectral parameter k culate these invariants. The recipe is as follows: if no contributions from the terms containing the matrix F are taken into account, the trace will be given by the trace of a product of the form (9), which in virtue of the periodicity constraint will yield simply the product N Tr(LN,N_I...L 2,ILI.O)=
fl (p,+p,÷1+a,_1—a,+1), aJ+N=aJ,
Pj+N=Pj-
Furthermore, for every contribution of a factor containing F we have to any twoover adjacent 2, replace k2), any trace one orfactors more ~
~+ a,_1
—
a,÷1,
...~
+p, + a,2
—
—
a, in this product by a factor (p 109
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
2
0
—1
Fig. 2. Projection onto the (X
1, X2, Y1) subspace of the orbit of
-2
I
-2
I
-1
2
1
the four-dimensional mapping (5), (6) forP= 3, ~=4.0, J= —0.5, andinitialvaluesX~=0.3,Y1=0.6,X2=0.4, Y2=—0.9.
matrices F in combination with strings of matrices L,~,,yielding either 0 due to the nilpotency of F or 1, apart from factors of the form given in eq. (9). We find as a result N
Tr{.~(aN,...,aI,aO)}=
fl
N
N
fl
(p~+p~+1+a~1—a,~1)+ ~ (k~—p~)
p_i
j~i
P#j—
2—p~)(k2—p~)
N
+
~
flN
(p~+p,+1+a,1—a9÷1) Li
(p~+p~+
(k
1+a~1—a~÷1)+....
(11)
v=I v,’i—I.i,J—l.J
(The exclusions v~1— 1, 1 and v~j—1, j in the products occurring in (11) are to be understood mod(N)). In the case of the McMillan map (N=2P=4), eq. (11) yields the integral of the mapping 2—X~)(e2—Y~)—2edX 5=(e 1Y1 , (12) which is an elliptic curve that can be parametrized in terms ofJacobi elliptic functions, thus leading to explicit solutions to the corresponding mapping. For N= 2P= 6, eq. (5) together with (6) yields a four-dimensional map in terms of variables X1, X2, Y1, Y2. The invariants are calculated from (11) as
2(x
J~=y~x1y1x~y2x3 —ed(x1ytx2y2 +x2y2x3y0 +x3y0x1y1)+ (ed)
1y1 +x2y2 +x3y0),
f2=y0x1y1x2y2x3+eö(y1x2y2x3+y0x1y2x3+y0x1y1x2)+(eö)
2x3+y0x1 +y1x2),
2(y
(13)
in which we have used the abbreviations x~me—X~, y,me—Y3
(1=1,2),
y0me+Y1+Y2, x3me+X1+X2.
By way of illustration we have plotted in fig. 2 the projection on the X1, Y1, 12 space of a typical orbit of the P= 3 mapping in fig. 2. The orbit lies on the intersection of the two hypersurfaces given by the level sets of the two invariants J~and .~.This and similar pictures suggest that it is a torus in four dimensions, which would correspond to the analogous situation for continuous finite-dimensional integrable systems. 110
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
3. The above given construction to obtain integrable multi-dimensional mappings from integrable partial difference (lattice) equations works equally well in cases of examples other than KdV, such as the modified KdV (mKdV) or the Toda equation with discrete time. Let us consider e.g. the equation 3] =V[ (q+r)O— (p+r)ii] (14) v[ (q—r)IY— (p—r)1 .
Eq. (14) is a mixed version ofthe lattice mKdV (r= 0) and a version ofthe discrete-time Toda equation (r=p), cf. e.g. ref. [1]. The parameters p, q denote as before the lattice parameters. Eq. (14) arises as the compatibility condition of the following linear system, (p—k)
( (k2_r2)/v p—r
kfIk!f’k
( 15
1) (p+r)~/v)~’
and a similar equation for the translations in the lattice direction indicated by that is obtained from (15) by replacing —i~” and p—4q. Eq. (14) is related to the lattice KdV (1) via a Miura transformation as follows, (16)
p—q+u—u= (p+r)i~—(q+r)O which on the level of the linear system reflects in a gauge transformation of the form Uk(k2
(k2—r2)!t’k=Ukøk,
~)‘
2
(17)
in which s= (p—ü)v— (p+r)1, and s= (p—r)v— (p+u)1, leading to UkFUk’=(k2—r2)
54=UkLkUkI,
Em(~
‘yE
~).
(18)
Assigning in precisely the same way as before initial data on staircases as the one depicted in fig. 1, namely ~(j,j)_—:b ~(j+l,j)=:b2~~1 v(1,j+l)=b,, v(1+l,j+l)=b’~,,÷1, 2~, we find by applying the lattice equation (14) I’.’ £~ L/2j—L/2j+I
(p+r)b2~+2+(q—r)b2~ I” ~s.
~
~
_
~
~p—r,~2~--~~,q-r-r~u21~2
~
U2j+1U2j+2.
Introducing now the reduced variables W= b~+1/b2~1,Z~=b2~+2/b2~ we obtain the following measure-preserving mapping,
w’-z i_i,
z’-w j_
Z,÷1-FPZ~,+a
2
in which p (q—r)/(p+r) and a= (p—r)/(q+r). The periodicity condition that we can now impose takes the following form, P
P—I
flW~=l, flZ1=l. j—.O
j=1
(21)
Eq. (20) together with the conditions (21) is as before a 2 (P— 1)-dimensional mapping that exhibits F— 1 invariants, that can be constructed in the same way as in the case of the lattice KdV by using the monodromy matrix The invanants are easily constructed from the ones of the KdV by using the Miura transformation (16). Alternatively, we can go through the same procedure as before, making use of the fact, that the Lax matrices that we need in the present case to calculate the monodromy matrix (7), replacing a,’s by b,’s, are of the form Ill ~.
Volume 147, number 2,3
PHYSICS LETTERS A
~
2 July 1990
M~,v=((+~/b
(the
bv÷iI(Pv+lr))
(22)
stands for the product of the off-diagonal entries), which have a similar property as (9), namely that
*
MN,N_IMNI,N
2...M2,IMI,o= fi(l+ ,=i
(23)
~ b~.±i)M p~+i—rb~_1
The monodromy matrix is easily expanded as in eq. (11), the leading term being again given by a product like the one in2—p1)b the r.h.s. of eq. (23), contributions of a factor containing the matrix E yielding an insertion of a factor (k 3+1/b~1,leading to Tr{.9~(bN,...,bl,bQ)}=
+~[(k2_pI)~t1
LI1
(pv÷ir+(pv+r~tL)]
v,&j— I,j
+
~
[(k2p~)(k2p1)~~’-±-L
b,_1 b~_1
fl
(pv+ir+(Pv+r)~t!)]+.... b~_1
(24)
v,&i— I,x,j— I,j
(The exclusions in the products occurring in the r.h.s. of eq. (24) are to be understood mod (N) as before.) For example, in the case of period N= 2P= 4 we find the invariant f=p(~+.~~)+a(WiZi+ W1Z)+(l+ap)(WI+~~+ZI+~~).
(25)
4. So far, we have been dealing with the case of even periodic conditions along a staircase as in fig. 1. In the case ofodd period N= 2P+ 1, some modifications of the scheme outlined above are necessary. A standard staircase corresponding to this situation is given in fig. 3. Note that in order for the periodicity of the data to be preserved when iterated by the mapping we must allow for a stairstep of length 2 at every point where a new sequel of data of the form a0, a1, ..., a2p÷1begins. Now, to obtain the mapping (we shall illustrate this case by means of the KdV lattice), we can use again the formulae (4) forj= 0, 1, ..., F— 1, except that now at the end of the string of data, we have to replace the formula by a’2~=a2~÷1—ô+ =a~—ô+ ,~ , —(ao—a2p) E—(ao—a2p,)
(26)
.
We use the same variables as before, i.e. X~=a2~+1—a2,1,j=l, F, and Y3=a2~+2—a2~,j=0, ..., P—l, but now we have no longer the periodicity conditions (6). Noticing that ...,
P
a2p—aj
=
P—I
E X~+~ f—_I j=0
we can reduce the system (4) together with (26) in terms of X~,Y~.The reduced mapping turns out to be 112
Volume 147, number 2,3
PHYSICS LETTERS A
a1
_____________
a’0
a~=
2 July 1990
0
a
a2
0
0
0
C
I a2p
a2p+i =
ua’~p
0
0
a0
a~
-e
-~
a1
-~
0
Fig. 3. Standard staircaseof periodic initial data on the lattice with odd period 2P+ I.
X~’=Y~,j=l,...,P—l, Yj=Xj+I+
——---v-, E—ij
~ E—~+i
C—
th Y~1=X~+ E~4p
—
Yo
_______
EIp1
.
(27)
We 2can use P again the formula trace ofFor theexample, monodromy matrix (11), when yields invariants underfor thethe mapping. in the case of P= which 1 we find theexpanded invariantin terms of k J=(e— Yo)(e—X 1)(e+X1 +Y0)+eô(X1 +Y0), (28) which can be parametrized in terms of Weierstrass elliptic functions. 5. In this Letter, a general prescription is given to obtain integrable mappings from the consideration of initial value problems of integrable lattice equations. (A different point of view, considering travelling-wavesolutions of partial difference equations is adopted in ref. [241.) The results obtained so far can be generalized to other 113
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
types of integrable lattice equations such as the lattice BSQ and MBSQ, cf. ref. [25], or KP and MKP, cf. ref. [26], equations. In all such cases one expects that integrable mappings can be obtained, with an explicit construction of their integrals in much the same way as we have done explicitly in the present note for KdV type of mappings. Work towards this direction is in progress. Another aspect that is of interest in connection with the mappings that we construct is the question of their canonical structure. Recently, a theory of Lagrangian and Hamiltonian mappingshas been developed by Veselov [6], cf. also ref. [71.It would be of interest to investigate whether this theory applies also to the mappings we introduce in the present Letter. In particular it would be of interest to give a characterization of the intersection of the level sets of the invariants that we find, in terms of multidimensional tori. This would lead to an explicit parametrization in terms of theta functions. This work is also under current investigation. Finally, we would like to point out that the mappings we construct pose a well-defined continuum limit, in much the same way as the original lattice equations do, cf. [1,2]. This limit yields integrable systems of ODEs which we will present elsewhere [27]. The authors (VGP and FWN) would like to thank Professors L. Kauffman, E. Previato and Y. Saridakis for their interest, and Professor S. Fulton for his help in producing fig. 2. VGP was partially supported by AFOSR Grant No. 86-0277.
References [1] F.W. Nijhoff, G.R.W. Quispel and H.W. Cape!, Phys. Lett. A 97 (1983)125. [21G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, Physica A 125 (1984) 344. [3]R. Hirota, J. Phys. Soc. Japan 43 (1977) 1424,2074,2079; 50 (1981) 3785. [4] E. Date, M. Jimbo and T. Miwa, J. Phys. Soc. Japan 51(1982) 4125; 52 (1983) 388, 766. [5] G.L. Wiersma and H.W. Capel, Physica A 142 (1987)199; 149 (1988) 49, 75; Phys. Lett. A 124 (1987)124. [6] A.P. Veselov, Funct. Anal. App!. 22 (1988) 83; Theor. Math. Phys. 71(1987) 446. [71J. Moser and A.P. Veselov, Preprint ETH (ZUrich), 1989. [8] M.L. Glasser, V.G. Papageorgiou and T.C. Bountis, SIAM J. App!. Math. 49 (1989) 692. [9] G.R.W. Quispel and H.W. Cape!, Local reversibility and local measure preservation in dynamical systems, Preprint Australian Nat!. Univ., May 1989. [10]T.Bountis,Part.Accel. 19(1986)181. [I!] R.H.G. Helleman, in: Long-time predictions in dynamics, eds. C. Horton, L. Reichl and V. Szebehely (Wiley—Interscience, New York, 1983). [121 M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett. 50 (1983) 1870; M. Kohmoto, B. Sutherland and K. Iguchi, Phys. Rev. Lett. 58 (1987) 2436. (1310. Gumbs and M.K. Ali, Phys. Rev. Lett. 60(1988) 1081;J. Phys. A 21(1988) L517. [14] F. Wijnands, J. Phys. A 22 (1989) 3267. [15] J. Bellissard, B. loehum, E. Scoppola and D. Testard, Commun. Math. Phys. 125 (1989) 527. [16]S. Aubry and 0. André, Ann. Israel Phys. Soc. 3 (1980) 133. [171 T.R. Taha and MJ. Ablowitz, J. Comput. Phys. 55 (1984) 192, 203. [181 B.M. Herbst and M.J. Ablowitz, Phys. Rev. Lett. 62 (1989) 2065. [19] C.G.J. Jacobi, see G. Birkhoff, Dynamical systems, Am. Math. Soc. Coil. Pub!. Vol. IX (1927). [20] E.M. McMillan, in: Topics in physics, eds. W.E. Brittin and H. Odabasi (Colorado Associated Univ. Press, Boulder, 1971) p. 219. [21] G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Physica D 34 (1989)183. [22] G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Phys. Lett. A 126 (1988) 419. [23] A.S. Fokas and M.J. Ablowitz, Phys. Rev. Lett. 47 (1981)1096. [24] G.R.W. Quispel, H.W. Capel, V.G. Papageorgiou and F.W. Nijhoff, in preparation. [25] F.W. Nijhoff, H.W. Capel and G.R.W. Quispel, lobe published. (26] F.W. NijhofT~H.W. Capel, G.L. Wiersma and G.R.W. Quispel, Phys. Leti. A 105 (1984) 267; H.W. Capel, G.L. Wiersma and F.W. Nijhoff, Physica A 138 (1986) 76. (271 V.0. Papageorgiou, F.W. Nijhoff and H.W. Capel, in preparation.
114
PHYSICS LETTERS A
2 July 1990
Integrable mappings and nonlinear integrable lattice equations V.G. Papageorgiou’, F.W. Nijhoff2
Department of Mathematicsand Computer Science and Institutefor NonlinearStudies, Clarkson University, Potsdam, NY 13676, USA
and H.W. Cape! Instituut voor TheoretischeFysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, The Netherlands Received 27 February 1990; accepted for publication 25 April 1990 Communicated by A.P. Fordy
Periodic initial value problems oftime and space discretizations of integrable partial differential equations give rise to multidimensional integrable mappings. Using the associated linear spectral problems (Lax pairs), a systematic derivation is given of the corresponding sets ofpolynomial invariants. The level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.
1. Nonlinear integrable lattices (i.e. integrable partial difference equations, cf. e.g. refs. [1—4]) are of fundamental importance for the study of classically integrable systems. They are generic in the sense that their various continuous limits give rise to hierarchies of integrable PDEs [5]. Furthermore, their study opens up some new points of view on classical integrability in general, in particular in connection with Hamiltonian structures for time-discrete systems [6,71. In this Letter we report on another application of such systems. In fact, we will show how these lattices give rise to nonlinear integrable mappings. Since many interesting problems in physics are close to systems that can be solved exactly, such mappings are useful for the study of bifurcations or transition to chaos, using perturbation techniques, cf. e.g. refs. [8,9]. In contrast to continuous systems, analytic nonlinear mappings have the advantage of exhibiting many of the complex phenomena of nonlinear dynamical systems while at the same time being computationally simpler. Concrete applications of mappings include the study of the stability of beam—beam collisions in storage rings [10,11], Anderson localization [12—151and incommensurate structures [16]. We also note their importance for the numerical integration of PDEs and the study of numerically induced chaos, cf. refs. [17,181. Probably the oldest nonlinear integrable mapping is the elliptic billiard due to Jacobi [19]. More recently McMillan found a four-parameter family ofrational mappings of the plane, together with their invanants [20]. An eighteen-parameter family generalizing the one of McMillan was presented by Quispel et al. in ref. [21]. Moreover, a connection with soliton equations of differential—differencetype was established, cf. also ref. [221. However, a spectral interpretation of the integrabiity of these mappings on the basis of a Lax pair was lacking. In this Letter we take a different point of view from the one expounded in refs. [21,22] by considering integrabte lattices rather than differential—difference equations as a starting point. This is convenient, because it allows us to obtain mappings from the consideration of an initial value problem on a two-dimensional planar 2
E-mail: [email protected]. E-mail: [email protected].
106
0375-9601/90/S 03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland)
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
lattice. Furthermore, we show that these mappings do indeed carry a spectral interpretation, and that the invariants can be calculated systematically from the monodromy matrix constructed from the Lax pair. 2. We shall use as prime example for the exposition of our ideas the lattice KdV. This equation (as well as other lattice analogues ofother integrable PDEs) was obtained in ref. [l} using a discrete version of the direct linearization method introduced in ref. [23]. Similar lattice equations have been proposed in the works of Hirota [3], cf. also ref. [4]. The lattice KdV we prefer to work with reads (1) (p_q+u_u)(p+q_i~+u)=p2_q2. (1) u=u(n, m) is the dynamical variable at the lattice site (n, m), n, me7L, and the and “are shorthand notations for translation on the lattice, i.e. ü=u(n+l, m), ü=u(n, m+ 1). p and q are the lattice parameters p, qeC. Eq. (1) arises as the compatibility condition ofa pair of linear problems (Lax pair) defining the shifts (translations) of an eigenfunction ~k in the n- and rn-directions, In
—
(2)
(p—k)~k=Lk~Pk, (q~k)r~k=Mk~k,
where Lk is given by
(
p—u
1
Lk=~k22+(+)() ~ (3) and where Mk is given by a similar matrix obtained from (3) by making the replacements p—’ q and The parameter k is the spectral parameter. Let us now consider initial value problems for (1) on the lattice. One way of doing this is to give initial date u( n, 0) on a horizontal line and consider either decaying, cf. ref. [5], or periodic boundary conditions, which in either case leads to a nonlocal scheme. Another way, namely the one that we will be concerned with in this Letter, which gives rise to a local iteration scheme, is to assign initial data on a “staircase” on the lattice. By a staircase we mean a sequence of neighbouring lattice sites with rn and n nondecreasing, as e.g. illustrated in fig. 1. From the fact that eq. (1) at each site involves only the four variables situated on the four lattice sites around a simple plaquette, it follows that the information on these staircases evolves diagonally through the lattice along “parallel” staircases. Furthermore, because of the “convexity” of the staircase the initial-value problem is well-posed. Although staircases of variable length and height stairsteps can be considered, for the sake of clarity we use two standard staircases: one for an even-periodic configuration of initial data, and the other for an odd one. In fact, one can show that several families of staircases give rise to equivalent mappings. Suppose we take the standard staircase through the origin (n, m) = (0, 0), and assign initial data on the lattice along this staircase as in fig. 1, ‘
~
u(j,j)=:a 2~, u(j+1,j)=:a2~+1 (jeZ) By applying the lattice equation (1) we can calculate the data on the next staircase one vertical step down, the (multiple) primes denoting the various iterations of this procedure. Thus, performing iterations by updating the lattice variables u along a vertical shift in the rn-direction, we define u(j,j+l)=a~,,,
uU+l,j+l)=a~,+1.
We then obtain from eq. (1) the mapping a~=a2~÷1—d+
~—a2J+2 +a2J
,
a~,+~=a2~÷2,
‘
(4)
in which o=p—q, E=p+q. By introducing now the differences Y~_=a2~÷2—a2,, .X,ssa2~+i—a~j, eq. (4) can be reduced to the measure-preserving rational mapping 107
Volume 147, number 2,3
PHYSICS LETTERS A
o
~a
2 July 1990
_____
0 a~
o
:
~-~-----------~
Fig. I. Standard staircase of periodic initial data on lattice, with even period 2P.
X~=Y~,Y=xj+~+~
—4.
(5)
It seems that the reduced variables Xi,, Y~have a deeper physical meaning than the unreduced ones, namely the a2~,a2~+1,because in the former we get rid of possible constant background contributions that come in via the lattice equation after iteration. We impose now the even penodicity condition a20+p)=a~,a2u+p)+I =a~÷i. Note that P (P=2, 3, ...) can be interpreted as the period along the two diagonals of the lattice corresponding to the staircase. It is easy to see that these periodic conditions are compatible with the lattice equation, and hence will be preserved after iteration of the mapping. This means that we have to supply (5) with the periodicity constraints ~‘ P
P—I
>~X~=0, ~Y1=0. J=I
(6)
J=O
~ It is possible to relax somewhat the condition of periodicity (6) by allowing fora constant on the right-hand side. This corresponds to a shift from a center-of-mass situation to one in which total momentum is this constant. We will not treat this situation in the present note.
108
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
Eq. (5) for j= 1, 2, P together with the constraint (6) is a 2 (P— 1)-dimensional integrable mapping. The simplest case P= 2 corresponds to the well-known McMillan mapping [20]. The McMillan map was also obtained as a reduction of the discrete nonlinear Schrodinger equation and also as a special case of a 18-parameter family of two-dimensional integrable mappings [21,22]. All mappings obtained in this way exhibit P— 1 nontrivial integrals which can be found in a straightforward way by exploiting the Lax representation (2) ofthe original lattice KdV. In order to do this we need to construct a monodromy rnatrix .~, i.e. an ordered product of Lax matrices Lk and Mk along the staircase over a oneperiod distance. Let us introduce the monodromy matrix in a way that is also suitable for the odd-period case. We work with lattice parameters p~at every joint of the staircase labeled by j (i.e. the jth link on the staircase counting from the origin) to be specified later. Thus, defining ...,
n fl L,,(a~f~,a,),
N- I
.~(aN,aNl,...,aI,ao)m
(7)
v=O
where the (1 indicates that the factors in the product are arranged from right to left, and in which the translation matrices Lk(a,+l, a,) represent either the Lax matrix in the n-direction (v even, p,~=p), i.e. Lk, or the Lax matrix in the rn-direction (v odd, p,~= q), i.e. Mk. They are of the form
Lk(a,+l,a,)=(k2—p~+I)F+L~+I,,F=(? ~),
(8)
in which the matrices
L
jp,+i—a,+i *
1 p~+a, 4_1
(* stands for the product of the diagonal entries) have the nice property that any product ofa connected string of them remains of the same form up to a factor, namely /N—I LN,N_I
LN_ 1,N_2...L2.I L1,0 =(
fl
(p, +P,+i
+ a,_1
—a,~1) )LNo.
(9)
\v=I
By construction, the monodromy matrix (7) has a particularly simple behaviour under application of the mapping, namely ...,
a’s, a’0) =Lk(a~,,~
)~(aN,
...,
aO)Lk(a~,a0)~=Lk(a~,ao).~(aN, aO)Lk(a~,a0)~ (10) ...,
where in the second equality use has been made of the compatibility condition of the Lax representation and in the last equality of the periodicity requirement. From eq. (10) follows that the trace Tr(~) is invariant under the mapping (5), andtohence we can mentioned generate F— 1 invanants by developing trace in powers of calthe 2. Due the property above for the matrices La,, it the is particularly easy to spectral parameter k culate these invariants. The recipe is as follows: if no contributions from the terms containing the matrix F are taken into account, the trace will be given by the trace of a product of the form (9), which in virtue of the periodicity constraint will yield simply the product N Tr(LN,N_I...L 2,ILI.O)=
fl (p,+p,÷1+a,_1—a,+1), aJ+N=aJ,
Pj+N=Pj-
Furthermore, for every contribution of a factor containing F we have to any twoover adjacent 2, replace k2), any trace one orfactors more ~
~+ a,_1
—
a,÷1,
...~
+p, + a,2
—
—
a, in this product by a factor (p 109
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
2
0
—1
Fig. 2. Projection onto the (X
1, X2, Y1) subspace of the orbit of
-2
I
-2
I
-1
2
1
the four-dimensional mapping (5), (6) forP= 3, ~=4.0, J= —0.5, andinitialvaluesX~=0.3,Y1=0.6,X2=0.4, Y2=—0.9.
matrices F in combination with strings of matrices L,~,,yielding either 0 due to the nilpotency of F or 1, apart from factors of the form given in eq. (9). We find as a result N
Tr{.~(aN,...,aI,aO)}=
fl
N
N
fl
(p~+p~+1+a~1—a,~1)+ ~ (k~—p~)
p_i
j~i
P#j—
2—p~)(k2—p~)
N
+
~
flN
(p~+p,+1+a,1—a9÷1) Li
(p~+p~+
(k
1+a~1—a~÷1)+....
(11)
v=I v,’i—I.i,J—l.J
(The exclusions v~1— 1, 1 and v~j—1, j in the products occurring in (11) are to be understood mod(N)). In the case of the McMillan map (N=2P=4), eq. (11) yields the integral of the mapping 2—X~)(e2—Y~)—2edX 5=(e 1Y1 , (12) which is an elliptic curve that can be parametrized in terms ofJacobi elliptic functions, thus leading to explicit solutions to the corresponding mapping. For N= 2P= 6, eq. (5) together with (6) yields a four-dimensional map in terms of variables X1, X2, Y1, Y2. The invariants are calculated from (11) as
2(x
J~=y~x1y1x~y2x3 —ed(x1ytx2y2 +x2y2x3y0 +x3y0x1y1)+ (ed)
1y1 +x2y2 +x3y0),
f2=y0x1y1x2y2x3+eö(y1x2y2x3+y0x1y2x3+y0x1y1x2)+(eö)
2x3+y0x1 +y1x2),
2(y
(13)
in which we have used the abbreviations x~me—X~, y,me—Y3
(1=1,2),
y0me+Y1+Y2, x3me+X1+X2.
By way of illustration we have plotted in fig. 2 the projection on the X1, Y1, 12 space of a typical orbit of the P= 3 mapping in fig. 2. The orbit lies on the intersection of the two hypersurfaces given by the level sets of the two invariants J~and .~.This and similar pictures suggest that it is a torus in four dimensions, which would correspond to the analogous situation for continuous finite-dimensional integrable systems. 110
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
3. The above given construction to obtain integrable multi-dimensional mappings from integrable partial difference (lattice) equations works equally well in cases of examples other than KdV, such as the modified KdV (mKdV) or the Toda equation with discrete time. Let us consider e.g. the equation 3] =V[ (q+r)O— (p+r)ii] (14) v[ (q—r)IY— (p—r)1 .
Eq. (14) is a mixed version ofthe lattice mKdV (r= 0) and a version ofthe discrete-time Toda equation (r=p), cf. e.g. ref. [1]. The parameters p, q denote as before the lattice parameters. Eq. (14) arises as the compatibility condition of the following linear system, (p—k)
( (k2_r2)/v p—r
kfIk!f’k
( 15
1) (p+r)~/v)~’
and a similar equation for the translations in the lattice direction indicated by that is obtained from (15) by replacing —i~” and p—4q. Eq. (14) is related to the lattice KdV (1) via a Miura transformation as follows, (16)
p—q+u—u= (p+r)i~—(q+r)O which on the level of the linear system reflects in a gauge transformation of the form Uk(k2
(k2—r2)!t’k=Ukøk,
~)‘
2
(17)
in which s= (p—ü)v— (p+r)1, and s= (p—r)v— (p+u)1, leading to UkFUk’=(k2—r2)
54=UkLkUkI,
Em(~
‘yE
~).
(18)
Assigning in precisely the same way as before initial data on staircases as the one depicted in fig. 1, namely ~(j,j)_—:b ~(j+l,j)=:b2~~1 v(1,j+l)=b,, v(1+l,j+l)=b’~,,÷1, 2~, we find by applying the lattice equation (14) I’.’ £~ L/2j—L/2j+I
(p+r)b2~+2+(q—r)b2~ I” ~s.
~
~
_
~
~p—r,~2~--~~,q-r-r~u21~2
~
U2j+1U2j+2.
Introducing now the reduced variables W= b~+1/b2~1,Z~=b2~+2/b2~ we obtain the following measure-preserving mapping,
w’-z i_i,
z’-w j_
Z,÷1-FPZ~,+a
2
in which p (q—r)/(p+r) and a= (p—r)/(q+r). The periodicity condition that we can now impose takes the following form, P
P—I
flW~=l, flZ1=l. j—.O
j=1
(21)
Eq. (20) together with the conditions (21) is as before a 2 (P— 1)-dimensional mapping that exhibits F— 1 invariants, that can be constructed in the same way as in the case of the lattice KdV by using the monodromy matrix The invanants are easily constructed from the ones of the KdV by using the Miura transformation (16). Alternatively, we can go through the same procedure as before, making use of the fact, that the Lax matrices that we need in the present case to calculate the monodromy matrix (7), replacing a,’s by b,’s, are of the form Ill ~.
Volume 147, number 2,3
PHYSICS LETTERS A
~
2 July 1990
M~,v=((+~/b
(the
bv÷iI(Pv+lr))
(22)
stands for the product of the off-diagonal entries), which have a similar property as (9), namely that
*
MN,N_IMNI,N
2...M2,IMI,o= fi(l+ ,=i
(23)
~ b~.±i)M p~+i—rb~_1
The monodromy matrix is easily expanded as in eq. (11), the leading term being again given by a product like the one in2—p1)b the r.h.s. of eq. (23), contributions of a factor containing the matrix E yielding an insertion of a factor (k 3+1/b~1,leading to Tr{.9~(bN,...,bl,bQ)}=
+~[(k2_pI)~t1
LI1
(pv÷ir+(pv+r~tL)]
v,&j— I,j
+
~
[(k2p~)(k2p1)~~’-±-L
b,_1 b~_1
fl
(pv+ir+(Pv+r)~t!)]+.... b~_1
(24)
v,&i— I,x,j— I,j
(The exclusions in the products occurring in the r.h.s. of eq. (24) are to be understood mod (N) as before.) For example, in the case of period N= 2P= 4 we find the invariant f=p(~+.~~)+a(WiZi+ W1Z)+(l+ap)(WI+~~+ZI+~~).
(25)
4. So far, we have been dealing with the case of even periodic conditions along a staircase as in fig. 1. In the case ofodd period N= 2P+ 1, some modifications of the scheme outlined above are necessary. A standard staircase corresponding to this situation is given in fig. 3. Note that in order for the periodicity of the data to be preserved when iterated by the mapping we must allow for a stairstep of length 2 at every point where a new sequel of data of the form a0, a1, ..., a2p÷1begins. Now, to obtain the mapping (we shall illustrate this case by means of the KdV lattice), we can use again the formulae (4) forj= 0, 1, ..., F— 1, except that now at the end of the string of data, we have to replace the formula by a’2~=a2~÷1—ô+ =a~—ô+ ,~ , —(ao—a2p) E—(ao—a2p,)
(26)
.
We use the same variables as before, i.e. X~=a2~+1—a2,1,j=l, F, and Y3=a2~+2—a2~,j=0, ..., P—l, but now we have no longer the periodicity conditions (6). Noticing that ...,
P
a2p—aj
=
P—I
E X~+~ f—_I j=0
we can reduce the system (4) together with (26) in terms of X~,Y~.The reduced mapping turns out to be 112
Volume 147, number 2,3
PHYSICS LETTERS A
a1
_____________
a’0
a~=
2 July 1990
0
a
a2
0
0
0
C
I a2p
a2p+i =
ua’~p
0
0
a0
a~
-e
-~
a1
-~
0
Fig. 3. Standard staircaseof periodic initial data on the lattice with odd period 2P+ I.
X~’=Y~,j=l,...,P—l, Yj=Xj+I+
——---v-, E—ij
~ E—~+i
C—
th Y~1=X~+ E~4p
—
Yo
_______
EIp1
.
(27)
We 2can use P again the formula trace ofFor theexample, monodromy matrix (11), when yields invariants underfor thethe mapping. in the case of P= which 1 we find theexpanded invariantin terms of k J=(e— Yo)(e—X 1)(e+X1 +Y0)+eô(X1 +Y0), (28) which can be parametrized in terms of Weierstrass elliptic functions. 5. In this Letter, a general prescription is given to obtain integrable mappings from the consideration of initial value problems of integrable lattice equations. (A different point of view, considering travelling-wavesolutions of partial difference equations is adopted in ref. [241.) The results obtained so far can be generalized to other 113
Volume 147, number 2,3
PHYSICS LETTERS A
2 July 1990
types of integrable lattice equations such as the lattice BSQ and MBSQ, cf. ref. [25], or KP and MKP, cf. ref. [26], equations. In all such cases one expects that integrable mappings can be obtained, with an explicit construction of their integrals in much the same way as we have done explicitly in the present note for KdV type of mappings. Work towards this direction is in progress. Another aspect that is of interest in connection with the mappings that we construct is the question of their canonical structure. Recently, a theory of Lagrangian and Hamiltonian mappingshas been developed by Veselov [6], cf. also ref. [71.It would be of interest to investigate whether this theory applies also to the mappings we introduce in the present Letter. In particular it would be of interest to give a characterization of the intersection of the level sets of the invariants that we find, in terms of multidimensional tori. This would lead to an explicit parametrization in terms of theta functions. This work is also under current investigation. Finally, we would like to point out that the mappings we construct pose a well-defined continuum limit, in much the same way as the original lattice equations do, cf. [1,2]. This limit yields integrable systems of ODEs which we will present elsewhere [27]. The authors (VGP and FWN) would like to thank Professors L. Kauffman, E. Previato and Y. Saridakis for their interest, and Professor S. Fulton for his help in producing fig. 2. VGP was partially supported by AFOSR Grant No. 86-0277.
References [1] F.W. Nijhoff, G.R.W. Quispel and H.W. Cape!, Phys. Lett. A 97 (1983)125. [21G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden, Physica A 125 (1984) 344. [3]R. Hirota, J. Phys. Soc. Japan 43 (1977) 1424,2074,2079; 50 (1981) 3785. [4] E. Date, M. Jimbo and T. Miwa, J. Phys. Soc. Japan 51(1982) 4125; 52 (1983) 388, 766. [5] G.L. Wiersma and H.W. Capel, Physica A 142 (1987)199; 149 (1988) 49, 75; Phys. Lett. A 124 (1987)124. [6] A.P. Veselov, Funct. Anal. App!. 22 (1988) 83; Theor. Math. Phys. 71(1987) 446. [71J. Moser and A.P. Veselov, Preprint ETH (ZUrich), 1989. [8] M.L. Glasser, V.G. Papageorgiou and T.C. Bountis, SIAM J. App!. Math. 49 (1989) 692. [9] G.R.W. Quispel and H.W. Cape!, Local reversibility and local measure preservation in dynamical systems, Preprint Australian Nat!. Univ., May 1989. [10]T.Bountis,Part.Accel. 19(1986)181. [I!] R.H.G. Helleman, in: Long-time predictions in dynamics, eds. C. Horton, L. Reichl and V. Szebehely (Wiley—Interscience, New York, 1983). [121 M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett. 50 (1983) 1870; M. Kohmoto, B. Sutherland and K. Iguchi, Phys. Rev. Lett. 58 (1987) 2436. (1310. Gumbs and M.K. Ali, Phys. Rev. Lett. 60(1988) 1081;J. Phys. A 21(1988) L517. [14] F. Wijnands, J. Phys. A 22 (1989) 3267. [15] J. Bellissard, B. loehum, E. Scoppola and D. Testard, Commun. Math. Phys. 125 (1989) 527. [16]S. Aubry and 0. André, Ann. Israel Phys. Soc. 3 (1980) 133. [171 T.R. Taha and MJ. Ablowitz, J. Comput. Phys. 55 (1984) 192, 203. [181 B.M. Herbst and M.J. Ablowitz, Phys. Rev. Lett. 62 (1989) 2065. [19] C.G.J. Jacobi, see G. Birkhoff, Dynamical systems, Am. Math. Soc. Coil. Pub!. Vol. IX (1927). [20] E.M. McMillan, in: Topics in physics, eds. W.E. Brittin and H. Odabasi (Colorado Associated Univ. Press, Boulder, 1971) p. 219. [21] G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Physica D 34 (1989)183. [22] G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Phys. Lett. A 126 (1988) 419. [23] A.S. Fokas and M.J. Ablowitz, Phys. Rev. Lett. 47 (1981)1096. [24] G.R.W. Quispel, H.W. Capel, V.G. Papageorgiou and F.W. Nijhoff, in preparation. [25] F.W. Nijhoff, H.W. Capel and G.R.W. Quispel, lobe published. (26] F.W. NijhofT~H.W. Capel, G.L. Wiersma and G.R.W. Quispel, Phys. Leti. A 105 (1984) 267; H.W. Capel, G.L. Wiersma and F.W. Nijhoff, Physica A 138 (1986) 76. (271 V.0. Papageorgiou, F.W. Nijhoff and H.W. Capel, in preparation.
114