Integrable equations on time scales

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Integrable equations on time scales

Metin Gürses

Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey

Gusein Sh. Guseinov

Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey Burcu Silindir

Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey

共Received 14 June 2005; accepted 19 September 2005; published online 15 November 2005兲

Integrable systems are usually given in terms of functions of continuous variables 共on R兲, in terms of functions of discrete variables 共on Z兲, and recently in terms of functions of q-variables共on Kq兲. We formulate the Gel’fand-Dikii 共GD兲 formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers 共difference equations兲 and over q-numbers 共q-difference equations兲. We formulate the GD formalism also in terms of shift operators for all regular-discrete time scales. We give a method allowing to construct the recursion operators for integrable systems on time scales. Finally, we give a trace formula on time scales and then construct infinitely many conserved quantities共Casimirs兲 of the integrable systems on time scales. © 2005 American Institute of Physics.

关DOI:10.1063/1.2116380兴

I. INTRODUCTION

Integrable systems are well studied and well understood in 1 + 1 dimensions.^1–3Here one of the dimensions denotes the time共evolution兲 variable and the other one denotes the space variable which is usually taken as continuous. There are also important examples where this variable takes values inZ, i.e., integer values. In both cases the Gel’fand-Dikii共GD兲 approach is quite effective.

One can generate hierarchies of integrable evolution equations, both onR and on Z共see Ref. 3 for GD applications and related references兲. In addition one can construct the conserved quantities, Hamilton operators, and recursion operators. Investigation of integrable systems on q-discrete intervals started in Refs. 4–6. They considered GD formalism on Kq and found q-integrable hierarchies including the q-KdV equation.

In this work we extend the Gel’fand-Dikii approach to time scales where R, Z, and Kq are special cases. In the next section we give a brief review of time scales calculus. See Refs. 7–13 for a more detailed review of the subject. In GD formalism, in obtaining integrable systems the essential tools are the differential and shift operators and their inverses. For extending the GD formulation to time scales we give the necessary means to construct in the sequel the algebra of pseudo-⌬-differential operators and the algebra of shift operators. In Sec. III we assume

⌬-differential Lax operators and derive the ⌬-Burgers hierarchy with its recursion operator. We present special cases of the Burgers equation for T = hZ and T = Kq. In Sec. IV, we consider the regular time scales where the inverse of jump operators can be defined. Here we assume a pseudo-

⌬-differential algebra and give the corresponding GD formulation. As an example we present a

⌬-KdV hierarchy. We first find n=1 member of the hierarchy and write out it explicitly for T

=R , Z , Kq and for T =共−⬁ ,0兲艛Kq. Then we give the n = 3 member and call it as the ⌬-KdV system. We call it ⌬-KdV equation, because the corresponding Lax operator is a second order

⌬-differential operator. It involves two fields u and v, but the second field v can be expressed in

46, 113510-1

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terms of the first filed u. WhenT = R, this system reduces to the standard KdV equation. In Sec. V, we consider the regular-discrete time scales and introduce the algebra of shift operators on them and give the corresponding GD formulation for all such time scales. Here several examples are presented. We first generalize the examples of discrete systems onZ given in Ref. 3共one field, two fields, and four fields examples in Ref. 3兲 to arbitrary discrete time scales. In all these examples whenT = Z we get the discrete evolutions given in Ref. 3. We construct the recursion operators of these systems on time scales. We generalize the Frenkel’s KdV system⁴ introduced on Kq to arbitrary discrete time scales and we construct its recursion operator. In this section, we finally give an example of the KP hierarchy on discrete time scales. In Sec. VI, we extend the standard way of constructing the conserved quantities of integrable systems to time scales by introducing a trace form on the algebra of ⌬-pseudo-differential operators. The trace form introduced in this section reduces, in particular cases, to the standard trace forms onR and Z. In the Appendix we give the recursion operators of two four-fields systems introduced in Sec. V. We end up with a conclusion.

II. TIME SCALE CALCULUS

The time scale calculus is developed mainly to unify differential, difference, and q-calculus. A time scale共T兲 is an arbitrary nonempty closed subset of the real numbers. The calculus of time scales was initiated by Aulbach and Hilger^7,8in order to create a theory that can unify and extend discrete and continuous analysis. The real numbers共R兲, the integers 共Z兲, the natural numbers 共N兲, the non-negative integers共N0兲, the h-numbers 共hZ=兵hk:k苸Z其, where h⬎0 is a fixed real num- ber兲, and the q-numbers 共Kq= q^Z艛兵0其⬅兵q^k: k苸Z其艛兵0其, where q⬎1 is a fixed real number兲 are examples of time scales, as are关0,1兴艛关2,3兴,关0,1兴艛N, and the Cantor set, where 关0,1兴 and 关2,3兴 are real number intervals. In Refs. 7 and 8 Aulbach and Hilger introduced also dynamic equations on time scales in order to unify and extend the theory of ordinary differential equations, difference equations, and quantum equations⁹共h-difference and q-difference equations based on h-calculus and q-calculus, respectively兲. For a general introduction to the calculus on time scales we refer the reader to the textbooks by Bohner and Peterson.^10,11 Here we give only those notions and facts connected to time scales which we need for our purpose in this paper.

Any time scale T is a complete metric space with the metric 共distance兲 d共x,y兲=兩x−y兩 for x , y苸T. Consequently, according to the well-known theory of general metric spaces, we have for T the fundamental concepts such as open balls 共intervals兲, neighborhood of points, open sets, closed sets, compact sets, and so on. In particular, for a given number r⬎0, the r-neighborhood U_r共x兲 of a given point x苸T is the set of all points y苸T such that d共x,y兲⬍r. By a neighborhood of a point x苸T is meant an arbitrary set in T containing an r-neighborhood of the point x. Also we have for functions f :T→R the concepts of the limit, continuity, and properties of continuous functions on general complete metric spaces 共note that, in particular, any function f :Z→R is continuous at each point of Z兲. The main task is to introduce and investigate the concept of derivative for functions f :T→R. This proves to be possible due to the special structure of the metric spaceT. In the definition of derivative, the so-called forward and backward jump operators play special and important roles.

Definition 1: For x苸T we define the forward jump operator ␴:T→T by

␴共x兲 = inf兵y 苸 T:y ⬎ x其, 共1兲

while the backward jump operator␳^:T→T is defined by

␳共x兲 = sup兵y 苸 T:y ⬍ x其. 共2兲

In this definition we set in addition ␴共max T兲=max T if there exists a finite max T, and

␳共min T兲=min T if there exists a finite min T. Obviously both ␴共x兲 and ␳共x兲 are in T when x 苸T. This is because of our assumption that T is a closed subset of R.

Let x苸T. If ␴共x兲⬎x, we say that x is right-scattered, while if ␳共x兲⬍x we say that x is left-scattered. Also, if x⬍max T and ␴共x兲=x, then x is called right-dense, and if x⬎min T and

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␳共x兲=x, then x is called left-dense. Points that are right-scattered and left-scattered at the same time are called isolated. Finally, the graininess functions␮^,␯^:T→关0, ⬁兲 are defined by

␮共x兲 =␴共x兲 − x, and␯共x兲 = x −␳共x兲 for all x 苸 T. 共3兲 Example 1: If T = R, then␴共x兲=␳共x兲=x and ␮共x兲=␯共x兲=0. If T=hZ, then ␴共x兲=x+h,␳共x兲

= x − h, and␮共x兲=␯共x兲=h. On the other hand, if T=Kq then we have

␴共x兲 = qx, ␳共x兲 = q⁻¹x, ␮共x兲 = 共q − 1兲x, and␯共x兲 = 共1 − q⁻¹兲x. 共4兲 LetT^␬denote Hilger’s above truncated set consisting ofT except for a possible left-scattered maximal point. Similarly, T_␬ denotes the below truncated set obtained from T by deleting a possible right-scattered minimal point.

Definition 2: Let f :T→R be a function and x苸T^␬. Then the delta derivative of f at the point x is defined to be the number f^⌬共x兲 (provided it exists) with the property that for each ␧⬎0 there exists a neighborhood U of x inT such that

兩f共␴共x兲兲 − f共y兲 − f^⌬共x兲关␴共x兲 − y兴兩艋 ␧兩␴共x兲 − y兩, 共5兲 for all y苸U.

Remark 1: If x苸T\T^␬, then f^⌬共x兲 is not uniquely defined, since for such a point x, small neighborhoods U of x consist only of x and besides we have␴共x兲=x. Therefore 共5兲 holds for an arbitrary number f^⌬共x兲. This is a reason why we omit a maximal left-scattered point.

We have the following:共i兲 If f is delta differentiable at x, then f is continuous at x. 共ii兲 If f is continuous at x and x is right-scattered, then f is delta differentiable at x with

f^⌬共x兲 =f共␴共x兲兲 − f共x兲

␮共x兲 . 共6兲

共iii兲 If x is right-dense, then f is delta differentiable at x iff the limit

lim

y→x

f共x兲 − f共y兲

x − y 共7兲

exists as a finite number. In this case f^⌬共x兲 is equal to this limit. 共iv兲 If f is delta differentiable at x, then

f共␴共x兲兲 = f共x兲 +␮共x兲f^⌬共x兲. 共8兲

Definition 3: If x苸T_␬, then we define the nabla derivative of f :T→R at x to be the number f^ⵜ共x兲 (provided it exists) with the property that for each ␧⬎0 there is a neighborhood U of x in T such that

兩f共␳共x兲兲 − f共y兲 − f^ⵜ共x兲关␳共x兲 − y兴兩艋 ␧兩␳共x兲 − y兩, 共9兲 for all y苸U.

We have the following:共i兲 If f is nabla differentiable at x, then f is continuous at x. 共ii兲 If f is continuous at x and x is left-scattered, then f is nabla differentiable at x with

f^ⵜ共x兲 =f共x兲 − f共␳共x兲兲

␯共x兲 . 共10兲

共iii兲 If x is left-dense, then f is nabla differentiable at x if and only if the limit

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lim

y→x

f共x兲 − f共y兲

x − y 共11兲

exists as a finite number. In this case f^ⵜ共x兲 is equal to this limit. 共iv兲 If f is nabla differentiable at x, then

f共␳共x兲兲 = f共x兲 −␯共x兲f^ⵜ共x兲. 共12兲

Example 2: IfT = R, then f^⌬共x兲= f^ⵜ共x兲= f⬘共x兲, the ordinary derivative of f at x. If T=hZ, then

f^⌬共x兲 = f共x + h兲 − f共x兲

h and f^ⵜ共x兲 = f共x兲 − f共x − h兲

h . 共13兲

IfT = Kq, then

f^⌬共x兲 = f共qx兲 − f共x兲

共q − 1兲x and f^ⵜ共x兲 = f共x兲 − f共q⁻¹x兲

共1 − q⁻¹兲x , 共14兲

for all x⫽0, and

f^⌬共0兲 = f^ⵜ共0兲 = lim

y→0

f共y兲 − f共0兲

y 共15兲

provided that this limit exists.

Among the important properties of the delta differentiation onT we have the Leibnitz rule, if f , g :T→R are delta differentiable functions at x苸T^␬, then so is their product fg and

共fg兲^⌬共x兲 = f^⌬共x兲g共x兲 + f共␴共x兲g^⌬共x兲 共16兲

= f共x兲g^⌬共x兲 + f^⌬共x兲g共␴共x兲兲. 共17兲 Also, if f , g :T→R are nabla differentiable functions at x苸T_␬, then so is their product fg and

共fg兲^ⵜ共x兲 = f^ⵜ共x兲g共x兲 + f共␳共x兲g^ⵜ共x兲, 共18兲

= f共x兲g^ⵜ共x兲 + f^ⵜ共x兲g共␳共x兲兲. 共19兲

In the next proposition we give a relationship between the delta and nabla derivatives 共see Ref.

12兲.

Proposition 4: (i) Assume that f :T→R is delta differentiable on T^␬. Then f is nabla differ- entiable at x and

f^ⵜ共x兲 = f^⌬共␳共x兲兲, 共20兲

for x苸T_␬such that␴共␳共x兲兲=x. If, in addition, f^⌬is continuous on T^␬, then f is nabla differen- tiable at x and (20) holds for any x苸T_␬.

(ii) Assume that f :T→R is nabla differentiable on T_␬. Then f is delta differentiable at x and

f^⌬共x兲 = f^ⵜ共␴共x兲兲, 共21兲

for x苸T^␬such that␳共␴共x兲兲=x. If, in addition, f^ⵜis continuous onT_␬, then f is delta differentiable at x and (21) holds for any x苸T^␬.

Now we introduce the concept of integral for functions f :T→R.

Definition 5: A function F :T→R is called a ⌬-antiderivative of f :T→R provided F^⌬共x兲

= f共x兲 holds for all x in T^␬. Then we define the⌬-integral from a to b of f by

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冕

a b

f共x兲⌬x = F共b兲 − F共a兲 for all a,b 苸 T. 共22兲 Definition 6: A function ⌽:T→R is called a ⵜ-antiderivative of f :T→R provided ⌽^ⵜ共x兲

= f共x兲 holds for all x in T_␬. Then we define theⵜ-integral from a to b of f by

冕

a b

f共x兲 ⵜ x = ⌽共b兲 − ⌽共a兲 for all a,b 苸 T. 共23兲

If a , b苸T with a艋b we define the closed interval 关a,b兴 in T by

关a,b兴 = 兵x 苸 T:a 艋 x 艋 b其. 共24兲

Open and half-open intervals, etc., are defined accordingly. Below all our intervals will be time scale intervals

Example 3: Let a , b苸T with a⬍b. Then we have the following.

共i兲 If f :T=R then

冕

a b

f共x兲⌬x =

冕

a b

f共x兲 ⵜ x =

冕

a b

f共x兲dx, 共25兲

where the integral on the right-hand side is the ordinary integral.

共ii兲 If 关a,b兴 consists of only isolated points, then

冕

a b

f共x兲⌬x =_x

The following relationship between the delta and nabla integrals follows from Definitions 5 and 6 by using Proposition 4.

Proposition 7: If the function f :T→R is continuous, then for all a,b苸T with a⬍b we have

冕

a b

f共x兲⌬x =

冕

a b

f共␳共x兲兲 ⵜ x and

冕

a b

f共x兲 ⵜ x =

冕

a b

a

b

f共x兲g^⌬共x兲⌬x. 共35兲

For more general treatment of the delta integral on time scales共Riemann and Lebesgue delta integrals on time scales兲 see Ref. 13 and Chap. 5 of Ref. 11.

III. BURGERS EQUATION ON TIME SCALES

The Gel’fand-Dikii approach is very effective in studying the symmetries, bi-Hamiltonian formulation, and in constructing the recursion operators of integrable nonlinear partial differential equations. In this approach one takes the Lax operator L in an algebra like a differential or pseudodifferential algebra, a matrix algebra, a polynomial algebra, or the Moyal algebra. In this section we take L in the algebra of delta-differential operators.

Let T be a time scale. We say that a function f : T→R is ⌬-smooth if it is infinitely

⌬-differentiable 共and hence infinitely ⵜ-differentiable兲. By ⌬ we denote the delta-differentiation operator which assigns to each⌬-differentiable function f :T→R its delta derivative ⌬共f兲 defined by

关⌬共f兲兴共x兲 = f^⌬共x兲 for x 苸 T^␬. 共36兲

The shift operator E is defined by the formula

共Ef兲共x兲 = f共␴共x兲兲 共37兲

for x苸T, where␴:T→T is the forward jump operator. It is convenient, in the operator relations to denote the delta-differentiation operator by␦ rather than by⌬. For example,␦f will denote the composition共product兲 of the delta-differentiation operator␦and the operator of multiplication by the function f. According to formula共16兲 we have

␦^{f = f}^⌬+ E共f兲␦^. 共38兲

Consider the Nth order␦-differential operator given by

L = aN␦^N^{+ a}N−1␦^N−1⁺ ¯ + a1␦^{+ a}0, 共39兲 where the coefficients a_i共i=0,1, ... ,N兲 are some ⌬-smooth functions of the variable x苸T. These functions are assumed to depend also on a continuous variable t苸R, however, we will not 共for simplicity兲 indicate explicitly the dependence on t.

Proposition 8: Let L be given as in (39) and A_n=共Lⁿ兲_⬎0 be the operator Lⁿ missing the␦⁰

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term. Then the Lax equation

dL dtn

=关An,L兴 = AnL − LAn 共40兲

for n = 1 , 2 , . . . produces a consistent hierarchy of coupled nonlinear evolutionary equations.

Example 4. Burgers equation on time scale: Let L =v␦+ u, where u andv are functions of x 苸T and t苸R. Then for an appropriate operator A the Lax equation

dL

dt =关A,L兴 共41兲

defines a system of two differential equations for the functions u andv. We find the operator A by using the Gelfand-Dikii formalism. Let us start with the second power of L and assume A

=共L²兲_⬎0, where

L²=vE共v兲␦²⁺关vv^⌬+vE共u兲 + uv兴␦⁺^vu^⌬^{+ u}²^. 共42兲 We can assume A = −共L²兲0共the part of −L² without the␦terms兲. With this choice, 共41兲 gives

dv

dt =␮v共vu^⌬+ u²兲^⌬, 共43兲

du

dt =v共vu^⌬+ u²兲^⌬, 共44兲

where␮共x兲=␴共x兲−x for x苸T.

Equations共43兲 and 共44兲 given above are not independent of each other. It is easy to see that v =␮u +␭, where ␭ is an arbitrary real function depending only on x苸T. Then these two equations reduce to a single equation, a Burgers equation on time scales,

du

dt =共␮u +␭兲关u²+共␮u +␭兲u^⌬兴^⌬. 共45兲 Let us present some special cases:共i兲 When T=R then␮^{= 0 and}␦= D, the usual differentiation.

Hence we can let ␭=1 and 共45兲 reduces to the standard Burgers equation on R. 共ii兲 When T

= hZ then␮共m兲=h and f^⌬共m兲=共1/h兲关f共m+h兲− f共m兲兴 for any f. Then taking ␭=0 in 共45兲 we find du共m兲

dt = u共m兲u共m + h兲关u共m + 2h兲 − u共m兲兴, 共46兲

where m苸hZ. The evolution equation given above in 共46兲 represents a difference version of the Burgers equation.共iii兲 Let T=q^Z, where q⫽1 and q⬎0. Then we have ␮共x兲=共q−1兲x and f^⌬共x兲

=关f共qx兲− f共x兲兴/共q−1兲x and taking ␭=0 we get from 共45兲 du共x兲

dt = u共x兲u共qx兲关u共q²x兲 − u共x兲兴. 共47兲

Taking An= −共Lⁿ兲0 with L given as in Example 4 we get a hierarchy of evolution equations 共Burgers hierarchy on time scales兲 from

dL dtn

= −关共Lⁿ兲0,v␦+ u兴 共48兲

for all n = 1 , 2 , 3 , . . . . Since共Lⁿ兲0is a scalar function, letting 共Lⁿ兲0=␳nwe obtain

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dv dtn

=␮v共␳n兲^⌬, 共49兲

du dtn

=v共␳n兲^⌬, 共50兲

where the first three␳nare given by

␳1= u, 共51兲

␳2=vu^⌬+ u², 共52兲

␳3=vE共v兲u^⌬⌬+关vv^⌬+vE共u兲 + uv兴u^⌬+共vu^⌬+ u²兲u. 共53兲 The above hierarchy reduces to a single evolution equation withv =␮u +␭,

du dtn

=共␮^{u +}␭兲共^˜␳n兲^⌬, n = 1,2, . . . , 共54兲 where^˜␳nis equal to␳nwithv =␮^{u +}␭. When T is a regular-discrete time scale, the first three␳^˜n

are given for␭=0 by

␳

˜₁= u, 共55兲

␳

˜₂= uE共u兲, 共56兲

␳

˜₃= uE共u兲E²共u兲. 共57兲

It is possible to construct the recursion operatorR by using the Lax representation.^14–16The hierarchy satisfies a recursion relation like

dL dt_n+1= LdL

dtn

+关Rn,L兴, n = 1,2, ... , 共58兲

where Rn is the remainder operator which has the same degree as the Lax operator L. We shall construct this operator for the Burgers equation with␭=0 on regular-discrete time scales. Choos- ing Rn=␣n␦ ^{we get}关by choosing v共x兲=␮共x兲u共x兲兴

R = uE + 关u共E共u兲 − u兲兴共E − 1兲⁻¹ E

E共u兲. 共59兲

One can generate the hierarchy共54兲 by application of the recursion operator R to the lowest order symmetry u₁= u共E共u兲−u兲,

du

dt_n=Rⁿ⁻¹u₁, n = 1,2, . . . . 共60兲

IV. ALGEBRA OF PSEUDO-DELTA-DIFFERENTIAL OPERATORS ON REGULAR TIME SCALES

Let us define the notion of regular time scales.

Definition 9: We say that a time scaleT is regular if the following two conditions are satisfied simultaneously:

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共i兲␴共␳共x兲兲 = x for all x 苸 T 共61兲 and

共ii兲␳共␴共x兲兲 = x for all x 苸 T, 共62兲

where␴^and ␳ denote the forward and backward jump operators, respectively.

From 共61兲 it follows that the operator ␴^:T→T is “onto” while 共62兲 implies that␴ ^{is “one-} to-one.” Therefore ␴ is invertible and␴⁻¹⁼␳. Similarly, the operator ␳^:T→T is invertible and

␳⁻¹⁼␴^ifT is regular.

Let us set x_*= minT if there exists a finite min T, and set x*= −⬁ otherwise. Also set x^*

= maxT if there exists a finite max T, and x^*=⬁ otherwise. It is not difficult to see that the following statement holds.

Proposition 10: A time scaleT is regular if and only if the following two conditions hold:

共i兲 The point x_*=T is right-dense and the point x^*= maxT is left-dense.

共ii兲 Each point ofT \兵x*, x^*其 is either two-sided dense or two-sided scattered.

In particular, R, hZ, and Kq are regular time scales, as are 关0,1兴, 关−1,0兴艛兵1/k:k 苸N其艛兵k/共k+1兲:k苸N其艛关1,2兴, and 共−⬁ ,0兴艛兵1/k:k苸N其艛兵2k:k苸N其, where 关−1,0兴, 关0,1兴, 关1,2兴, 共−⬁ ,0兴 are real line intervals.

If f :T→R is a function we define the functions f^␴:T→R and f^␳:T→R by

f^␴共x兲 = f共␴共x兲兲 and f^␳共x兲 = f共␳共x兲兲 for all x 苸 T. 共63兲 Defining the shift operator E by the formula Ef = f^␴ we have

共Ef兲共x兲 = f^␴共x兲 = f共␴共x兲兲 for all x 苸 T. 共64兲 The inverse E⁻¹exists only in case of regular time scales and is defined by

共E⁻¹f兲共x兲 = f共␴⁻¹共x兲兲 = f共␳共x兲兲 for all x 苸 T. 共65兲 In the operator relations, for convenience, we will denote the shift operator byE rather than by E.

For example,Ef will denote the composition 共product兲 of the shift operator E and the operator of multiplication by the function f. Obviously, for any integer m苸Z, we have

E^mf =共E^mf兲E^m. 共66兲

Remember that␦denotes the delta-differentiation operator acting in the operator relations by

␦^{f = f}^⌬+ E共f兲␦. The following proposition is an immediate consequence of the formulas 共8兲 and 共16兲.

Proposition 11: The operator formulas

E = I +␮␦ 共67兲

and

␦^{f = f}^⌬+ E共f兲␦ 共68兲

hold, where the function ␮^:T→R is defined by ␮共x兲=␴共x兲−x for all x苸T, and I denotes the identity operator.

In this section we will assume that all our considered functions fromT to R are⌬-smooth and tend to zero sufficiently rapidly together with their⌬-derivatives as x goes to x* or x^*, where x_*

= minT if there exists a finite min T and x_*= −⬁ otherwise, x^*= maxT if there exists a finite max T and x^*=⬁ otherwise. The inverse operator ␦⁻¹ exists on such functions. If g :T→T is such a function, then

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关⌬⁻¹共g兲兴共x兲 =

冕

x_* x

g共y兲⌬y. 共69兲

Proposition 12: Let f :T→R be a ⌬-smooth function such that f and all its ⌬-derivatives vanish rapidly at x_*and x^*. Then the operator␦⁻¹f being the composition (product) of␦⁻¹^{and f} has the form of the formal series in powers of␦⁻¹^,

␦⁻¹^{f =}␣0␦⁻¹⁺␣1␦⁻²⁺ ¯ , 共70兲 where␣0= E⁻¹f, and␣k=共−1兲^k共E⁻¹f兲^ⵜ^kfor k = 1 , 2 , . . . .

Proof: Multiplying共68兲 on the left and right by␦⁻¹ ^{we obtain}

␦⁻¹E共f兲 = f␦⁻¹⁻␦⁻¹^f^⌬␦⁻¹^. 共71兲 Replacing here f by E⁻¹f we get

␦⁻¹^{f =}共E⁻¹f兲␦⁻¹⁻␦⁻¹共E⁻¹f兲^⌬␦⁻¹^. 共72兲 Further, applying this rule to the function共E⁻¹f兲^⌬and taking into account that by Proposition 4共i兲

E⁻¹共E⁻¹f兲^⌬=共E⁻¹f兲^ⵜ, 共73兲

we find

␦⁻¹共E⁻¹f兲^⌬=共E⁻¹f兲^ⵜ␦⁻¹⁻␦⁻¹共共E⁻¹f兲^ⵜ兲^⌬␦⁻²^. 共74兲 Substituting this into the second term on the right-hand side of共72兲 we obtain

␦⁻¹^{f =}共E⁻¹f兲␦⁻¹⁻共E⁻¹f兲^ⵜ␦⁻²⁺␦⁻¹共共E⁻¹f兲^⌬兲^ⵜ␦⁻²^. 共75兲 Continuing this procedure repeatedly we arrive at the statement of the proposition.

Definition 13: By⌳ we denote the algebra of pseudo-delta-differential operators. Any opera- tor K苸⌳ of order k has the form

K =_ᐉ=−⬁

兺

^k ^a^ᐉ^␦^ᐉ^, ^共76兲

where a_ᐉ’s are ⌬-smooth functions of x苸T. For K given by (76) we will use the following notations:

K_艌0=

兺

ᐉ=0 k

a_ᐉ␦^ᐉ ^{and K}⬍0=

兺

−⬁

−1

a_ᐉ␦^ᐉ^. 共77兲

As an example we let

L = a_N␦^N^{+ a}N−1␦^N−1⁺ ¯ + a1␦^{+ a}0, 共78兲 where ai共i=0,1, ... ,N兲 are some ⌬-smooth functions on T. Then we have the following.

Proposition 16: Let L be given in (78). For each fixed N the Lax equation dL

dtn

=关An,L兴, An=共L^n/N兲_艌0, 共79兲

for n = 1 , 2 , . . . not divisible by N, produces a (consistent) hierarchy of evolution equations (a KdV hierarchy on time scales).

Proof: Since共L^n/N兲_艌0= L^n/N−共L^n/N兲_⬍0, we get

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dL dtn

=关共L^n/N兲_艌0,L兴 = − 关L^n/N兲_⬍0,L兴 . 共80兲 Evidently the commutator 关共L^n/N兲_艌0, L兴 involves only non-negative powers of␦, while the com- mutator关共L^n/N兲_⬍0, L兴 has the form 兺_j=−⬁^N−1bj␦^j. Therefore, we get by共80兲 that, for all n not divisible by N, 共79兲 produces nontrivial consistent N+1-number of evolutionary coupled ⌬-differential equations for ai, i = 0 , 1 , . . . , N. Note that aNturns out to be a fixed共i.e., time independent兲 function of x.

Example 5: A KdV hierarchy on time scales. Let

L =␦²⁺^v␦^{+ u,} 共81兲

where u andv are⌬-smooth functions. It is straightforward to find that

L^1/2=␦⁺␣0+␣1␦⁻¹⁺␣2␦⁻²⁺ ¯ , 共82兲 where

E共␣0兲 +␣0=v, 共83兲

E共␣1兲 +␣1+共␣0兲^⌬+共␣0兲²= u, 共84兲

E共␣2兲 +␣2+␣1E⁻¹共␣0兲 + 共␣1兲^⌬= 0. 共85兲 Choosing n = 1 , 3 , . . . we get the members of the KdV hierarchy.

(1) Let n = 1. Then Lax equation 共79兲 becomes dv

dt␦⁺^du

dt =关共L^1/2兲_艌0,L兴共86兲

and gives coupled equations for u andv, du

dt = u^⌬−v共␣0兲^⌬−共␣0兲^⌬⌬, 共87兲

dv

dt =v^⌬+ E共u兲 − u − v关E共␣0兲 −␣0兴 − E共␣0⌬兲 − E共␣0兲^⌬=␮共u^⌬−v共␣0兲^⌬−共␣0兲^⌬⌬兲. 共88兲 Comparing the above equations we get

dv dt −␮^du

dt = 0, 共89兲

and therefore

v =␮^{u +}␭, 共90兲

where␭ is an arbitrary real function depending only on x苸T. Thus, two equations 共87兲 and 共88兲 reduce to the following single equation:

du

dt = u^⌬−共␮^{u +}␭兲共␣0兲^⌬−共␣0兲^⌬⌬, 共91兲 where␣0is expressed, according to 共83兲, from

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E共␣0兲 +␣0=␮^{u +}␭. 共92兲 If we take␭=0, then 共91兲 and 共92兲 become

du

dt = u^⌬−␮^u共␣0兲^⌬−共␣0兲^⌬⌬, 共93兲

E共␣0兲 +␣0=␮^u. 共94兲

We shall now give␣0, for illustration, for particular cases ofT.

共i兲 In the case T=R we have␮^{= 0 and}共94兲 gives␣0= 0 and共93兲 becomes du

dt =du

dx, 共95兲

which is a linear equation explicitly solvable,

u共x,t兲 =␸共x + t兲, 共96兲

where␸ is an arbitrary differentiable function.

共ii兲 In the case T=Z we have ␮^{= 1 and} 共94兲 is satisfied by

␣0共n兲 = −_k=−⬁

兺

ⁿ⁻¹ ^{共− 1兲}^n+ku共k兲, n 苸 Z 共97兲

and therefore the Eq.共93兲 becomes du共n兲

dt = − u²共n兲 + 2u共n兲 + 2共− 1兲ⁿ关2 + u共n兲兴

兺

k=−⬁ n−1

共− 1兲^ku共k兲, 共98兲

for n苸Z.

(iii) In the caseT = Kqwe have␮共x兲=共q−1兲x and 共94兲 is satisfied by␣0共0兲=0 and

␣0共x兲 = − 共q − 1兲

兺

y苸共0,q⁻¹x兴

共− 1兲^log^q^共xy兲yu共y兲 共99兲

for x苸Kqand x⫽0. Substituting 共99兲 into 共93兲 we can get an evolution equation for u.

(iv) Let T =共−⬁ ,0兲艛Kq=共−⬁ ,0兴艛q^Z. In this case ␮共x兲=0 if x苸共−⬁ ,0兴 and ␮共x兲=共q

− 1兲x if x苸q^Z. The equation 共94兲 is satisfied by the function␣0given by

␣0共x兲 =

再

⁰ ^x苸共− ⬁ ,0兴,

−共q − 1兲⌺y苸共0,q⁻¹x兴共− 1兲^log^q^共xy兲yu共y兲 x 苸 q^Z. 共100兲 Therefore共93兲 will yield an evolution equation coinciding on 共−⬁ ,0兴 and q^Zwith the evolution equations described in the examples 共i兲 and 共iii兲, respectively. Now an essential complementary point is that the solution u must satisfy at x = 0 the smoothness conditions

u共0⁻兲 = u共0⁺兲, u⬘共0⁻兲 = u^⌬共0⁺兲. 共101兲 (2) Letting n = 3, first we get

L^3/2=␦³^{+ p}␦²^{+ q}␦^{+ r +}共terms with negative powers of␦兲, 共102兲 where

p =␣0+ E共v兲, 共103兲

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q =v^⌬+ E共u兲 +␣0v +␣1, 共104兲 r = u^⌬+␣0u +␣1E⁻¹共v兲 +␣2, 共105兲 and the Lax equation

dv dt␦⁺^du

dt =关共L^3/2兲_艌0,L兴, 共106兲

gives the coupled equations for u andv, du

dt = u^⌬⌬⌬+ pu^⌬⌬+ qu^⌬− r^⌬⌬−vr^⌬, 共107兲 dv

dt =v^⌬⌬⌬+ E共u^⌬⌬兲 + 共E共u^⌬兲兲^⌬+ E共u兲^⌬⌬+ p关v^⌬⌬+ E共u^⌬兲 + E共u兲^⌬兴 + q共v^⌬+ E共u兲 − u兲 + rv − q^⌬⌬

− E共r^⌬兲 − E共r兲^⌬−vq^⌬−vE共r兲. 共108兲

As in the first member of the hierarchy共n=1 case兲, the above ⌬-KdV equations reduce to a single equation for the function u. Below in Corollary 23 we found that v =␮共x兲u+␭共x兲. Letting ␭=

constant we get

du

dt = u^⌬⌬⌬+ pu^⌬⌬+ qu^⌬− r^⌬⌬−vr^⌬. 共109兲 It is possible to write the above equation more explicitly in terms of u forT = R, T = Z, and for T = Kq but they are quite lengthy. For the discrete case we give a KdV hierarchy in Example 8, next section.

V. SHIFT LAX OPERATORS ON REGULAR-DISCRETE TIME SCALES

LetT be a time scale. Let us set x_*= minT if there exists a finite min T and x_*= −⬁ otherwise.

Also set x^*= maxT if there exists a finite max T and x^*=⬁ otherwise. We will briefly write x*

= minT and x^*= maxT.

Definition 17: We say that a time scaleT is regular-discrete if the following two conditions are satisfied:

共i兲 The point x_*is right-dense and the point x^*is left-dense.

共ii兲 Each point ofT \兵x*, x^*其 is two-sided scattered (isolated).

The shift operator E is defined on functions f :T→R by the formula

共Ef兲共x兲 = f共␴共x兲兲 for x 苸 T, 共110兲

where␴^:T→T is the forward jump operator.

In this section we deal only with regular-discrete time scalesT. For such time scales T we have

␮共x兲 =␴共x兲 − x ⫽ 0 for all x 苸 T \ 兵x*,x^*其共111兲 and, therefore, on functions given onT \兵x*, x^*其 we have the operator relationship

␦⁼ ¹

␮^{共E − 1兲.} ^共112兲

All our functions will be assumed to be defined onT \兵x*, x^*其 and tends to zero sufficiently rapidly as x goes to x_* or x^*.

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This shift operator E, should be quite useful in the application of the Gel’fand-Dikii formal- ism. The reason is that for any integer m we have the simple product rule

E^mu =共E^mu兲E^m. 共113兲

Hence, for regular-discrete time scales, we can define an algebra ofE operators.

Definition 18: An algebra,⌳_⑀, ofE operators satisfying the operator equation (113) is defined as follows: Any operator K in⌳_⑀with degree k is of the form

K =

兺

_−⬁^k ^a^ᐉ^E^ᐉ^, ^共114兲

where a_ᐉare functions of x苸T that depend also on t苸R.

Hence we can form Lax operators in⌳_⑀, and produce integrable equations on regular-discrete time scales. Following³we obtain two classes of Lax representations.

Proposition 19: The Lax equation dL

dt_ᐉ=关共L^ᐉ兲_艌k,L兴, k = 0,1 共115兲

produces consistent hierarchy of equations forᐉ=1,2,... with the following suitable Lax opera- tors:

L =E^␣+n+ u_␣+n−1E^␣+n−1+ ¯ + u_␣E^␣, 共116兲

L =v_␣+nE^␣+n+v_␣+n−1E^␣+n−1+ ¯ + v_␣+1E^␣+1+ E^␣, 共117兲 for k = 0 and k = 1, respectively. Here u_i and v_i are functions defined onT and the integer ␣ ^is restricted to satisfy the inequality −n⬍␣艋−1.

Remark: Lax operators above and the following examples are given on any regular-discrete time scaleT 共we can take in particular T=Z or Kq兲. This means that for any function u on such a time scale E共u兲=u共␴共x兲兲 where ␴ is the jump operator defined in the second section. Hence our examples and results should be considered as more general than those considered in Ref. 3. In the case of Ref. 3 time scale is just the integers共T=Z兲 where E共u共n兲兲=u共n+1兲.

Example 6: Two field equations. Let k = 0, ␣^{= −1 and}

L = u₋₁E⁻¹+ u₀+E ⬅ vE⁻¹+ u +E. 共118兲 Then we find

ᐉ = 1 dv

dt₁=v共u − E⁻¹共u兲兲, 共119兲

du

dt₁= E共v兲 − v, 共120兲

ᐉ = 2 dv

dt₂= u²v + E共v兲v − vE⁻¹共v兲 − vE⁻¹共u²兲, 共121兲

du

dt₂= uE共v兲 + E共u兲E共v兲 − vE⁻¹共u兲 − uv, 共122兲

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ᐉ = 3, dv

dt₃= uv²+ u³v − vE⁻¹共v兲E⁻²共u兲 − 2vE⁻¹共u兲E⁻¹共v兲 − vE⁻¹共u³兲 + 2uvE共v兲 − v²E⁻¹共u兲

+vE共u兲E共v兲, 共123兲

du

dt₃= E共v兲关u²+ uE共u兲 + E共u²兲 + E共v兲 + 共E²共v兲兲兴 − v关E⁻¹共v兲 + E⁻¹共u²兲 + uE⁻¹共u兲 + u²+v兴.

共124兲 This is a Toda hierarchy on discrete time scales. The recursion relation between the n + 1th and nth elements of the hierarchy is given by

v_n+1= uv_n+vu_n+vE⁻¹共un兲 + v共E⁻¹共u兲 − u兲共1 − E兲⁻¹v_n

v, 共125兲

u_n+1= E共vn兲 + uun+v共1 − E兲⁻¹vn

v − E共v兲共1 − E兲⁻¹Evn

v. 共126兲

From this recursion relation the recursion operator of the hierarchy follows.

Example 7: Four-field system on time scale. We give two examples which are studied in Ref.

3 for the caseT = Z.

(1) Let k = 0 and␣^{= −2 and}

L =E²+ wE + v + uE⁻¹+ pE⁻². 共127兲

Then we get the four-field equations

ᐉ = 1 dp

dt₁=vp − pE⁻²共v兲, 共128兲

du

dt₁=vu + wE共p兲 − pE⁻²共w兲 − uE⁻¹共v兲, 共129兲

dv

dt₁= wE共u兲 + E²共p兲 − uE⁻¹共w兲 − p, 共130兲

dw

dt₁= E²共u兲 − u. 共131兲

(2) Let k = 1 and␣^{= −2 and}

L = q¯E²+ w¯E + v¯ + u¯E⁻¹+E⁻². 共132兲 Then we get another four-field equations,

ᐉ = 1 du¯

dt₁= w¯ − E⁻²共w¯兲, 共133兲

dv¯

dt₁= w¯ E共u¯兲 + q¯ − E⁻²共q¯兲 − u¯E⁻¹共w¯兲, 共134兲

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dw¯

dt₁= w¯ E共v¯兲 + q¯E²共u¯兲 − u¯E⁻¹共q¯兲 − v¯w¯, 共135兲

dq¯

dt₁= q¯E²共v¯兲 − v¯q¯. 共136兲

So far we considered the hierarchies coming from Proposition 19 with integer powers of the Lax operators. Now we consider the rational powers of the Lax operator.

Proposition 22: Let

L = wE^N+ u_N−1E^N−1+ ¯ + u0, 共137兲 where w共x兲 is a function of x which is not a dynamical variable dw/dt=0, ui, i = 0 , 1 , . . . , N − 1 are functions of t and x苸T. Then

dL dtn

=关共L^n/N兲_艌0,L兴, n = 1,2, ... 共138兲

produces hierarchies of integrable systems. Here n is a positive integer not divisible by N. Fur- thermore the function u₀ is also not dynamical, i.e., u₀= u₀共x兲, not depending on t.

Corollary 23: When N = 2 and w =共1/␮兲E共1/␮兲 then the ⌬-KdV Lax operator (81) reduces to the above form with

u₀= − v

␮⁺ 1

␮²^{+ u,} ^共139兲

u₁= − 1

␮

冋

^E

冉

␮¹

冊

⁺␮¹

册

⁺␮^v^. ^共140兲

Hence in part (2) of Example 5 we have a single equation withv = −␮^u0+共1/␮兲+␮^u.

In the following example we study the N = 2 case in more detail.

Example 8: KdV on discrete time scales. Let

L = wE共w兲E²+ uE + v. 共141兲

Then

L^1/2= wE +␣0+␣1E⁻¹+␣2E⁻²+ ¯ , 共142兲 where first three␣iare given as

w共E共␣0兲 +␣0兲 = u, 共143兲

wE共␣1兲 + E⁻¹共w兲␣1=v −共␣0兲², 共144兲

wE共␣2兲 + E⁻²共w兲␣2= −␣1E⁻¹共u兲

E⁻¹共w兲 . 共145兲

Then we calculate L^3/2by

L^3/2= wE共w兲E²共w兲E³+ p₂E²+ p₁E + p0+ negative powers ofE, 共146兲 where

p₂= E共w兲关wE²共␣0兲 + u兴, 共147兲

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p₁= wE共w兲E²共␣1兲 + uE共␣0兲 + wv, 共148兲

p₀= wE共w兲E²共␣2兲 + uE共␣1兲 + v␣0, 共149兲

=wE⁻¹共w兲关E⁻¹共w兲 + E共w兲E兴⁻¹

冉

^E共^␣¹^兲w^u

冊

⁺^{v共1 + E兲}⁻¹w^u. 共150兲 Then共138兲 with N=2 produces a hierarchy of evolution equations. It turns out that v becomes a constant in the whole hierarchy. We give the first two members of the hierarchy共for n=1 and n

= 3兲,

ut₁= u共1 − E兲共1 + E兲⁻¹u

w, 共151兲

ut₃= u共1 − E兲p0, 共152兲

where p₀is given above. The next members of the hierarchy can be found by taking n = 5 in共138兲 or by applying the recursion operatorR to ut₃. ForT = Kq and w = 1 the above hierarchy and its Hamilton formulation were given by Frenkel.⁴ The recursion operator of this hierarchy with w

= 1 can be found by using共58兲 with Rn=␣nE+␤n. We find that

共E²− 1兲␣n= E²共un兲, 共153兲

共E²− 1兲␤n= uE共un兲 + E共u兲␣n− uE共␣n兲共154兲 and the equation which determines the recursion operator is

u_n+1=vu_n− u共E − 1兲␤n, n = 0,1,2, . . . . 共155兲 We find that

R = v − u共E + 1兲⁻¹关− u + E共u兲E兴共E²− 1兲⁻¹E. 共156兲 When the Lax operator is of degree one and has an infinite power series in operatorE⁻¹the corresponding system is called the KP hierarchy.

Proposition 24: Let

L =E + u0+ u₁E⁻¹+ u₂E⁻²+ ¯ . 共157兲 Then

dL dtn

=关共Lⁿ兲_艌0,L兴, n = 1,2, ... , 共158兲

produces the following hierarchy:

n = 1 du₀

dt₁ =共E − 1兲u1, 共159兲

du₁

dt₁ =共E − 1兲u2+ u₁关u0− E⁻¹共u0兲兴, 共160兲 du_k

dt₁ =共E − 1兲uk+1+ u_k关u0− E^−k共u0兲兴, k = 0,1, ... . 共161兲

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n = 2 du₀

dt₂ =共E²− 1兲u2− u₁E⁻¹共E + 1兲u0+ E共u1兲共E共u0兲 + u0兲, 共162兲 du₁

dt₂ =共E²− 1兲u3+␣1E共u2兲 − u2E⁻²共␣1兲 +␣0u₁− u₁E⁻¹共␣0兲, 共163兲 duk

dt₂ =共E²− 1兲uk+2+␣1E共uk+1兲 − uk+1E^−k−1共␣1兲 +␣0uk− ukE^−k共␣0兲, k = 0,1, ... , 共164兲 where␣0=共E+1兲u1+共u0兲²and␣1=共E+1兲u0. The caseT = Z of this hierarchy is discussed in Ref.

3共see also the references therein兲 and the case T=Kqis discussed in Refs. 4 and 5.

VI. TRACE FUNCTIONAL AND CONSERVATION LAWS

LetT be a regular time scale and⌳ be the algebra of pseudo-delta-differential operators. Any operator F苸⌳ of order k has the form

F = ak␦^k^{+ a}k−1␦^k−1⁺ ¯ + a1␦^{+ a}0+ a₋₁␦⁻¹^{+ a}−2␦⁻²⁺ ¯ , 共165兲 where a_ᐉ’s are⌬-smooth functions of x苸T 共they are also functions of t苸R兲. The coefficients a0

and a₋₁we call, respectively, the free term共zero order term兲 and the residue of F associated with its “␦-expansion” 共165兲 and write

Free_␦F = a₀共x兲 and Res_␦F = a₋₁共x兲. 共166兲 In case of regular-discrete time scalesT we have

␦⁼ ¹

␮^{共E − I兲 =} 1

␮^{E −} 1

␮ ^共167兲

and therefore the same operator F can be expanded in series with respect to the powers ofE of the form

F = bkE^k+ b_k−1E^k−1+ ¯ + b1E + b0+ b₋₁E⁻¹+ b₋₂E⁻²+ ¯ . 共168兲 We write

Free_EF = b₀共x兲 and Res_EF = b₋₁共x兲. 共169兲 Substituting共167兲 and

␦⁻¹⁼共E − I兲⁻¹␮=共E⁻¹+E⁻²+ ¯ 兲␮= E⁻¹共␮兲E⁻¹+ E⁻²共␮兲E⁻²+ ¯ , 共170兲 into共165兲 and taking into account that

E⁻¹共␮兲 =␮共␳共x兲兲 =␴共␳共x兲兲 −␳共x兲 = x −␳共x兲 =␯共x兲, 共171兲 we find that

Res_EF =␯^Res_␦^F. 共172兲

Definition 25: The trace of an operator F苸⌳ is defined by Tr共F兲 =

冕

T

Res_␦兵F共I +␮␦兲⁻¹其 ⵜ x, 共173兲

where the nabla integral is defined according to Sec. II.

Proposition 26: Let F be given as in (165). In case of regular-discrete time scales we have