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Lax pairs, symmetries and conservation laws of a differential-difference equation—Sato's approach
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Lax Pairs, Symmetries and Conservation Laws of a Differential-Difference Equation-Sato’s Approach S. KANAGA
VEL
and K. M. TAMIZHMANI
Department of Mathematics, Pondicherry University, Kalapet, Pondicherry 605 014, India (Accepted 5 November 19%)
on the known elementary introduction of Sato theory for differential equations. the differential-difference equation which belongs to the single component KP family has been considered in the framework of Sato theory. We show that in this natural framework, Lax pairs, symmetries and conservation laws can be obtained in a systematic way. 0 1997 Elsevier Science Ltd
Abstract-Based
1. INTRODUCTION
Following its discovery, the inverse scattering transform (ET) technique has proved to be an effective method for solving non-linear partial differential equations having soliton solutions [l, 21. These types of equations possess common many properties: existence of N-soliton solutions, infinitely many conservation laws, symmetries, Backlund transformations, Hamiltonian structures, Hirota bilinear forms, Painleve property etc., and have been classified as completely integrable systems [l-17]. Many integrable systems in (1 + 1) and (2 + 1) dimensions have now been found [l, 2, 18,191. Following this, attempts have also been made to formulate the above ideas for differential-difference (DA) equations [l&35]. The search was made for the existence of a unified approach to derive these equations and their integrability properties. Sato presented a remarkable theory which clarified the algebraic structures involving the integrable systems [36, 371. His approach, based on considering the pseudo differential operators in an infinite number of dependent variables, treats integrable systems as a flow in terms of an infinite set of time variables on a Grassmanian manifold and shows that the bilinear form is nothing but the Plucker relations for the Wronskian. Following this discovery, Date et al. developed a group theoretical framework for soliton equations and made a large classification of bilinear equations of soliton type [38, 393. In fact, using the same group theoretical framework under various assumptions, Date et al. also derived integrable equations in continuous, differential-difference and fully discrete cases WIRecently, much attention has been paid to utilizing the best possible level of Sato’s theory [41-471. For example, in an excellent paper, Ohta et al. explained the Sato theory in an elementary way [41]. More specifically, KP, multicomponent KP, constrained KP etc., have been discussed in Refs [42-451. and the Sato equation has been used to derive symmetries and conservation laws of KP and multicomponent KP equations. Motivated by these works [41-431, in this paper we study the DA version of Lax equations, symmetries and conservation laws through Sato’s theory. At this point it is important to mention the work of Ueno and Takasaki on a Toda lattice [47] and Oevel’s work on lattice systems [48]. 917
S. KANAGA
918
VEL
and K. M. TAMIZHMANI
The Sato theory presented in the following sections for the differential-difference equation is developed in the same sense as that of the continuous case [41, 421.
2. PRELIMINARIES
We begin by introducing the pseudo difference operator of the form W = 1 + w,A-’ + w2A-’ + W~A-~+ w4AP4+ wsAPs+ q-1
(1)
where A-’ is the formal inverse of the forward difference operator A defined by At(x) = j-(x + 1) -f(x). Th e coefficients, w,, are functions of {x:x E Z}. We also look for the inverse operator W -’ in the form of a pseudo difference operator which is expressed by W-’ = 1 + u,A-’ + u~A-~ + u,AP” + u4AP4+ usAP5+ . . ..
(2)
Now it is easy to compute the relations between the coefficients, w,, of W and ui of W ’ using the fact WW-’ = 1 holds. First we list a few of them below: u,= -w, u2= W’E-‘WI - w2
(3)
ug = - W’E-‘w, + WlE-2Wl - W,E-‘WlE-2Wl + w,.!-‘w,
+ W2E-*Wl - WJ
where E-’ is . the inverse of the shift operator E defined by Ef(x) =f(x + 1). Note that A and E are related by A = E - 1. Now we restrict ourselves to the finite number of terms in W. Accordingly we denote W,=~+W,A-‘+W~A-~+~~~+W,,,A~~.
(4)
Now it is obvious that using the operator W,, multiplying A” from the right, we can construct an mth order linear difference equation of the form W,Amf(x) = (Am + wlAm-’ + e.. + w&x)
= 0.
This equation has m linearly independent solutions f”‘(~), fC2’(x); * +,f’“‘(x). gives Am-‘f(‘)W, + &7-*f(‘)W2
+
...
+ f”‘wn,
=
+ A”-2f@“)W2 +
...
+ f’““w,,
= _ Amf(m).
c-9 Equation (5)
- Amf(‘)
(6) A’-‘f(m)Wl
It is easy to find the solution to these simultaneous equations in the form - A”f”’
...
f(l)
.. .
.
Am-‘f(l)
1..
Am-‘f(m)
._.
_ Amf(m)
. . . f(i)
A”-‘f(l)
... ... ...
Am-/f(‘)
. . . f(l) ... . . . . ,,i
...
w, = -
A+
Am-&)
(7)
Lax pairs, symmetries and conservation laws of a differential-difference equation
and hence
919
f (HI) A-,’
&,f(m, A-’ A”f’“’
1
(8)
f (ml
. ..
...
We assume that the set of linearly independent solutions f’(x) of the mth order linear difference equations can be expanded using the Newton-Gregory formula (9) where A’f”)(O) = 5, and XC, = x(x - l)...(x - r + 1)/r!. Now, the difference equation can be . written m the form) W,,A”(lxC,xCz...)@ = 0 (10) where (11) Let A be the shift operator given by 0
0 1 0
...
On using this we write
i: xc*nr = r=o Then we define.
1 xc,
xc2
1
xc, . 1:‘.
i 0
...
1
... . . ...
H(x) = i: xC,A’@ = r=O
(13)
(14)
The determinant of the first m rows of H(x) is nothing but the denominator in W, which is a Casorati determinant for the solution of the difference equation (5). 3. TIME EVOLUTION
In this section, we introduce infinite number of time variables t = (t,,f2,f3; . .) in w; = w;(x,t), (j = 1,2; . e). As a consequence of this, we have
f (A= f”‘(x;t) = f qx,t, ,t2;..).
(15)
S. KANAGA
920
VEL
and K. M. TAMIZHMANI
We consider the time evolution of H(x) in the form W-w)
= (i: (CK)) r=O
exp 77(uW
(16)
where n(t,A) = X~=,t,h”. We write formally, exp n(t,A) = 2 pnAn. n=O
(5 WN)) r =0
When we compare the coefficients of equal powers of A on both sides we get PO=1
(17)
(18)
pj =x+r,
x(x - 1) P2= ___2r + xr, +; P3 =
(r: + 2t,)
x(x - 1)(x - 2) +x(x - 1) tl + b (t: + 2r-J~ + k (t: + 6r, r2 + 6t3) 2! 3!
p = -4x - 1)(x - 2)(x - 3) + x(x - 1)(x - 2) r + 1_ (rf + 2r )X(X - 1) 2 4 2! ’ 2! 3! 4! + $ (t; + 6t,tz + 6t3)x + -jj (t’: + 1215, + 12r2,+ 24t,r, + 24r,).
These polynomials are called Schur polynomials or Plucker relations. It can be seen that aPI3 ;tt,=Pn-nrJ
pn=O, foralln
(19)
and (20) APn = Pn-1. Now, it is possible to express the function H(x;r) in terms of pi, which is written in the form
H(x;r) =
It is noted that and h!j)(x;r) =
ahg)(x;r) = A”h&j)(x;r). ar n
(23)
We infer from this h(x;r) = l@(x;r), n = 1,2; . . and j = 1,2; . ., m is the solution of a set of linear partial differential-difference equation ct-
8” h(x;r) = 0, n = L2,e.a
>
(24)
Lax pairs, symmetries and conservation laws of a differential-difference equation
921
with h(x;O) =f@(x). Then, we have WmAmhhi)= 0, j = 1,2;..,n.
(25)
Solving the system of equation (25) we get, A*-‘/&”
...
- A”#’
...
h&”
Am-&Tl,
:::
: - A”h,f’“’
:::
hr’’
A*-‘/&‘)
...
A’+/&‘)
...
hi,”
A”-$/)*’
.. . ...
A”-‘/.@;“’ ’
:::
’ hhmT”
w, =
and hence
hb” A,&($ A”‘#
. ..
hv’
A-”
1:: A”- i hh*;“’ A-’: . ..
Amhhm”
w*= g)”
(26)
...
1 I (27)
hb””
Now wj and W, are completely given in terms of Schur polynomials p,,. It is clear that pn play a crucial role for further developments of the Sato theory.
4. SATO, LAX AND ZAKHAROV-SHABAT
EQUATIONS
It is well known that the integrability of the non-linear systems is associated with finding of the appropriate Lax or Zakharov-Shabat equations [l, 21. In this section, we present the Lax and Zakharov-Shabat equations for DA formalism from Sato’s equation. By differentiating equation (25) with respect to t,, we obtain dW* A”/#’ at n
+ W,,A” %
= 0, n
(28)
Using the property (dh,fj’)/(dc,,) = A”hg’(x;t), we get c
aw,m at A + WmAmAnhj/‘(x;t) = 0. ) n
(29)
The operator in parentheses can be factorized as 8W --E A” + W, AmA” = B, WmA”‘, %I where B, is an nth order difference operator. B,s can be obtained by applying A-“W;’ the right-hand side of the equation B,=L
dW at W,’ + W,A”W-’m n
(30) to
(31)
922
S. KANAGA
VEL and K. M. TAMIZHMANI
From equation (31) we can write
aw = B, W, - WmA”. at,
m
This is true for W defined in equation (1) and hence
aw
-=B,W-WA” at, which is the famous Sato equation. The B,s in the Sato equation can be computed from B, =(WA'T')+ (34) where ( )’ denotes the non-negative powers of A only. Hence, we have discarded the first term of equation (31) because it involves only negative powers of A, where as B, consists only of non-negative powers of A. Below we give the first few B,‘s explicitly: B,= A-Aw, B2 = A'-(2Aw, +A'w,)A +(-2Aw,A2w, +Aw,A*w, +~(Aw,)~ + w,A2w,+2w,Aw,-A2w2-2Aw2) B3= A3-(3Aw,+3A2w,+A3w,)A2+(-3Aw,-6A2w,-3A3w,+A3w,A2w, +2A3w,Aw,+w,+A3w,+3(A2w,)2+9A2w,Aw,+3w,A2w,+6(Aw,)2 +3w,Aw,-A3w23A2W23Aw2)A+(-3A*w,-3A3w,+2A3w,A2w, +~A'~,A~,+~w,A~w,+~(A~~,)~+~~Aw,A*w, +6w,A2w,+6Aw,A2w, + ~(Aw,)~ + ~w~AwI - 3A3wz- 6A2w2- 3AW2 - Aw~A~w~A’w~ (35) -3Aw,(A2w,)2-2A3w,(Aw,)2-9A2w,(Aw,)2-6(Aw,)3-9w,(Aw,)2 -12w,Aw,A2~,-~,A2~,A3~,-3w,(A2~,)2-3~,A~,A3~,-3w:A~, -3w:A2w,-w:A3w,+A3w,A2 w2+2A3w,Aw2+w2A3w,+3A2w,A2w2 +6A2w,Aw2+ 3W2Li2W,+ 6Aw,A2w2+9Aw,Aw2+ 3W,fiW,+ A3w2Aw, +w,A'w,+ 3W,h2W,+3W,i\W2A3w3-3A2w3 -3Aw3) To determine the Lax equation we define L = WAW-‘. (36) Differentiating equation (36) with respect to t,, we get:
aL aw -=--AW-‘+WAc at,
at,
at,,
= B,L - LB, = P,,Ll where we have used the Sato equation and WW-’ = 1. Also note that B, = (I.“)‘. (37) is called the generalized Lax equation. From equation (37) we have $= It is evident that
al,“’ at,
n
--at=
(37) Equation
[B,,L”‘], m,n = 1,2;...
(38)
aLn [B,,L”‘] - [B,,,L”]. “I
(39)
We denote B’, = B, - L” which contains only terms with A-j, j > 0. Using this relation in equation (39), we arrived at: y-F= ”
m
[B,,B,,] - [BCn,BC,,J.
(40)
Lax pairs, symmetries and conservation laws of a differential-difference equation
923
Considering the difference part on the both sides we find that aB ,-aB,= at,
[B,,B,]
at,
(41)
which is the Zakharov-Shabat equation.
5. DIFFERENTIAL-DIFFERENCE
EQUATION
IN A SINGLE
COMPONENT
KP FAMILY
The power of the Sato theory explained in the previous sections will be demonstrated in this section by deriving explicitly the DA equation which belongs to a single component KP family [40] and its Lax pairs. The first step towards achieving this goal is to write down the operator L defined in equation (36) in terms of Uj, j = 0,1,2; . . . For this purpose, we substitute W and W-’ defined in equations (1) and (2) in L and obtain: L = A + u0 + u,A-’ + u~A-~ + ...
(42)
where /A,,= - Aw, u, = - Aw, - Awz + w,Aw, u2 = - Aw2 - Aw3 + Aw,E-‘w, + w2AwI + E-‘w,Aw2 - w,Aw,E-‘w, u3 = - Aw‘, - Aw3 + w3Aw, + Aw3E-‘w, - Aw2E-‘w, + ~Aw~E-~w,
(43)
- Aw,E-‘w, E-*w, + Aw~E-‘w2 - Aw, E-‘wZ - w2Aw,E-2~, - w,Aw, E-‘w, - Aw, E-‘w, + Aw, E-*w, - Aw, E-‘w,E-~w, + w,Aw,E-‘w, - w,Aw,E-‘w, + w,Aw,E-‘w,E-~w,. From B, = (L”)+, it is now obvious that B, =A+uO B, = A* + (2u,, + Au,JA + (Au0 + U; + 2u, + AU,) B3 = A” + (3uo + 3Au, + A2u0)A2+ (2A2u0+ 3Au, + 3~; + 3u,,Au,, + (Aw,)~ + 3u, + 3Au, (44) + A2u,)A + (A2u0+ 5u,u, + 3u0Au0 + u30+ (Auo)’ + Au,,Au, + 3u&, + UIAUO + u,E-‘u.
+ 2A2u, + 3Au, + 3U2
+ 3h42
+
A2u2).
Now for the given L and B,, we can find the hierarchy of equations. For n = 1 we find the following equations:
$=Au, I
+Au2+u0u,
-u,E-‘u.
F= Au, + Au3 + uDu2+ u, E-lug - u, E-2~o - u2E-*u, 1
(45)
S. KANAGA
924
VEL
and K. M. TAMIZHMANI
Also, for II = 2, we have: -
= A2u, + 2Au, + A2u2+ u,AuO + 2u0Au, + AuOAu, + uOu, - u,E-‘u,
(32
-
= A2u, + 2Au2 + 2A2u2+ 2Au3 + A2u3+ 2u,,Au, + AuOAu, + 2u,,u,
(46)
at2
+ u2Au0 + 2u0Au2+ AuzAuO+ u,Au” + u,u; + u: + u,Au, - U,E-224” + u,E-‘uo - u,(E-‘uO)2 - u,I!-‘u,
- U*E’4)
-
U2E-2Uo.
Now we consider the first two equations from the set given in equation (45) and the first equation in equation (46). Solving these equations for u0 we arrive at the DA equation in a single component KP family $+2$-2u,? 2
=(2+A)%. 1 >
I
(47) I
This equation is also been derived by Date et al. in a different approach [40]. It is obvious that equation (47) can also be derived from the Zakharov-Shabat equation for B, and B2.
6. CONSERVED
QUANTITIES
AND GENERALIZED
SYMMETRIES
As we mentioned in Section 1, the existence of an infinite number of conservation laws and generalized symmetries is the basic property for integrable equations in a continuous case [3-81. This is also true in the DA case [25-291. Recently, this property has been established for a two-dimensional Toda lattice [44]. In this section we present a detailed investigation for finding the conserved quantities and generalized symmetries for the DA equation (47). Before doing so, we give a brief review of definitions in this theory. We consider an evolution equation: u,=K(u) (48) where K is a vector field defined over the space M defined by M = {u = (u,, u2; . ., u,)]u: 2 + R”}. We call the functional S(U) a symmetry of equation (48) if it satisfies the linearized equation given by: St = K’tu)Pl,
(49)
where the Frechet derivative K’(u) is defined by: K’(u)[S]
=;
K(u + ES) r=O
It can be shown that a symmetry S must satisfy: [S,K] = S’[K] - K’[S] = 0.
This means that any symmetry S commutes with K(u).
(51)
Lax pairs, symmetries and conservation laws of a differential-difference equation
925
The linearized equation of the DA equation (47) is: $B2,0$2~$+2!$(2+~~)~-$=0. 2
1
1
(52) 1
I
The solutions of the above equation are nothing but the symmetries of equation (47). We use the eigenvalue problem and its adjoint to find the symmetries instead of solving equation (52) directly. In the continuous case, the recursion operator has been obtained by many authors using the squared eigenfunctions of the eigenvalue problem [6, 71. It is natural to expect in the DA case that we can also use the eigenfunctions of the linear problem and its adjoint to derive symmetries and conserved quantities [42-451. In this paper, we follow the procedure developed in [42]. Consider the following linear eigenvalue problem: L*=A+
(53)
$=B& n
(54)
with A[,,= 0. Notice that L = WAW-‘. Hence, we rewrite equation (53) in the form: (55)
Wo = Wo
where I+$,= W-‘I/J. The above equation (55) is just a first-order difference equation whose solution is given by: $” = g(t1,f2,**.;A)(1 + A) (56) where g(t,,t,; - -; A) is an integration function. From this result it follows that: $ = Wg(t,,t,;..;A)(l
+ A)
= (1 + w,A-’ + w~A-~ + .a.)(1 + A)xg(t,,t2;.-;A)
*=(
(57)
(1 + A)"g(t,,t2;..;A)
In deriving the above we have used the fact: A-‘(l+A)“=(1+
1..
j=12 A"
".
Again we can rewrite equation (54) using B, = L” + B’, in the form: 2 = (L” + B’,Ji,/i n We recall that B”, consists only of the negative powers of A. After tedious calculations, it is possible to express A-j, i > 0, in terms of the negative powers of L. Then $ = (L” + q&’ n
+ q2P
+ ..)$
where qj are appropriate functions of A,tl,t2; . *. On using LI,!I = AJI we obtain:
a* A”+&+%+... -= A A2 at,
(61)
S. KANAGA
926
VEL and K. M. TAMIZHMANI
This implies
a1%* -=*“+&+42+ at, h
A2
(62)
.**
This is true for any integer n > 0. On integrating the set of equations, we finally find (63) where qj again are appropriate functions. Comparing equation (57) with equation (63) at t, = 0, Vj = 1,2; . *, we get
*=( I+:+:+...
)(l + A)” eXp
(i*jtj). ,=I
Also, the adjoint of IJJcan be obtained by
**=(1+.+F+ wf ...)(l
+ A)-“exp ( -
Ch’f,). /=I
In order to derive the conservation laws of the DA hierarchy, presented here we express B, in powers of L. That is B, = L” + u(l”‘L-’ Therefore,
+ 4mm)p + . . ..
(*I) up a* /y+“‘+-+ -= ..* q/. A A2 1 at, (
(66)
(67)
From which we get &n) --_ a@% ‘4) at,
h”
(68)
where Jm) = c+)h-i. J=I
(69)
Differentiating equation (68) with respect to the time variable t,,, we will arrive at the conservation law (70) We list the first few conserved quantities corresponding to B,: (i) B,: fll
(1) =
up=
-u, -u,E-‘uo-242
a$“= - u,(E-‘u~)~ + u,E-‘u, - u,E-~u~ - u,E-‘IA, - u~E-~u, - u,E-‘u,, - u3 CT!‘)= - u,(E-‘u~)~ + ~u,(E-‘u~)~ - 2u,E-‘u,E-‘u, - u,(E-~u~)~ - u,E-~u~E-‘u, - u,E-‘u, + ~u,E-~u~ - u,E-~u~ + u,E-‘u, - u,E-~u, - u,E-‘u2 - u,E-‘u~E-~u~ - u~(E-~u~)~ - u~E-‘u~E-~u~ - uZ(E-‘u,,)~ + u2E-‘uO - u2E-‘u’ + u2E-‘uO - ~u~E-~u~ - u~E-~u’ - u~E-~u~ + u~E-~u~ + u3E-‘u,, - u4
(71)
Lax pairs, symmetries and conservation laws of a differential-difference equation
927
(ii) B,: CT:‘)= - Au, - 2u2 - Au2 - uou, - u, E-'u. CT:*‘= -Au, E-'u,-2uZE-'uO-
Au,E-‘u,-uou,E-‘uo-
~,(E-'u~)~
E-'~~-~,E-*u,-u,E%,-u~E-~u~ E-'uo- Au’ E-2~o- Au, E-'~,-~u~(E-'u,)*
-Au2-2u3-Auj-uouz+u, a!:‘=
- Au,(E-‘~~)~+hu,
+2u2E-'u~~-2~2E-2~o-2u2E-'~,-
Au~(E-‘u~)*-~Au~E-~u~
(iii) B,: a{3) = - ( A*U, + 3Au2 + 2A2u2 + 2u,Au, + u,Auo + AuoAu, - ul E-‘u. + 3uou2
+ u, E-'u, + 2uoAu2 + u2Auo + Au,,Auz + Au, E-ho + 224:+ u,Au, +2u2E-'uo+ Au2E-‘u,+~;~,+~O~, E-'uo+u,E-*u,,+u,(E-'uo)* +u2E-*uo+3u3+
3Au,+ A2u3)
ui3’ = - (A2u2 + 3Au3 + 2A‘u3 + uoAu2 + u,Au,, + AuoAu2+ Au, E-*u, +u’E-‘uo+u,E-‘~2+2~c,A~3+~3A~~+A~~A~~-A~,E-’~o + Au,E-‘u,
(73)
+ u,u2 + u2Au2 - 2u2E-'u. - AuzE-‘u,, + 2A~zE-~ug
+ Au2E-‘u, + 2u3E-*uO+ Au-,E-*uO + uoAu2 + 3u0u, + U&Z
-uou,E-'uo+uou,E-2~o+~O~lE-'u,+uou~E-2uo-2u,E-2~o +u'E-3~o-~,(E-'~o)2+~l(E-2~0)2+~'E-2~,-u,E-1u,+ +u,E-'uoE-'u,+u,E-'uoE-2uo+2~2E-3~o+~2(E-2~o)2+
u1u2 ~2E-*u1
+2u2E-'u' +u~E-~u~+~u~+ 3Auq+A2uJ. The conserved densities of the DA equation (47) can be computed from u(“~. Since the DA equation (47) is written only in terms of u. we have to eliminate u,, u2; . . from Us. It is known from the Lax equation with n = 1 that we have: $=Au, I
9
1
au2
= Au, + Au2 + uoul - u’ E-ho
-=Au2+Aug+uou2+u,
at,
E-'uo-u,E-2~o-~2E-2u0
(74)
S. KANAGA
928
VEL
and K. M. TAMIZHMANI
From the above equations we can express u,, u2, u3; . . in terms of ug and we list the first few U, for i = 1,2; . +: u,=A
-,au, atI
+ A-‘(E-‘u,A-’
9)
- 2A-‘( u. z)
+ A-~(u,E-~u,~)
+ A-‘@
2)
E-*UoE-‘UoA-’ !%? . at1
Now substituting the values of u,, u2; * . in C#‘)~we obtain the conserved densities of the DA equation (47). We some of them list below: a'l'=
-*-I!!.!! at1
#=
&!%?+A-‘$A-’
at:
1 2 A-‘?$+
E-‘uoA-’ 1
_
E-zUoA-’
A-‘$+A-‘!$-
(76) 1
E-2UoA*?!$+
E-‘uoE-2uoA-’
$
1
1
+
5-
1
E-zUoA-’
$
I
$
-
E-zUoA-’
1
.
1
To derive symmetries we consider the linear eigenvalue problem and its adjoint given by:
a* = A2$ + (2uo + Au,)A+ + at Au0 + u; + (2 + A)A-’ 5)$ 2
1
-w* = AE-‘+* - uoqj* atI 2 = - A2E-2,)* + AE-‘(2uo$* + Auo$*> 2
+ (Au, + u; + (2 + A)A-’ z)$*.
(77)
Lax pairs, symmetries and conservation laws of a differential-difference equation
920
Using equation (77) we can see that $J/* satisfies
~-*~,~+2~-(2+A)A-‘~=o. 2
(78)
1
1
I
Now it is obvious that J/&,($+*) satisfies the linearized DA equation (52) and hence it is a symmetry. The infinite number of symmetries can be computed systematically. From equations (64) and (65), we find:
*l+b*= 5 &A-“,
(79)
S, = ~ WjW*,-j. I=0
(80)
where
In the above we have used:
v=( I+:+$+
***) eXp ( -stjA-j)(l
+ A)-“.
Below we list few wj+: wi= -Ew 1 w; = - EAw, + Ew, E’w, - E2w2. Now the symmetries are: S,=$s,,,n=0,1,2;~~. 1 We present below the first few generalized symmetries:
(81)
so = $ (1) = 0 1 s, = ;
1
(L&J= 3 1
S, = $ (-u. 1 s,=$
+ u; + (2 + A)A-’ $)
1
=$
+ $(82) 2
1
~+A-1~+3A-2~-4A-1~+3uoA-‘~+3A-’ 1
2
1
1
1
1
+uo~+~o-2u:fu: 1
)
etc. These are the solutions of the linearized DA equation (52).
7. CONCLUSIONS
In this paper we have explicitly presented Sato’s theory for differential-difference equations. As a consequence, we found the Lax pairs, conserved quantities and symmetries of the differential-difference equation which belongs to the single component KP family [40]. At this stage, many questions remain to be answered. For example, it would be interesting to find the recursion operator and master symmetries of the differential-difference equation of
S. KANAGA
930
VEL and K. M. TAMIZHMANI
a single component KP family and its connection with the r functions. Moreover, two-component Sato theory and reductions are very important problems to be considered. Similar analysis for differential-difference BKP and CKP are also of great interest. Work in the above areas is in progress and will be presented elsewhere. express our sincere thanks to Y. Ohta, .I. Satsuma, B. Grammaticos, A. Ramani, J. Hiterinta, J. J. C. Nimmo and W. Oevel for fruitful discussions and encouragement at various stages of this research. S. K. would also like to thank the Council for Scientific and Industrial Research, India for financial support in the form of a Senior Research Fellowship. K. M. T. is supported by the Indo-French Project 1201-l.
Acknowledgements-We
REFERENCES 1. Ablowitz,
M. J. and Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991. 2. Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons. Springer, New York, 1987. 3. Olver, P. J., Applications of Lie Groups to Differential Equations. Springer, New York, 1986. 4. Chen, H. H., Lee, Y. C. and Liu, J. E., On a new hierarchy of symmetries for the Kadomtsev-Petviashvilli equation. fhysica, 9D (1983) 439-445. 5. Fokas, A. S. and Anderson, R. L., On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems. J. Math. Phys., 23 (1982) 10661073. 6. Fokas, A. S. and Fuchssteiner, B., The hierarchy of the Benjamin-On0 equation. Phys. Left., 86A (1981) 341. 7. Fokas, A. S. and Santini, P. M., The recursion operator of the Kadomtsev-Petviashvilh equation and the squared eigen functions of the Schrodinger operator. Stud. Appl. Math., 75 (1986) 179-185. 8. Konopelchenko, B., The two dimensional matrix spectral problem: general structure of the integrable equations and their Backlund transformations. Phys. Lett., 89A (1981) 346-350. 9. Matsuno, J., Bilinear Transformation Method. Academic Press, London, 1984. 10. Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Let?., 27 (1971) 1192-1194. 11. Hirota, R. and Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equations. Phys. Lett, 85A (1981) 407-408. 12. Hietarinta, J., A search for bilinear equations passing Hirota’s three-soliton condition. I KdV type bilinear equations. J. Math. fhys., 2$ (1987) 1732-1742. 13. Satsuma, J., N-soliton solution of the two dimensional Korteweg-de Vries equation. J. Phys. Sot. Jpn. 40 (1976) 286-290. 14. Satsuma, J., A Wronskian representation of n-soliton solutions of nonlinear evolution equations. J. Phys. Sot. Jpn, 46 (1979) 359-360. 15. Freeman, N. C. and Nimmo, J. J. C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvilh equations: the Wronskian technique. fhys. Lett., 95A (1983) l-3. 16. Weiss, J., Tabor, M. and Carnevale, G., The Painlevt property for partial differential equations. J. Math. fhys., 24 (1983) 522-526.
17. Ramani, A., Grammaticos, B. and Bountis, T. C., The Painleve property and singularity analysis of integrable and non integrable systems. Phys. Rep., 180 (1989) 159-245. 18. Konopelchenko, B., Nonlinear Integrable Equations. Lecture Notes in Physics. Vol. 270. Springer, New York, 1987. 19. Boiti, M., Leon, J. J. P., Martina, L. and Pempinelli, F. Integrable nonlinear evolution in (2 + 1) dimensions with non-analytic dispersion relations. J. Phys. A: Math. Gen., 21 (1981) 3611-3627. 20. Toda, M.. Studies of a nonlinear lattice. Phys. Rep., 18 (1975) 1-112. 21. Flaschka, H., The Toda lattice. I Existence of integrals. Phys. Rev., B9 (1974) 1924-1925. 22. Flaschka, H., On the Toda lattice.11 Inverse scattering solution. Prog. Theor. fhys. Suppl., 51 (1974) 703-716. 23. Ablowitz, M. J. and Ladik, J. F., Nonlinear differential-difference equations. J. Math. Phys., 16 (1975) 598-603. 24. Ablowitz, M. J. and Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys., 17 (1976) 1011-1018. 25. Oevel, W., Zhang, H. and Fuchssteiner, B., Symmetries, conserved quantities and hierarchies for Some lattice svstems with soliton structure. J. Math. Phvs.. 32 (1991) 1908-1918. 26. bevel, W., Fuchssteiner, B., Zhang, H. and Ragnisco: O., Master symmetries, angle variables and recursion operator of the relativistic Toda lattice. J. Math. Phys., 30 (1989) 2664-2670. 27. Oevel, W., Zhang, H. and Fuchssteiner, B., Master symmetries and multi-Hamiltonian formulations for some integrable lattice systems. Prog. Theor. Phys., 81 (1989) 294-308. 28. Toda, M. and Wadati, M., Bkklund transformations for the exponential lattice. J. Phys. Sot. Jpn. 39 (1975) 1196-1203. 29. Kuperschmidt, B. A., Asterisque 123, 1985.
Lax pairs, symmetries and conservation laws of a differential-difference equation
931
30. Ramani, A., Grammaticos, B. and Tamizhmani, K. M., J. Whys.A., 25 (1992) L883. 31. Quispel, G. R. W., Capel, H. W. and Sahadevan, R., Continuous symmetries of difference equations; the Kac-Van Moerbeke equation and PainlevC reduction. Phys. Lett., 170A (1992) 379-383. 32. Levi, D. and Winternitz, P., Symmetries and conditional symmetries of differential-difference equations. J. Math. Phys., 34 (1993) 3713-3730. 33. Wiersma, G. and Capel, H. W., Lattice equations, hierarchies and Hamiltonian structures. Physica, 142A (1987) 199-244. 34. Nijhoff, F. W. and Capel, H. W., The direct linearisation approach to hierarchies of integrable PDE’s in 2 + 1 dimensions. I Lattice eauations and the differential-difference hierarchies. Inu. Prob.. 6 (1990) 567-590. 35. Grammaticos, B., Ram&i, A. and Moreira, I. C., Delay-differential equations and the Pain&e transcendents. fhysica, l%A (1993) 574-590. 36. Sato, M., Soliton equations as dynamical systems on infinite Grassmann manifold. RIMS, Kokyuroku, 439 (1981) 30-46. 37. Sato, M. and Sato, Y., Soliton equations as dynamical systems on infinite Grassmann manifold. In Nonlinear Partial Differential Equations in Applied Science, eds H. Fujita, P. D. Lax and G. Strang, pp. 259-271. Kinokuniya/North-Holland, Tokyo, 1983. 38. Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., Transformation groups for soliton equations. Proc. RIMS Symp. Nonlinear Integrable Systems, Classical and Quantum Theory (Kyoto 1981). pp. 39-119. World Scientific, Singapore, 1983. 39. Jimbo, M. and Miwa, T., Solitons and infinite dimensional Lie algebras. fubl. RIMS. Kyoto Univ.. 19 (1983) 943-1001. 40. Date, E., Jimbo, M. and Miwa, T., Method for generating discrete soliton equations I-V. J. fhys. Sot. Jpn, 51, 4116-4124, 4125-4131, 1982; 52,388-393, 761-765, 766-771, 1983. 41. Ghta, Y., Satsuma, J., Takahashi, D. and Tokihiro, T., An elementary introduction to Sato theory. frog. Theor. fhys. Suppl., 94 (1988) 210-241. 42. Matsukidaira, J., Satsuma, J. and Strampp, W., Conserved quantities and symmetries of KP hierarchy. J. Math. Phys., 31 (1990) 1426. 43. Kajiwara, K., Matsukidaira, J. and Satsuma, J., Conserved quantities of two component KP hierarchy. fhys. Left., 146A (1990) 115. 44. Kajiwara, K. and Satsuma, J., The conserved quantities and symmetries of the two-dimensional Toda lattice hierarchy. J. Math. Phys., 32 (1991) 506. 45. Sidorenko, J. and Strampp, W., Symmetry constraints of the KP hierarchy. Inu. Prob., 7 (1991) L37. 46, Konopelchenko, B. and Strampp, W., The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inu. Prob.. 7 (1991) L17-L24. 47. IJeno, K. and Takasaki, K., Group representations and systems of differential equations. Adv. Stud. in Pure Math., Konikuniya Kinokuniya, Tokyo, 4, p. 1, 1984. 48. Gevel, W., Poisson brackets for integrable lattice systems. In Algebraic Aspects of Integrable Systems: In Memory of Irne Dorfman. eds A. S. Fokas and I. M. Gelfand, pp. 261-283. Birkhauser, Boston, 1996.
m Great
Bntam. All rights reserved oYho-n77Y/Y7 $17.00 + n.m
PII: SO%O-0779(%)00142-7
Lax Pairs, Symmetries and Conservation Laws of a Differential-Difference Equation-Sato’s Approach S. KANAGA
VEL
and K. M. TAMIZHMANI
Department of Mathematics, Pondicherry University, Kalapet, Pondicherry 605 014, India (Accepted 5 November 19%)
on the known elementary introduction of Sato theory for differential equations. the differential-difference equation which belongs to the single component KP family has been considered in the framework of Sato theory. We show that in this natural framework, Lax pairs, symmetries and conservation laws can be obtained in a systematic way. 0 1997 Elsevier Science Ltd
Abstract-Based
1. INTRODUCTION
Following its discovery, the inverse scattering transform (ET) technique has proved to be an effective method for solving non-linear partial differential equations having soliton solutions [l, 21. These types of equations possess common many properties: existence of N-soliton solutions, infinitely many conservation laws, symmetries, Backlund transformations, Hamiltonian structures, Hirota bilinear forms, Painleve property etc., and have been classified as completely integrable systems [l-17]. Many integrable systems in (1 + 1) and (2 + 1) dimensions have now been found [l, 2, 18,191. Following this, attempts have also been made to formulate the above ideas for differential-difference (DA) equations [l&35]. The search was made for the existence of a unified approach to derive these equations and their integrability properties. Sato presented a remarkable theory which clarified the algebraic structures involving the integrable systems [36, 371. His approach, based on considering the pseudo differential operators in an infinite number of dependent variables, treats integrable systems as a flow in terms of an infinite set of time variables on a Grassmanian manifold and shows that the bilinear form is nothing but the Plucker relations for the Wronskian. Following this discovery, Date et al. developed a group theoretical framework for soliton equations and made a large classification of bilinear equations of soliton type [38, 393. In fact, using the same group theoretical framework under various assumptions, Date et al. also derived integrable equations in continuous, differential-difference and fully discrete cases WIRecently, much attention has been paid to utilizing the best possible level of Sato’s theory [41-471. For example, in an excellent paper, Ohta et al. explained the Sato theory in an elementary way [41]. More specifically, KP, multicomponent KP, constrained KP etc., have been discussed in Refs [42-451. and the Sato equation has been used to derive symmetries and conservation laws of KP and multicomponent KP equations. Motivated by these works [41-431, in this paper we study the DA version of Lax equations, symmetries and conservation laws through Sato’s theory. At this point it is important to mention the work of Ueno and Takasaki on a Toda lattice [47] and Oevel’s work on lattice systems [48]. 917
S. KANAGA
918
VEL
and K. M. TAMIZHMANI
The Sato theory presented in the following sections for the differential-difference equation is developed in the same sense as that of the continuous case [41, 421.
2. PRELIMINARIES
We begin by introducing the pseudo difference operator of the form W = 1 + w,A-’ + w2A-’ + W~A-~+ w4AP4+ wsAPs+ q-1
(1)
where A-’ is the formal inverse of the forward difference operator A defined by At(x) = j-(x + 1) -f(x). Th e coefficients, w,, are functions of {x:x E Z}. We also look for the inverse operator W -’ in the form of a pseudo difference operator which is expressed by W-’ = 1 + u,A-’ + u~A-~ + u,AP” + u4AP4+ usAP5+ . . ..
(2)
Now it is easy to compute the relations between the coefficients, w,, of W and ui of W ’ using the fact WW-’ = 1 holds. First we list a few of them below: u,= -w, u2= W’E-‘WI - w2
(3)
ug = - W’E-‘w, + WlE-2Wl - W,E-‘WlE-2Wl + w,.!-‘w,
+ W2E-*Wl - WJ
where E-’ is . the inverse of the shift operator E defined by Ef(x) =f(x + 1). Note that A and E are related by A = E - 1. Now we restrict ourselves to the finite number of terms in W. Accordingly we denote W,=~+W,A-‘+W~A-~+~~~+W,,,A~~.
(4)
Now it is obvious that using the operator W,, multiplying A” from the right, we can construct an mth order linear difference equation of the form W,Amf(x) = (Am + wlAm-’ + e.. + w&x)
= 0.
This equation has m linearly independent solutions f”‘(~), fC2’(x); * +,f’“‘(x). gives Am-‘f(‘)W, + &7-*f(‘)W2
+
...
+ f”‘wn,
=
+ A”-2f@“)W2 +
...
+ f’““w,,
= _ Amf(m).
c-9 Equation (5)
- Amf(‘)
(6) A’-‘f(m)Wl
It is easy to find the solution to these simultaneous equations in the form - A”f”’
...
f(l)
.. .
.
Am-‘f(l)
1..
Am-‘f(m)
._.
_ Amf(m)
. . . f(i)
A”-‘f(l)
... ... ...
Am-/f(‘)
. . . f(l) ... . . . . ,,i
...
w, = -
A+
Am-&)
(7)
Lax pairs, symmetries and conservation laws of a differential-difference equation
and hence
919
f (HI) A-,’
&,f(m, A-’ A”f’“’
1
(8)
f (ml
. ..
...
We assume that the set of linearly independent solutions f’(x) of the mth order linear difference equations can be expanded using the Newton-Gregory formula (9) where A’f”)(O) = 5, and XC, = x(x - l)...(x - r + 1)/r!. Now, the difference equation can be . written m the form) W,,A”(lxC,xCz...)@ = 0 (10) where (11) Let A be the shift operator given by 0
0 1 0
...
On using this we write
i: xc*nr = r=o Then we define.
1 xc,
xc2
1
xc, . 1:‘.
i 0
...
1
... . . ...
H(x) = i: xC,A’@ = r=O
(13)
(14)
The determinant of the first m rows of H(x) is nothing but the denominator in W, which is a Casorati determinant for the solution of the difference equation (5). 3. TIME EVOLUTION
In this section, we introduce infinite number of time variables t = (t,,f2,f3; . .) in w; = w;(x,t), (j = 1,2; . e). As a consequence of this, we have
f (A= f”‘(x;t) = f qx,t, ,t2;..).
(15)
S. KANAGA
920
VEL
and K. M. TAMIZHMANI
We consider the time evolution of H(x) in the form W-w)
= (i: (CK)) r=O
exp 77(uW
(16)
where n(t,A) = X~=,t,h”. We write formally, exp n(t,A) = 2 pnAn. n=O
(5 WN)) r =0
When we compare the coefficients of equal powers of A on both sides we get PO=1
(17)
(18)
pj =x+r,
x(x - 1) P2= ___2r + xr, +; P3 =
(r: + 2t,)
x(x - 1)(x - 2) +x(x - 1) tl + b (t: + 2r-J~ + k (t: + 6r, r2 + 6t3) 2! 3!
p = -4x - 1)(x - 2)(x - 3) + x(x - 1)(x - 2) r + 1_ (rf + 2r )X(X - 1) 2 4 2! ’ 2! 3! 4! + $ (t; + 6t,tz + 6t3)x + -jj (t’: + 1215, + 12r2,+ 24t,r, + 24r,).
These polynomials are called Schur polynomials or Plucker relations. It can be seen that aPI3 ;tt,=Pn-nrJ
pn=O, foralln
(19)
and (20) APn = Pn-1. Now, it is possible to express the function H(x;r) in terms of pi, which is written in the form
H(x;r) =
It is noted that and h!j)(x;r) =
ahg)(x;r) = A”h&j)(x;r). ar n
(23)
We infer from this h(x;r) = l@(x;r), n = 1,2; . . and j = 1,2; . ., m is the solution of a set of linear partial differential-difference equation ct-
8” h(x;r) = 0, n = L2,e.a
>
(24)
Lax pairs, symmetries and conservation laws of a differential-difference equation
921
with h(x;O) =f@(x). Then, we have WmAmhhi)= 0, j = 1,2;..,n.
(25)
Solving the system of equation (25) we get, A*-‘/&”
...
- A”#’
...
h&”
Am-&Tl,
:::
: - A”h,f’“’
:::
hr’’
A*-‘/&‘)
...
A’+/&‘)
...
hi,”
A”-$/)*’
.. . ...
A”-‘/.@;“’ ’
:::
’ hhmT”
w, =
and hence
hb” A,&($ A”‘#
. ..
hv’
A-”
1:: A”- i hh*;“’ A-’: . ..
Amhhm”
w*= g)”
(26)
...
1 I (27)
hb””
Now wj and W, are completely given in terms of Schur polynomials p,,. It is clear that pn play a crucial role for further developments of the Sato theory.
4. SATO, LAX AND ZAKHAROV-SHABAT
EQUATIONS
It is well known that the integrability of the non-linear systems is associated with finding of the appropriate Lax or Zakharov-Shabat equations [l, 21. In this section, we present the Lax and Zakharov-Shabat equations for DA formalism from Sato’s equation. By differentiating equation (25) with respect to t,, we obtain dW* A”/#’ at n
+ W,,A” %
= 0, n
(28)
Using the property (dh,fj’)/(dc,,) = A”hg’(x;t), we get c
aw,m at A + WmAmAnhj/‘(x;t) = 0. ) n
(29)
The operator in parentheses can be factorized as 8W --E A” + W, AmA” = B, WmA”‘, %I where B, is an nth order difference operator. B,s can be obtained by applying A-“W;’ the right-hand side of the equation B,=L
dW at W,’ + W,A”W-’m n
(30) to
(31)
922
S. KANAGA
VEL and K. M. TAMIZHMANI
From equation (31) we can write
aw = B, W, - WmA”. at,
m
This is true for W defined in equation (1) and hence
aw
-=B,W-WA” at, which is the famous Sato equation. The B,s in the Sato equation can be computed from B, =(WA'T')+ (34) where ( )’ denotes the non-negative powers of A only. Hence, we have discarded the first term of equation (31) because it involves only negative powers of A, where as B, consists only of non-negative powers of A. Below we give the first few B,‘s explicitly: B,= A-Aw, B2 = A'-(2Aw, +A'w,)A +(-2Aw,A2w, +Aw,A*w, +~(Aw,)~ + w,A2w,+2w,Aw,-A2w2-2Aw2) B3= A3-(3Aw,+3A2w,+A3w,)A2+(-3Aw,-6A2w,-3A3w,+A3w,A2w, +2A3w,Aw,+w,+A3w,+3(A2w,)2+9A2w,Aw,+3w,A2w,+6(Aw,)2 +3w,Aw,-A3w23A2W23Aw2)A+(-3A*w,-3A3w,+2A3w,A2w, +~A'~,A~,+~w,A~w,+~(A~~,)~+~~Aw,A*w, +6w,A2w,+6Aw,A2w, + ~(Aw,)~ + ~w~AwI - 3A3wz- 6A2w2- 3AW2 - Aw~A~w~A’w~ (35) -3Aw,(A2w,)2-2A3w,(Aw,)2-9A2w,(Aw,)2-6(Aw,)3-9w,(Aw,)2 -12w,Aw,A2~,-~,A2~,A3~,-3w,(A2~,)2-3~,A~,A3~,-3w:A~, -3w:A2w,-w:A3w,+A3w,A2 w2+2A3w,Aw2+w2A3w,+3A2w,A2w2 +6A2w,Aw2+ 3W2Li2W,+ 6Aw,A2w2+9Aw,Aw2+ 3W,fiW,+ A3w2Aw, +w,A'w,+ 3W,h2W,+3W,i\W2A3w3-3A2w3 -3Aw3) To determine the Lax equation we define L = WAW-‘. (36) Differentiating equation (36) with respect to t,, we get:
aL aw -=--AW-‘+WAc at,
at,
at,,
= B,L - LB, = P,,Ll where we have used the Sato equation and WW-’ = 1. Also note that B, = (I.“)‘. (37) is called the generalized Lax equation. From equation (37) we have $= It is evident that
al,“’ at,
n
--at=
(37) Equation
[B,,L”‘], m,n = 1,2;...
(38)
aLn [B,,L”‘] - [B,,,L”]. “I
(39)
We denote B’, = B, - L” which contains only terms with A-j, j > 0. Using this relation in equation (39), we arrived at: y-F= ”
m
[B,,B,,] - [BCn,BC,,J.
(40)
Lax pairs, symmetries and conservation laws of a differential-difference equation
923
Considering the difference part on the both sides we find that aB ,-aB,= at,
[B,,B,]
at,
(41)
which is the Zakharov-Shabat equation.
5. DIFFERENTIAL-DIFFERENCE
EQUATION
IN A SINGLE
COMPONENT
KP FAMILY
The power of the Sato theory explained in the previous sections will be demonstrated in this section by deriving explicitly the DA equation which belongs to a single component KP family [40] and its Lax pairs. The first step towards achieving this goal is to write down the operator L defined in equation (36) in terms of Uj, j = 0,1,2; . . . For this purpose, we substitute W and W-’ defined in equations (1) and (2) in L and obtain: L = A + u0 + u,A-’ + u~A-~ + ...
(42)
where /A,,= - Aw, u, = - Aw, - Awz + w,Aw, u2 = - Aw2 - Aw3 + Aw,E-‘w, + w2AwI + E-‘w,Aw2 - w,Aw,E-‘w, u3 = - Aw‘, - Aw3 + w3Aw, + Aw3E-‘w, - Aw2E-‘w, + ~Aw~E-~w,
(43)
- Aw,E-‘w, E-*w, + Aw~E-‘w2 - Aw, E-‘wZ - w2Aw,E-2~, - w,Aw, E-‘w, - Aw, E-‘w, + Aw, E-*w, - Aw, E-‘w,E-~w, + w,Aw,E-‘w, - w,Aw,E-‘w, + w,Aw,E-‘w,E-~w,. From B, = (L”)+, it is now obvious that B, =A+uO B, = A* + (2u,, + Au,JA + (Au0 + U; + 2u, + AU,) B3 = A” + (3uo + 3Au, + A2u0)A2+ (2A2u0+ 3Au, + 3~; + 3u,,Au,, + (Aw,)~ + 3u, + 3Au, (44) + A2u,)A + (A2u0+ 5u,u, + 3u0Au0 + u30+ (Auo)’ + Au,,Au, + 3u&, + UIAUO + u,E-‘u.
+ 2A2u, + 3Au, + 3U2
+ 3h42
+
A2u2).
Now for the given L and B,, we can find the hierarchy of equations. For n = 1 we find the following equations:
$=Au, I
+Au2+u0u,
-u,E-‘u.
F= Au, + Au3 + uDu2+ u, E-lug - u, E-2~o - u2E-*u, 1
(45)
S. KANAGA
924
VEL
and K. M. TAMIZHMANI
Also, for II = 2, we have: -
= A2u, + 2Au, + A2u2+ u,AuO + 2u0Au, + AuOAu, + uOu, - u,E-‘u,
(32
-
= A2u, + 2Au2 + 2A2u2+ 2Au3 + A2u3+ 2u,,Au, + AuOAu, + 2u,,u,
(46)
at2
+ u2Au0 + 2u0Au2+ AuzAuO+ u,Au” + u,u; + u: + u,Au, - U,E-224” + u,E-‘uo - u,(E-‘uO)2 - u,I!-‘u,
- U*E’4)
-
U2E-2Uo.
Now we consider the first two equations from the set given in equation (45) and the first equation in equation (46). Solving these equations for u0 we arrive at the DA equation in a single component KP family $+2$-2u,? 2
=(2+A)%. 1 >
I
(47) I
This equation is also been derived by Date et al. in a different approach [40]. It is obvious that equation (47) can also be derived from the Zakharov-Shabat equation for B, and B2.
6. CONSERVED
QUANTITIES
AND GENERALIZED
SYMMETRIES
As we mentioned in Section 1, the existence of an infinite number of conservation laws and generalized symmetries is the basic property for integrable equations in a continuous case [3-81. This is also true in the DA case [25-291. Recently, this property has been established for a two-dimensional Toda lattice [44]. In this section we present a detailed investigation for finding the conserved quantities and generalized symmetries for the DA equation (47). Before doing so, we give a brief review of definitions in this theory. We consider an evolution equation: u,=K(u) (48) where K is a vector field defined over the space M defined by M = {u = (u,, u2; . ., u,)]u: 2 + R”}. We call the functional S(U) a symmetry of equation (48) if it satisfies the linearized equation given by: St = K’tu)Pl,
(49)
where the Frechet derivative K’(u) is defined by: K’(u)[S]
=;
K(u + ES) r=O
It can be shown that a symmetry S must satisfy: [S,K] = S’[K] - K’[S] = 0.
This means that any symmetry S commutes with K(u).
(51)
Lax pairs, symmetries and conservation laws of a differential-difference equation
925
The linearized equation of the DA equation (47) is: $B2,0$2~$+2!$(2+~~)~-$=0. 2
1
1
(52) 1
I
The solutions of the above equation are nothing but the symmetries of equation (47). We use the eigenvalue problem and its adjoint to find the symmetries instead of solving equation (52) directly. In the continuous case, the recursion operator has been obtained by many authors using the squared eigenfunctions of the eigenvalue problem [6, 71. It is natural to expect in the DA case that we can also use the eigenfunctions of the linear problem and its adjoint to derive symmetries and conserved quantities [42-451. In this paper, we follow the procedure developed in [42]. Consider the following linear eigenvalue problem: L*=A+
(53)
$=B& n
(54)
with A[,,= 0. Notice that L = WAW-‘. Hence, we rewrite equation (53) in the form: (55)
Wo = Wo
where I+$,= W-‘I/J. The above equation (55) is just a first-order difference equation whose solution is given by: $” = g(t1,f2,**.;A)(1 + A) (56) where g(t,,t,; - -; A) is an integration function. From this result it follows that: $ = Wg(t,,t,;..;A)(l
+ A)
= (1 + w,A-’ + w~A-~ + .a.)(1 + A)xg(t,,t2;.-;A)
*=(
(57)
(1 + A)"g(t,,t2;..;A)
In deriving the above we have used the fact: A-‘(l+A)“=(1+
1..
j=12 A"
".
Again we can rewrite equation (54) using B, = L” + B’, in the form: 2 = (L” + B’,Ji,/i n We recall that B”, consists only of the negative powers of A. After tedious calculations, it is possible to express A-j, i > 0, in terms of the negative powers of L. Then $ = (L” + q&’ n
+ q2P
+ ..)$
where qj are appropriate functions of A,tl,t2; . *. On using LI,!I = AJI we obtain:
a* A”+&+%+... -= A A2 at,
(61)
S. KANAGA
926
VEL and K. M. TAMIZHMANI
This implies
a1%* -=*“+&+42+ at, h
A2
(62)
.**
This is true for any integer n > 0. On integrating the set of equations, we finally find (63) where qj again are appropriate functions. Comparing equation (57) with equation (63) at t, = 0, Vj = 1,2; . *, we get
*=( I+:+:+...
)(l + A)” eXp
(i*jtj). ,=I
Also, the adjoint of IJJcan be obtained by
**=(1+.+F+ wf ...)(l
+ A)-“exp ( -
Ch’f,). /=I
In order to derive the conservation laws of the DA hierarchy, presented here we express B, in powers of L. That is B, = L” + u(l”‘L-’ Therefore,
+ 4mm)p + . . ..
(*I) up a* /y+“‘+-+ -= ..* q/. A A2 1 at, (
(66)
(67)
From which we get &n) --_ a@% ‘4) at,
h”
(68)
where Jm) = c+)h-i. J=I
(69)
Differentiating equation (68) with respect to the time variable t,,, we will arrive at the conservation law (70) We list the first few conserved quantities corresponding to B,: (i) B,: fll
(1) =
up=
-u, -u,E-‘uo-242
a$“= - u,(E-‘u~)~ + u,E-‘u, - u,E-~u~ - u,E-‘IA, - u~E-~u, - u,E-‘u,, - u3 CT!‘)= - u,(E-‘u~)~ + ~u,(E-‘u~)~ - 2u,E-‘u,E-‘u, - u,(E-~u~)~ - u,E-~u~E-‘u, - u,E-‘u, + ~u,E-~u~ - u,E-~u~ + u,E-‘u, - u,E-~u, - u,E-‘u2 - u,E-‘u~E-~u~ - u~(E-~u~)~ - u~E-‘u~E-~u~ - uZ(E-‘u,,)~ + u2E-‘uO - u2E-‘u’ + u2E-‘uO - ~u~E-~u~ - u~E-~u’ - u~E-~u~ + u~E-~u~ + u3E-‘u,, - u4
(71)
Lax pairs, symmetries and conservation laws of a differential-difference equation
927
(ii) B,: CT:‘)= - Au, - 2u2 - Au2 - uou, - u, E-'u. CT:*‘= -Au, E-'u,-2uZE-'uO-
Au,E-‘u,-uou,E-‘uo-
~,(E-'u~)~
E-'~~-~,E-*u,-u,E%,-u~E-~u~ E-'uo- Au’ E-2~o- Au, E-'~,-~u~(E-'u,)*
-Au2-2u3-Auj-uouz+u, a!:‘=
- Au,(E-‘~~)~+hu,
+2u2E-'u~~-2~2E-2~o-2u2E-'~,-
Au~(E-‘u~)*-~Au~E-~u~
(iii) B,: a{3) = - ( A*U, + 3Au2 + 2A2u2 + 2u,Au, + u,Auo + AuoAu, - ul E-‘u. + 3uou2
+ u, E-'u, + 2uoAu2 + u2Auo + Au,,Auz + Au, E-ho + 224:+ u,Au, +2u2E-'uo+ Au2E-‘u,+~;~,+~O~, E-'uo+u,E-*u,,+u,(E-'uo)* +u2E-*uo+3u3+
3Au,+ A2u3)
ui3’ = - (A2u2 + 3Au3 + 2A‘u3 + uoAu2 + u,Au,, + AuoAu2+ Au, E-*u, +u’E-‘uo+u,E-‘~2+2~c,A~3+~3A~~+A~~A~~-A~,E-’~o + Au,E-‘u,
(73)
+ u,u2 + u2Au2 - 2u2E-'u. - AuzE-‘u,, + 2A~zE-~ug
+ Au2E-‘u, + 2u3E-*uO+ Au-,E-*uO + uoAu2 + 3u0u, + U&Z
-uou,E-'uo+uou,E-2~o+~O~lE-'u,+uou~E-2uo-2u,E-2~o +u'E-3~o-~,(E-'~o)2+~l(E-2~0)2+~'E-2~,-u,E-1u,+ +u,E-'uoE-'u,+u,E-'uoE-2uo+2~2E-3~o+~2(E-2~o)2+
u1u2 ~2E-*u1
+2u2E-'u' +u~E-~u~+~u~+ 3Auq+A2uJ. The conserved densities of the DA equation (47) can be computed from u(“~. Since the DA equation (47) is written only in terms of u. we have to eliminate u,, u2; . . from Us. It is known from the Lax equation with n = 1 that we have: $=Au, I
9
1
au2
= Au, + Au2 + uoul - u’ E-ho
-=Au2+Aug+uou2+u,
at,
E-'uo-u,E-2~o-~2E-2u0
(74)
S. KANAGA
928
VEL
and K. M. TAMIZHMANI
From the above equations we can express u,, u2, u3; . . in terms of ug and we list the first few U, for i = 1,2; . +: u,=A
-,au, atI
+ A-‘(E-‘u,A-’
9)
- 2A-‘( u. z)
+ A-~(u,E-~u,~)
+ A-‘@
2)
E-*UoE-‘UoA-’ !%? . at1
Now substituting the values of u,, u2; * . in C#‘)~we obtain the conserved densities of the DA equation (47). We some of them list below: a'l'=
-*-I!!.!! at1
#=
&!%?+A-‘$A-’
at:
1 2 A-‘?$+
E-‘uoA-’ 1
_
E-zUoA-’
A-‘$+A-‘!$-
(76) 1
E-2UoA*?!$+
E-‘uoE-2uoA-’
$
1
1
+
5-
1
E-zUoA-’
$
I
$
-
E-zUoA-’
1
.
1
To derive symmetries we consider the linear eigenvalue problem and its adjoint given by:
a* = A2$ + (2uo + Au,)A+ + at Au0 + u; + (2 + A)A-’ 5)$ 2
1
-w* = AE-‘+* - uoqj* atI 2 = - A2E-2,)* + AE-‘(2uo$* + Auo$*> 2
+ (Au, + u; + (2 + A)A-’ z)$*.
(77)
Lax pairs, symmetries and conservation laws of a differential-difference equation
920
Using equation (77) we can see that $J/* satisfies
~-*~,~+2~-(2+A)A-‘~=o. 2
(78)
1
1
I
Now it is obvious that J/&,($+*) satisfies the linearized DA equation (52) and hence it is a symmetry. The infinite number of symmetries can be computed systematically. From equations (64) and (65), we find:
*l+b*= 5 &A-“,
(79)
S, = ~ WjW*,-j. I=0
(80)
where
In the above we have used:
v=( I+:+$+
***) eXp ( -stjA-j)(l
+ A)-“.
Below we list few wj+: wi= -Ew 1 w; = - EAw, + Ew, E’w, - E2w2. Now the symmetries are: S,=$s,,,n=0,1,2;~~. 1 We present below the first few generalized symmetries:
(81)
so = $ (1) = 0 1 s, = ;
1
(L&J= 3 1
S, = $ (-u. 1 s,=$
+ u; + (2 + A)A-’ $)
1
=$
+ $(82) 2
1
~+A-1~+3A-2~-4A-1~+3uoA-‘~+3A-’ 1
2
1
1
1
1
+uo~+~o-2u:fu: 1
)
etc. These are the solutions of the linearized DA equation (52).
7. CONCLUSIONS
In this paper we have explicitly presented Sato’s theory for differential-difference equations. As a consequence, we found the Lax pairs, conserved quantities and symmetries of the differential-difference equation which belongs to the single component KP family [40]. At this stage, many questions remain to be answered. For example, it would be interesting to find the recursion operator and master symmetries of the differential-difference equation of
S. KANAGA
930
VEL and K. M. TAMIZHMANI
a single component KP family and its connection with the r functions. Moreover, two-component Sato theory and reductions are very important problems to be considered. Similar analysis for differential-difference BKP and CKP are also of great interest. Work in the above areas is in progress and will be presented elsewhere. express our sincere thanks to Y. Ohta, .I. Satsuma, B. Grammaticos, A. Ramani, J. Hiterinta, J. J. C. Nimmo and W. Oevel for fruitful discussions and encouragement at various stages of this research. S. K. would also like to thank the Council for Scientific and Industrial Research, India for financial support in the form of a Senior Research Fellowship. K. M. T. is supported by the Indo-French Project 1201-l.
Acknowledgements-We
REFERENCES 1. Ablowitz,
M. J. and Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991. 2. Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons. Springer, New York, 1987. 3. Olver, P. J., Applications of Lie Groups to Differential Equations. Springer, New York, 1986. 4. Chen, H. H., Lee, Y. C. and Liu, J. E., On a new hierarchy of symmetries for the Kadomtsev-Petviashvilli equation. fhysica, 9D (1983) 439-445. 5. Fokas, A. S. and Anderson, R. L., On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems. J. Math. Phys., 23 (1982) 10661073. 6. Fokas, A. S. and Fuchssteiner, B., The hierarchy of the Benjamin-On0 equation. Phys. Left., 86A (1981) 341. 7. Fokas, A. S. and Santini, P. M., The recursion operator of the Kadomtsev-Petviashvilh equation and the squared eigen functions of the Schrodinger operator. Stud. Appl. Math., 75 (1986) 179-185. 8. Konopelchenko, B., The two dimensional matrix spectral problem: general structure of the integrable equations and their Backlund transformations. Phys. Lett., 89A (1981) 346-350. 9. Matsuno, J., Bilinear Transformation Method. Academic Press, London, 1984. 10. Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Let?., 27 (1971) 1192-1194. 11. Hirota, R. and Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equations. Phys. Lett, 85A (1981) 407-408. 12. Hietarinta, J., A search for bilinear equations passing Hirota’s three-soliton condition. I KdV type bilinear equations. J. Math. fhys., 2$ (1987) 1732-1742. 13. Satsuma, J., N-soliton solution of the two dimensional Korteweg-de Vries equation. J. Phys. Sot. Jpn. 40 (1976) 286-290. 14. Satsuma, J., A Wronskian representation of n-soliton solutions of nonlinear evolution equations. J. Phys. Sot. Jpn, 46 (1979) 359-360. 15. Freeman, N. C. and Nimmo, J. J. C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvilh equations: the Wronskian technique. fhys. Lett., 95A (1983) l-3. 16. Weiss, J., Tabor, M. and Carnevale, G., The Painlevt property for partial differential equations. J. Math. fhys., 24 (1983) 522-526.
17. Ramani, A., Grammaticos, B. and Bountis, T. C., The Painleve property and singularity analysis of integrable and non integrable systems. Phys. Rep., 180 (1989) 159-245. 18. Konopelchenko, B., Nonlinear Integrable Equations. Lecture Notes in Physics. Vol. 270. Springer, New York, 1987. 19. Boiti, M., Leon, J. J. P., Martina, L. and Pempinelli, F. Integrable nonlinear evolution in (2 + 1) dimensions with non-analytic dispersion relations. J. Phys. A: Math. Gen., 21 (1981) 3611-3627. 20. Toda, M.. Studies of a nonlinear lattice. Phys. Rep., 18 (1975) 1-112. 21. Flaschka, H., The Toda lattice. I Existence of integrals. Phys. Rev., B9 (1974) 1924-1925. 22. Flaschka, H., On the Toda lattice.11 Inverse scattering solution. Prog. Theor. fhys. Suppl., 51 (1974) 703-716. 23. Ablowitz, M. J. and Ladik, J. F., Nonlinear differential-difference equations. J. Math. Phys., 16 (1975) 598-603. 24. Ablowitz, M. J. and Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys., 17 (1976) 1011-1018. 25. Oevel, W., Zhang, H. and Fuchssteiner, B., Symmetries, conserved quantities and hierarchies for Some lattice svstems with soliton structure. J. Math. Phvs.. 32 (1991) 1908-1918. 26. bevel, W., Fuchssteiner, B., Zhang, H. and Ragnisco: O., Master symmetries, angle variables and recursion operator of the relativistic Toda lattice. J. Math. Phys., 30 (1989) 2664-2670. 27. Oevel, W., Zhang, H. and Fuchssteiner, B., Master symmetries and multi-Hamiltonian formulations for some integrable lattice systems. Prog. Theor. Phys., 81 (1989) 294-308. 28. Toda, M. and Wadati, M., Bkklund transformations for the exponential lattice. J. Phys. Sot. Jpn. 39 (1975) 1196-1203. 29. Kuperschmidt, B. A., Asterisque 123, 1985.
Lax pairs, symmetries and conservation laws of a differential-difference equation
931
30. Ramani, A., Grammaticos, B. and Tamizhmani, K. M., J. Whys.A., 25 (1992) L883. 31. Quispel, G. R. W., Capel, H. W. and Sahadevan, R., Continuous symmetries of difference equations; the Kac-Van Moerbeke equation and PainlevC reduction. Phys. Lett., 170A (1992) 379-383. 32. Levi, D. and Winternitz, P., Symmetries and conditional symmetries of differential-difference equations. J. Math. Phys., 34 (1993) 3713-3730. 33. Wiersma, G. and Capel, H. W., Lattice equations, hierarchies and Hamiltonian structures. Physica, 142A (1987) 199-244. 34. Nijhoff, F. W. and Capel, H. W., The direct linearisation approach to hierarchies of integrable PDE’s in 2 + 1 dimensions. I Lattice eauations and the differential-difference hierarchies. Inu. Prob.. 6 (1990) 567-590. 35. Grammaticos, B., Ram&i, A. and Moreira, I. C., Delay-differential equations and the Pain&e transcendents. fhysica, l%A (1993) 574-590. 36. Sato, M., Soliton equations as dynamical systems on infinite Grassmann manifold. RIMS, Kokyuroku, 439 (1981) 30-46. 37. Sato, M. and Sato, Y., Soliton equations as dynamical systems on infinite Grassmann manifold. In Nonlinear Partial Differential Equations in Applied Science, eds H. Fujita, P. D. Lax and G. Strang, pp. 259-271. Kinokuniya/North-Holland, Tokyo, 1983. 38. Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., Transformation groups for soliton equations. Proc. RIMS Symp. Nonlinear Integrable Systems, Classical and Quantum Theory (Kyoto 1981). pp. 39-119. World Scientific, Singapore, 1983. 39. Jimbo, M. and Miwa, T., Solitons and infinite dimensional Lie algebras. fubl. RIMS. Kyoto Univ.. 19 (1983) 943-1001. 40. Date, E., Jimbo, M. and Miwa, T., Method for generating discrete soliton equations I-V. J. fhys. Sot. Jpn, 51, 4116-4124, 4125-4131, 1982; 52,388-393, 761-765, 766-771, 1983. 41. Ghta, Y., Satsuma, J., Takahashi, D. and Tokihiro, T., An elementary introduction to Sato theory. frog. Theor. fhys. Suppl., 94 (1988) 210-241. 42. Matsukidaira, J., Satsuma, J. and Strampp, W., Conserved quantities and symmetries of KP hierarchy. J. Math. Phys., 31 (1990) 1426. 43. Kajiwara, K., Matsukidaira, J. and Satsuma, J., Conserved quantities of two component KP hierarchy. fhys. Left., 146A (1990) 115. 44. Kajiwara, K. and Satsuma, J., The conserved quantities and symmetries of the two-dimensional Toda lattice hierarchy. J. Math. Phys., 32 (1991) 506. 45. Sidorenko, J. and Strampp, W., Symmetry constraints of the KP hierarchy. Inu. Prob., 7 (1991) L37. 46, Konopelchenko, B. and Strampp, W., The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inu. Prob.. 7 (1991) L17-L24. 47. IJeno, K. and Takasaki, K., Group representations and systems of differential equations. Adv. Stud. in Pure Math., Konikuniya Kinokuniya, Tokyo, 4, p. 1, 1984. 48. Gevel, W., Poisson brackets for integrable lattice systems. In Algebraic Aspects of Integrable Systems: In Memory of Irne Dorfman. eds A. S. Fokas and I. M. Gelfand, pp. 261-283. Birkhauser, Boston, 1996.