Characteristics of conservation laws for difference equations
CCHARACTERISTICS OF CONSERVATION LAWS FOR DIFFERENCEEQUATIONS
TIMOTHY J. GRANT AND PETER E. HYDON
Communicated by Evelyne Hubert
Abstract.
Each conservation law of a given partial differential equation is determined (up to equiva-lence) by a function known as the characteristic. This function is used to find conservation laws, to proveequivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferringthese results to difference equations is nontrivial, largely because difference operators are not derivationsand do not obey the chain rule for derivatives. We show how these problems may be resolved and illus-trate various uses of the characteristic. In particular, we establish the converse of Noether’s Theoremfor difference equations, we show (without taking a continuum limit) that the conservation laws in theinfinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation lawsfor the potential Lotka–Volterra equation. Introduction
Current research in symmetry methods owes a tremendous debt to Peter Olver. In particular, hisremarkable text, “Applications of Lie Groups to Differential Equations,” remains pre-eminent after morethan a quarter of a century. It is a masterpiece of scholarship that is notable for the lucidity of itsexposition and the precision of its proofs. The first (1986) edition was the first text to describe theconditions under which the converse of Noether’s Theorem [18] holds. A cornerstone of this result isthe proof that, for any system of partial differential equations (PDEs) in Kovalevskaya form that islocally analytic, there is a bijection between equivalence classes of conservation laws and equivalenceclasses of characteristics. A simpler proof of this result, due to Alonso [14], is incorporated in the secondedition [19] of Olver’s text.Here is a summary of the main definitions for scalar PDEs . For a given PDE, ∆ = 0, a conservationlaw (CLaw) is a divergence expression that vanishes on solutions of the equation, so thatDiv F = 0 when ∆ = 0 . A CLaw is trivial of the first kind if F vanishes on solutions of the PDE; it is trivial of the second kind ifDiv F ≡
0. A CLaw is trivial if it is a linear combination of the two kinds of trivial CLaws. Two CLawsare equivalent if and only if they differ by a trivial CLaw. If the PDE is totally nondegenerate (see [19])– for instance, if it is in Kovalevskaya form – the CLaw can be integrated by parts to find an equivalentCLaw in characteristic form , that is, with Div ˜F = Q ∆ . (1)The multiplier Q is called the characteristic of the CLaw.For example, the KdV equation, ∆ ≡ u t + uu x + u xxx = 0 , has a CLaw withDiv F = D t (cid:18) u − u x (cid:19) + D x (cid:18) u + u u xx − u x u xxx + u xx − u x u (cid:19) = (cid:0) u − u x D x (cid:1) ∆ . (2)Integration by parts yields the characteristic form of (2) :Div ˜F = (cid:0) u + 2 u xx (cid:1) ∆ . Date : August 21, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Difference equations, conservation laws, Noether’s Theorem. For simplicity, we restrict attention to scalar equations throughout this paper; the corresponding results for systemsare contained in the first author’s PhD thesis [7]. a r X i v : . [ m a t h . NA ] J a n TIMOTHY J. GRANT AND PETER E. HYDON So Q = (cid:0) u + 2 u xx (cid:1) is the characteristic and (2) is equivalent to the CLaw D t (cid:18) u − u x (cid:19) + D x (cid:18) u + u u xx + 2 u x u t + u xx (cid:19) = 0 . (3)A characteristic is said to be trivial if it vanishes on solutions of the PDE. By definition, the set ofcharacteristics is a vector space; two characteristics are equivalent if they differ by a trivial characteristic.Therefore, the correspondence between equivalence classes of characteristics and CLaws makes it easyto identify when two seemingly different CLaws are equivalent: one only needs to compare their char-acteristics. Given a nontrivial characteristic, it is usually easy to reconstruct a corresponding CLaw byinspection; this can also be achieved systematically with the aid of a homotopy operator. In particular,where Noether’s Theorem applies, each characteristic that arises from a one-parameter (local) Lie groupof variational symmetries can be used to construct an associated CLaw.Until now, Alonso’s result has not been transferred to difference equations. Yet one might wish toapproximate a given PDE by a finite difference scheme that preserves difference analogues of severalCLaws, particularly those that have a clear physical interpretation . This raises the question: is there afunction that characterizes each equivalence class of difference CLaws?Of course, a given difference equation may be interesting in its own right, whether or not it is anapproximation to a differential equation. Recent work has shown that CLaws of partial difference equa-tions (P∆Es) have many features in common with CLaws of PDEs. For instance, Dorodnitsyn [5, 6] hasformulated a finite difference analogue of Noether’s theorem. Hydon & Mansfield [12] studied variationalproblems whose symmetry generators constitute an infinite-dimensional Lie algebra and derived a dif-ference analogue of Noether’s Second Theorem. CLaws of a given P∆E can be found directly (whetheror not the P∆E is an Euler–Lagrange equation); see [10] for an algorithmic approach that works for anyP∆E in Kovalevskaya form (see below) and see [22–24] for applications of this approach to integrablequad-graph equations. The main shortcoming of the direct construction method is that the algebraiccomplexity of the computations grows exponentially with the order of the CLaw; in practice, therefore,the method is restricted to low-order CLaws.Given a difference equation, ∆ = 0, one cannot obtain the characteristics of its CLaw in the same wayone does as for a PDE. For PDEs, the chain rule ensures that each CLaw is linear in the highest-orderderivatives through which the dependence on ∆ occurs. Integration by parts is then used to find thecharacteristic. The analogue of integration by parts for difference equations is summation by parts.However there is no analogue of the chain rule and therefore CLaws typically depend nonlinearly on ∆and its shifts. Consequently, it is not possible to construct a characteristic merely by summing by parts.In the current paper, we show how this difficulty can be surmounted.We also derive and use a difference analogue of Alonso’s result. By pulling the characteristic backto a specified set of initial conditions, one can determine a function (the root) which labels the distinctequivalence classes of conservation laws. We show how the root is calculated in practice; examplesinclude an integrable P∆E with infinitely many CLaws. It is well-known that integrable PDEs haveinfinite hierarchies of CLaws which can be found using recursion operators, mastersymmetries, or Gardnertransformations. Recently, Mikhailov and co-workers [16,17] and Rasin and co-workers [20,21] have shownthat the same is true for integrable quad-graph equations. Having used the Gardner transformation toconstruct such a hierarchy, Rasin & Schiff [21] took a continuum limit in order to show that these CLawsare distinct. By using the difference analogue of Alonso’s result, we show how to determine directlywhen CLaws are distinct, irrespective of whether they are preserved in any continuum limit or whetherthe underlying P∆E is integrable.To establish the necessary results, it is helpful to begin by looking at scalar O∆Es ( § §
3, the definition of a characteristic is extended to CLaws of P∆Es; we prove that there isa bijection between equivalence classes of CLaws and characteristics. This result has several immediateapplications. In § § This is one of the oldest branches of geometric integration but, by exploiting the growing power of computer algebrasystems, some new strategies for doing this have been developed recently [7].
IFFERENCE CONSERVATION LAW CHARACTERISTICS 3 Scalar ordinary difference equations
Although the main focus of this paper is on P∆Es, it is instructive to look at scalar O∆Es first. Theindependent variable is n ∈ Z and the dependent variable is u ∈ R . It is convenient to regard n as afree variable and to denote the shifts of u from a fixed but unspecified n by u i := u ( n + i ). In order toevaluate first integrals on solutions of the O∆E, one must be able to eliminate the highest (or lowest)shift of u . Therefore, we restrict attention to explicit K th -order O∆Es, which are of the form∆ := u K − γ ( n, u , . . . , u K − ) = 0 , ∂γ∂u (cid:54) = 0 , . (4)The set of initial conditions is the set of values z = { n, u , . . . , u K − } from which all u i , i ≥ K , can becalculated.A first integral of (4) is a non-constant function, φ ( n, u , . . . , u K − ), that is constant on solutions. Itis helpful to introduce the forward shift operator, S n , and the identity operator, I , which are defined by S n : ( n, f ( n ) , u i ) (cid:55)→ ( n + 1 , f ( n + 1) , u i +1 ) , I : ( n, f ( n ) , u i ) (cid:55)→ ( n, f ( n ) , u i );here f is any function that is defined at n and n + 1. In terms of these operators, φ is constant onsolutions if and only if the following difference CLaw holds:( S n − I ) φ = 0 when ∆ = 0 . (5)It is useful to refer to constant solutions of (5) as trivial first integrals, by analogy with trivial CLaws ofthe second kind. (Trivial CLaws of the first kind cannot occur when φ depends only on n, u , . . . , u K − .)A nontrivial first integral must depend on u K − , otherwise ( S n − I ) φ does not depend on ∆ (in whichcase, the only way for φ to be a first integral is to be identically constant). Therefore, the CLaw can bewritten as C ( z , ∆) := φ ( n +1 , u , . . . , u K − , ∆ + γ ( n, u , . . . , u K − )) − φ ( n, u , . . . , u K − ) , (6)where C ( z ,
0) = 0. We now use the Fundamental Theorem of Calculus to write the CLaw in the form C ( z , ∆) = (cid:90) λ =0 ddλ C ( z , λ ∆) dλ =∆ (cid:90) λ =0 φ ,K ( n + 1 , u , . . . , u K − , λ ∆ + γ ( n, u , . . . , u K − )) dλ. (Throughout this paper, the partial derivative of a function, f , with respect to its i th continuous argumentis denoted by f ,i ). By analogy with differential equations, we define the characteristic to be the multiplier Q ( z , ∆) := (cid:90) λ =0 ∂C ( z , λ ∆) ∂λ ∆ dλ = ( S n φ ) | u K =∆+ γ − ( S n φ ) | u K = γ ∆ . (7)As with differential equations, a trivial characteristic is one that vanishes on solutions, so that Q ( z ,
0) = 0.A trivial first integral is a constant, so C ( z , ∆) ≡
0; therefore any trivial first integral has a trivialcharacteristic. To show that any trivial characteristic corresponds to a trivial first integral, it is helpfulto define the root of the characteristic to be the function Q ( z ) := lim µ → Q ( z , µ ∆) = lim µ → ( S n φ ) | u K = µ ∆+ γ − ( S n φ ) | u K = γ µ ∆= φ ,K ( n + 1 , u , . . . , u K − , γ ( n, u , . . . , u K − )); (8)to do this, we require that S n φ is differentiable in its K th continuous argument at u K = γ . If thecharacteristic is trivial, its root is zero, so0 = φ ,K ( n + 1 , u , . . . , u K − , γ ( n, u , . . . , u K − )) . (9)As γ , (cid:54) = 0, there is only one way for (9) to be satisfied identically in u : the CLaw (6) cannot dependon ∆, so φ must be a trivial first integral.Having dealt with the question of triviality, we now show how to reconstruct the first integral fromthe root. For clarity, we begin with a second-order example, but the same procedure applies in general.The O∆E∆ := u − γ ( n, u , u ) = 0 , γ ( n, u , u ) = 14 (cid:16) u + 5 u + 3 (cid:8) u + u ) (cid:9) / (cid:17) , (10)has a first integral φ ( n, u , u ) = 2 − n (cid:16) u + u + (cid:8) u + u ) (cid:9) / (cid:17) . (11) TIMOTHY J. GRANT AND PETER E. HYDON
Consequently, the characteristic is Q ( z , ∆) = 2 − ( n +1) (cid:16) ∆ + (cid:8) γ ( n, u , u ) + u ) (cid:9) / − (cid:8) γ ( n, u , u ) + u ) (cid:9) / (cid:17) / ∆ , and so the root is Q ( z ) = 2 − ( n +1) (cid:16) (cid:8) γ ( n, u , u ) + u (cid:9) (cid:8) γ ( n, u , u ) + u ) (cid:9) − / (cid:17) (12)= 2 − n (cid:16) u + u ) + 2( u + u ) (cid:8) u + u ) (cid:9) / (cid:17)
25 + 16( u + u ) . (13)To reconstruct the first integral from the root, one must reverse this process. First use the O∆E (10)to eliminate u from (13), obtaining (12). Now treat u and u as the continuous variables in (8), whichamounts to ∂∂u φ ( n + 1 , u , u ) = 2 − ( n +1) (cid:16) (cid:0) u + u (cid:1) (cid:8) u + u ) (cid:9) − / (cid:17) . Solving this and then applying S − n gives φ ( n, u , u ) = 2 − n (cid:16) u + (cid:8) u + u ) (cid:9) / (cid:17) + f ( n, u ) , where f ( n, u ) is yet to be determined. The determining equation is (5), which amounts (after simplifi-cation) to f ( n + 1 , u ) − f ( n, u ) = 2 − ( n +1) ( u − u ) . (14)Differentiating this with respect to u gives f , ( n, u ) = 2 − n , and so f ( n, u ) = 2 − n u + g ( n ) . Thus, (14) yields the O∆E g ( n + 1) − g ( n ) = 0 . Consequently, g ( n ) is an (irrelevant) arbitrary constant, which can be set to zero without loss of generality.This completes the reconstruction of the first integral (11).The process of reconstructing the first integral for a general scalar O∆Es is similar. For any firstintegral, φ , of the K th -order O∆E (4), the root is Q ( z ) := Q ( z ,
0) = φ ,K ( n + 1 , u , . . . , u K − , γ ( z )) . (15)Given a root, eliminate u in favour of u K to obtain Q ( n, u ( n, u , . . . , u K ) , u , . . . , u K − ) = ∂∂u K φ ( n + 1 , u , . . . , u K − , u K ) . Integrating this and applying S − n yields φ ( n, u , . . . , u K − ) = (cid:90) Q ( n − , u , . . . , u K − ) du K − + f ( n, u , . . . , u K − ) . All that remains is to find f . Substitute φ into (5) and simplify to obtain the determining equationfor f . Differentiate this with respect to u , then integrate to obtain f up to an arbitrary function g ( n, u , . . . , u K − ). Obtain the determining equation for g , apply S − n , and repeat the whole processwith S − n g replacing f . Continue in the way until the remaining function to be determined depends on n only. The determining equation for this function (which we call h ) is of the form h ( n + 1) − h ( n ) = H ( n ) , where H ( n ) is given. The solution is obtained by summation; this completes the reconstruction of theCLaw. IFFERENCE CONSERVATION LAW CHARACTERISTICS 5 m n u u u u ω Figure 1.
A P∆E in Kovalevskaya form: u = ω ( m, n, u , u , u ). The box enclosesthe values u ij on which ω depends; these are represented by crosses. The dashed linesrepresent the initial conditions that would be required to obtain all u ij in the upperhalf-plane. 3. Partial Difference Equations
We now generalize the ideas from the last section to scalar P∆Es for u ∈ R with two independentvariables n = ( m, n ) ∈ Z . Again, we regard the independent variables as being free; given ( m, n ), let u ij := u ( m + i, n + j ). [The indices i and j may be negative: each minus sign in the subscript shouldbe treated as being attached to the following digit. For instance, u − denotes u ( m + 1 , n − m, n ) induces an action on every function f ( m, n ), and inparticular on u ij , as follows: S m : ( m, n, f ( m, n ) , u ij ) (cid:55)→ ( m + 1 , n, f ( m + 1 , n ) , u ( i +1) j ) ,S n : ( m, n, f ( m, n ) , u ij ) (cid:55)→ ( m, n + 1 , f ( m, n + 1) , u i ( j +1) ) ,I : ( m, n, f ( m, n ) , u ij ) (cid:55)→ ( m, n, f ( m, n ) , u ij ) . A P∆E is written as ∆( m, n, [ u ]) = 0 , (16)where [ · ] denotes the argument · and a finite number of its shifts.A CLaw of a P∆E is a divergence expression that vanishes on solutions of the PDE:Div F := ( S m − I ) F + ( S n − I ) G = 0 when [∆] = . (17)The functions F := F ( m, n, [ u ]) and G := G ( m, n, [ u ]) are the densities of the CLaw. In the same wayas for PDEs, a CLaw of a P∆E is trivial if and only if it is a linear combination of the following twokinds of trivial CLaws. First kind : F | [∆]= = . The densities vanish on solutions, so we call these trivial densities . Second kind : Div F ≡
0, without reference to the equation ∆ = 0 and its shifts. This occurs ifthere exists a function H such that F = ( S n − I ) H and G = − ( S m − I ) H .3.1. Kovalevskaya Form.
A crucial step in dealing with first integrals is to replace u K by ∆ + γ . Wecan do something similar for a K th-order P∆E (16) if it is in Kovalevskaya form ,∆ := u K − ω ( m, n, u , u , . . . , u K − ) = 0 , (18)where u i = { u ij : j ∈ Z } and there exists j such that ∂ω/∂u j (cid:54) = 0. A schematic example of a 2-D scalarP∆E in Kovalevskaya form is shown in Figure 1. Let z = { m, n, u , u , . . . , u K − } be the (minimal)initial conditions from which shifts of (18) can be used to find any point of the form u lj for l ≥ K . Thefunction ω := ω ( z ) depends on a finite subset of these points. It is convenient to denote shifts of ω by ω ij := S im S jn ω .A scalar P∆E with two independent variables is explicit if it can be transformed into Kovalevskayaform by an admissible change of independent variables, that is, by a bijective linear map from Z toitself. The new independent variables are (cid:18) ˜ m ˜ n (cid:19) = A (cid:18) mn (cid:19) + b where A ∈ GL (2 , Z ) , det( A ) = ± , and b ∈ Z . (19) The corresponding results for systems of difference equations with arbitrarily many independent variables are obtained mutatis mutandis ; see [7]. For differential equations on R N and difference equations on Z N , the set of divergence expressions is the kernel of theEuler–Lagrange operator [13]. For a given PDE in Kovalevskaya form, any equation that holds on solutions of the PDE can be pulled back to anidentity on the initial conditions; with the above definition, the same is true for P∆Es.
TIMOTHY J. GRANT AND PETER E. HYDON u u u u ˜ u ˜ u ˜ u ˜ u m n ˜ m ˜ n Figure 2.
Transformation of a quad-graph equation into Kovalevskaya form. Thedashed lines show the initial conditions that would be required to determine all u ij (respectively ˜ u ij ) in the upper-right (respectively upper) half-plane.Although the value of the u at each point is unchanged, the coordinates of the point have changed, so itis helpful to define ˜ u ( ˜ m, ˜ n ) := u ( m ( ˜ m, ˜ n ) , n ( ˜ m, ˜ n )) . (20)Now fix ( m, n ); using the shorthand u ij = u ( m + i, n + j ) and setting ˜ u = u , we obtain u ij = S im S jn u = ˜ u ( ˜ m ( m + i, n + j ) , ˜ n ( m + i, n + j )) = ˜ u { ˜ m ( m + i,n + j ) − ˜ m ( m,n ) }{ ˜ n ( m + i,n + j ) − ˜ n ( m,n ) } . For instance, the shear A = (cid:18) (cid:19) , b = (cid:18) (cid:19) (21)transforms any quad-graph equation, u = ω ( m, n, u , u , u ) , into the Kovalevskaya form ˜ u = ω ( m ( ˜ m, ˜ n ) , n ( ˜ m, ˜ n ) , ˜ u , ˜ u , ˜ u ) . In particular, the dpKdV equation ( H1 in the ABS classification [1]), u = u + β − αu − u , (22)is transformed into ˜ u = ˜ u + β − α ˜ u − ˜ u . (23) Lemma 3.1.
When an explicit scalar P ∆ E is transformed according to (19) and (20) , there is a bijectivecorrespondence between equivalence classes of CLaws of the original P ∆ E and the transformed P ∆ E.Proof.
The transformation is of the form (cid:18) ˜ m ˜ n (cid:19) = (cid:18) a bc d (cid:19) (cid:18) mn (cid:19) + (cid:18) ef (cid:19) , ad − bc = ± , a, b, c, d, e, f ∈ Z , so the effect of the original shift operators on the transformed independent variables is S m ˜ m = a ( m + 1) + bn + e = ˜ m + a, S m ˜ n = c ( m + 1) + dn + f = ˜ n + c,S n ˜ m = am + b ( n + 1) + e = ˜ m + b, S n ˜ n = cm + d ( n + 1) + f = ˜ n + d. Thus S m = S a ˜ m S c ˜ n , S n = S b ˜ m S d ˜ n . Therefore given a CLaw with densities F and G ,( S m − I ) F + ( S n − I ) G = ( S a ˜ m S c ˜ n − I ) ˜ F + ( S b ˜ m S d ˜ n − I ) ˜ G = (cid:110) ( S a ˜ m − I ) ˜ F + ( S b ˜ m − I ) ˜ G (cid:111) + (cid:110) ( S c ˜ n − I ) S a ˜ m ˜ F + ( S d ˜ n − I ) S b ˜ m ˜ G (cid:111) = ( S ˜ m − I ) ˆ F + ( S ˜ n − I ) ˆ G, (24)where ˜ F and ˜ G denote the densities in terms of the transformed variables and the last equality isobtained by factorizing each expression in braces. If the original CLaw is trivial of the second kind thenit vanishes identically, so the transformed CLaw must also vanish identically. If the original densities F IFFERENCE CONSERVATION LAW CHARACTERISTICS 7
Table 1.
Densities for the three and five point CLaws of transformed dpKdV ˆ F = ( − m (2 u ( u − u ) + α − β )ˆ G = ( − m (cid:16) u (cid:16) u − + β − αu − − u (cid:17) − α (cid:17) ˆ F = ( u − u ) ( u u − α ) − ( u − u ) ( u u − β )ˆ G = (cid:16) u − u − − β − αu − − u (cid:17) (cid:16) u (cid:16) u − + β − αu − − u (cid:17) − α (cid:17) ˆ F = ( − m (cid:0) u ( u − u )( u + u + u ) + α ( u + u ) − β ( u + u ) (cid:1) ˆ G = ( − m (cid:16) u + u − + β − αu − − u (cid:17) (cid:16) u (cid:16) u − + β − αu − − u (cid:17) − α (cid:17) ˆ F = ( − m +1 (cid:0) u ( u − u ) + 4 u ( βu − αu ) + α − β (cid:1) ˆ G = ( − m (cid:18) u (cid:16) u − + β − αu − − u (cid:17) − α u (cid:16) u − + β − αu − − u (cid:17) + α (cid:19) ˆ F = − ln (cid:16) β − αu − u (cid:17) + ln (cid:16) u − − u + β − αu − − u (cid:17) ˆ G = ln (cid:18) u − − u + ( β − α ) (cid:16) u − − u − + β − αu − − u − − β − αu − − u (cid:17) − (cid:19) ˆ F = − ln (cid:16) u − u − + β − αu − u (cid:17) + ln (cid:16) β − αu − − u (cid:17) ˆ G = ln (cid:18) ( β − α ) (cid:16) u − − u − + β − αu − − u − − β − αu − − u (cid:17) − (cid:19) ˆ F = ( m − n ) ˆ F + n ˆ F ˆ G = ( m − n + 1) ˆ G + n ˆ G and G are trivial then so are ˜ F and ˜ G ; consequently, the new densities ˆ F and ˆ G will also be trivial. Asthe transformation is invertible, the converse is also true, which completes the proof. (cid:3) For example, when a quad-graph equation is transformed by (21), the transformed equation has CLawswith the densities ˆ F = ˜ F + ˜ G, ˆ G = S ˜ m ˜ G. (25)To transform back to the quad-graph, use (cid:18) mn (cid:19) = (cid:18) −
10 1 (cid:19) (cid:18) ˜ m ˜ n (cid:19) . (26)Then the densities for the CLaws are F = ˆ F − S − m ˆ G, G = S − m ˆ G. (27)Generally speaking, the new densities will depend on variables other than the transformed initial condi-tions; however, the difference equation can be used to pull them back to expressions that depend onlyon the transformed initial conditions.For later use, the dpKdV equation has four three-point and three five-point CLaws [24]; the trans-formed equation (23) has corresponding CLaws whose densities are listed in Table 1 (where the tildeshave been dropped to prevent clutter).3.2. The characteristic.
The characteristic of a given CLaw of a P∆E in Kovalevskaya form (18)is defined as follows. Any u ij may be written in terms of the initial conditions, z , and [∆]. Givena differentiable function, C ( z , [∆]), that satisfies C ( z , [0]) = 0, the Fundamental Theorem of Calculusyields C ( z , [∆]) = (cid:90) λ =0 ddλ C ( z , [ λ ∆]) dλ = (cid:90) λ =0 (cid:88) i,j ( S im S jn ∆) ∂C ( z , [ λ ∆]) ∂ ( S im S jn λ ∆) dλ. Summing by parts gives C ( z , [∆]) =∆ (cid:90) λ =0 { E ∆ ( C ( z , [∆])) } (cid:12)(cid:12) ∆ (cid:55)→ λ ∆ dλ + Div F , (28) TIMOTHY J. GRANT AND PETER E. HYDON where E ∆ is the difference Euler–Lagrange operator that corresponds to variations in ∆, E ∆ ( C ( z , [∆])) := (cid:88) i,j S − im S − jn ∂C ( z , [∆]) ∂ ( S im S jn ∆) , (29)and where F has two components, each of which vanishes on solutions of the P∆E. (The generalizationto P∆Es with more than two independent variables is obvious.) If, in addition, C ( z , [∆]) is a divergenceexpression (so that it is a CLaw) then, by subtracting the trivial CLaw Div F from both sides of (28),one obtains the equivalent CLaw˜ C ( z , [∆]) = C ( z , [∆]) − Div F = Q ( z , [∆]) · ∆where Q ( z , [∆]) := (cid:90) λ =0 { ( E ∆ ( C ( z , [∆]))) } (cid:12)(cid:12) ∆ (cid:55)→ λ ∆ dλ. (30)The function Q is a multiplier of the difference equation, so to be consistent with the continuous caseand [15], it is defined to be the characteristic of the CLaw.Just as for O∆Es, a trivial characteristic is one that vanishes on solutions, so Q ( z , [0]) = 0. To factorout trivial characteristics, we define the root of Q to be Q ( z ) := lim µ → Q ( z , [ µ ∆]) = lim µ → (cid:90) λ =0 { E µ ∆ ( C ( z , [ µ ∆])) } (cid:12)(cid:12) ∆ (cid:55)→ λ ∆ dλ = { E ∆ ( C ( z , [∆])) } (cid:12)(cid:12) [∆]= . (31)For PDEs, the characteristic of a CLaw is trivial if and only if the CLaw is trivial. We will now showthat, with our definition of the characteristic (30), the same is true for P∆Es in Kovalevskaya form.Hence, we need to prove that a CLaw is trivial if and only if Q ( z ) = 0.3.3. A trivial CLaw implies a trivial characteristic.
A trivial CLaw of the second kind (Div F ≡ F ( z , [0]) = . To deal with this we use the identity E ∆ (Div F ( z , [ ∆ ])) = E ∆ (cid:16) ( S m − I ) (cid:0) F ( z , [∆]) − F ( z , [0]) (cid:1) + ( S n − I ) (cid:0) G ( z , [∆]) − G ( z , [0]) (cid:1)(cid:17) = (cid:88) i,j S − im S − jn (cid:32)(cid:88) k ∂S m z k ∂ ( S im S jn ∆) · S m ∂ { F ( z , [∆]) − F ( z , [0]) } ∂z k (cid:33) − (cid:88) i,j S − im S − jn (cid:32) ∂F ( z , [∆]) ∂S im S jn ∆ l + ∂G ( z , [∆]) ∂ ( S im S jn ∆) (cid:33) + (cid:88) i,j S − im S − jn (cid:88) l,k ∂S m S lm S kn ∆ ∂S im S jn ∆ · S m ∂F ( z , [∆]) ∂S lm S kn ∆ + ∂S n S lm S kn ∆ ∂S im S jn ∆ · S n ∂G ( z , [∆]) ∂S lm S kn ∆ , (32)where z k ∈ z and we have used the fact that (for equations in Kovalevskaya form) S n z k ∈ z . The identity (cid:88) i,j,k,l S − im S − jn (cid:18) ∂S m S lm S kn ∆ ∂S im S jn ∆ · S m ∂F ( z , [∆]) ∂S lm S kn ∆ (cid:19) = (cid:88) k,l S − m S − lm S − kn S m ∂F ( z , [∆]) ∂S lm S kn ∆ , together with a similar identity for the G term, simplifies (32) to E ∆ (Div F ( z , [∆])) = (cid:88) i,j,k S − im S − jn (cid:18) ∂S m z k ∂S im S jn ∆ · S m ∂∂z k ( F ( z , [∆]) − F ( z , [0])) (cid:19) . From this, it immediately follows that (cid:8) E ∆ (Div F ( z , [∆])) (cid:9)(cid:12)(cid:12)(cid:12) [∆]= = 0 , so the root is zero.3.4. A trivial characteristic implies a trivial CLaw.
Having proved that the characteristic of atrivial CLaw is a trivial characteristic, it is now clear that trivial densities may be added to the densitiesof any CLaw without affecting the root. In particular, the components of F ( z , [ ∆ ]) − F ( z , [ ]) are trivialdensities, so any explicit dependence on [ ∆ ] in the densities can be removed. Therefore, we now assumethat the densities of any particular CLaw for (18) do not depend explicitly on [ ∆ ]; with this assumption,only the second kind of triviality can occur. Consequently, the CLaw has densities of the form F := F ( m, n, u , . . . , u K − ) , and G := G ( m, n, u , . . . , u K − ) . IFFERENCE CONSERVATION LAW CHARACTERISTICS 9 m u u u ω R ω ω − R n S m FS − Rn S m F Figure 3.
A graphical representation of the terms in the characteristic for a scalarP∆E with two independent variables. The solid black box encloses points on which S m F depends and the dash-dot box encloses points on which S − Rn S m F depends. Thedashed boxes enclose points on which ω − R , ω and ω R depend.Clearly, S n G does not depend on u K , so the root is Q = R (cid:88) j =0 S − jn ∂∂S jn ∆ F ( m + 1 , n, u , . . . , u K − , [ S jn ∆ + ω j ]) (cid:12)(cid:12) [∆]=0 . Here we have assumed that F depends on { u K − ,j , j = 0 , . . . , R } and on no other u K − ,j ; there alwaysexists an R ≥ m, n ) relative to which all u ij are compared. The shifts of S m F on which Q depends are shown inFigure 3; here and henceforth (for brevity), we refer to the values u ij that occur in any expression as‘points’ on which the expression depends.Let u L be the leftmost point in u on which ω = ω depends. Then ∂ω j ∂u L + j ) (cid:54) = 0 , j ∈ Z ; ∂ω i ∂u L + j ) = 0 , i > j i, j ∈ Z . On solutions of the P∆E, each point u L + j ) (such as the square points in Figure 3) may be replaced by ω j (represented by the discs in Figure 3) as an independent variable in Q , because the determinant ofthe Jacobian ∂ ( ω − R ,...,ω R ) ∂ ( u L − R ) ,...,u L + R ) ) is nonzero. Therefore Q = R (cid:88) j =0 S − jn ∂∂ω j F ( m + 1 , n, u , . . . , u K − , [ ω j ]) =: E ω ( S m F ) . (33)This is a restricted difference Euler operator corresponding to variations in ω ; in effect, it treats m andthe u i terms as parameters. The kernel of this operator is made up of total divergences of the form( S n − I ) H , and functions of z \ u only (see Lemma 3.2 below). Thus if the characteristic is zero onsolutions of the P∆E then F ( m +1 , n, u , . . . , u K − , [ ω j ]) = ( S n − I ) H ( m +1 , n, u , . . . , u K − , [ ω j ]) + f ( m +1 , n, u , . . . , u K − ) , for some f , and so F ( z ) = ( S n − I ) H ( m, n, u , . . . , u K − , [ u ( K − j ]) + f ( m, n, u , . . . , u K − ) . Adding the trivial CLaw F T = − ( S n − I ) H, G T = ( S m − I ) H, (34)to the original densities gives the equivalent densities˜ F = f ( m, n, u , . . . , u K − ) , ˜ G = G + ( S m − I ) H. The ˜ G density may contain ∆ terms, but these can be removed by adding a trivial density. Thus thedivergence expression for these densities cannot contain any ∆ terms, so in order for it to be a CLaw itmust vanish identically and is thus trivial.The following lemma identifies the kernel of the restricted Euler operator; part of the proof will beused in § Lemma 3.2.
The kernel of E ω consists of sums of functions that are independent of ω and its shifts,together with total divergences in the n direction.Proof. In the following, we use the notation u = { u ij : 1 ≤ i ≤ K − } and ω λ := λω + (1 − λ ) g ( m, n, u ) , Table 2.
Roots of the transformed dpKdV Q = 2 ( − m +1 ( u − u ) Q = ( u − u )( u + u − ω ) + α − βQ = ( − m (cid:0) ( u − u )( u + u + 2 ω ) + α − β (cid:1) Q = 4 ( − m (cid:0) ω ( u − u ) + αu − β u (cid:1) Q = ( β − α ) ( ω − − ω ) − (cid:16) u − u + β − αω − − ω (cid:17) − − ( β − α ) ( ω − ω ) − (cid:16) u − u + β − αω − ω (cid:17) − + ( ω − ω ) − − ( ω − − ω ) − Q = ( β − α ) ( ω − ω ) − (cid:16) u − u + β − αω − ω (cid:17) − − ( β − α ) ( ω − − ω ) − (cid:16) u − u − + β − αω − − ω (cid:17) − + ( ω − − ω ) − − ( ω − ω ) − Q = ( β − α )( ω − ω ) − (cid:26)(cid:16) u − u + β − αω − ω (cid:17) − + (cid:16) u − u + β − αω − ω (cid:17) − (cid:27) + ( m − n ) Q + nQ + 2 ( ω − ω ) − where g is any convenient function (usually, g = 0). For any differentiable function f = f ( m, n, u , [ ω ]), ddλ f ( m, n, u , [ ω λ ]) = (cid:88) j ∂f∂S jn ω λ S jn ( ω − g ( n , u ))=( ω − g ) E ω λ ( f ( m, n, u , [ ω λ ])) + ( S n − I ) h ( m, n, u , [ ω ] , λ ) , (35)for some function h . Integrating (35) with respect to λ , we obtain f ( m, n, u , [ ω ]) = f ( m, n, u , [ g ])+( ω − g ) (cid:90) λ =0 E ω λ ( f ( m, n, u , [ ω λ ])) dλ +( S n − I ) (cid:90) λ =0 h ( m, n, u , [ ω ] , λ ) dλ If f ∈ ker( E ω ) then the result follows. (cid:3) Using roots to detect equivalence
For P∆Es, the fact that a CLaw may involve a large number of points can make it difficult toidentify its underlying order. The results of § maple , we have calculatedthe roots of the CLaws of (23) that are listed in Table 1. These roots are displayed in Table 2, expressedin terms of the functions ω ij (because this is more compact than pulling back to write each Q i in termsof z ). The transformed dpKdV equation also has the following CLaw F = ( − m +1 (cid:18) u (cid:18) u + β − αu − u (cid:19) − β (cid:19) + ( − m +1 (cid:18) α − u (cid:18) u + β − αu − u (cid:19)(cid:19) ,G = 2 u ( − m +1 y (cid:32)(cid:18) u + β − αu − u (cid:19) − (cid:18) u + β − αu − u (cid:19) (cid:18) u − + β − αu − − u (cid:19)(cid:33) ++ ( − m +1 y (cid:18) α (cid:18) u + β − αu − u (cid:19) + α (cid:18) u − + β − αu − − u (cid:19) − β (cid:18) u + β − αu − u (cid:19)(cid:19) , where y = u − u − − β − αu − − u + β − αu − u . This apparently high-order CLaw’s root is − m ( u − u ) , which shows that actually it is equivalentto a multiple of the first CLaw in Table 1. IFFERENCE CONSERVATION LAW CHARACTERISTICS 11 F S m F G S n G S − m S n G u u m mn n Figure 4.
The figure on the left shows the densities for the third CLaw in the hierarchy;on the right the extreme shifts of the densities in the characteristic are shown.4.1.
The Gardner CLaws for dpKdV.
In [21], Rasin and Schiff used a discrete version of the Gardnertransformation to construct an infinite number of CLaws for the dpKdV equation (22). (Rasin [20] hassubsequently used the same approach to generate an infinite hierarchy of CLaws for all the equations inthe ABS classification and one asymmetric equation.) They showed that these CLaws were distinct bytaking a continuum limit and showing that the resulting CLaws for the continuous equation are distinct.By using roots, one can prove that their CLaws are distinct without having to take a continuum limit.In this section, we consider the dpKdV equation on the quad-graph rather than in Kovalevskaya form .The dpKdV equation can be solved for any of the points on the quad-graph; hence we choose the initialconditions z = { m, n, u i , u − j , u k | i, j, ∈ N , k ∈ Z } which are shown by dashed lines in Figure 4.The densities for the CLaws generated by the Gardner transformation are the functions F i and G i inthe expansion of F = − ln( u − u ) − ln (cid:32) u − u ∞ (cid:88) i =1 v ( i )00 (cid:15) i (cid:33) = ∞ (cid:88) i =0 F i (cid:15) i , (36) G = ln (cid:15) − ln( u − u ) + ln (cid:32) v (1)00 ∞ (cid:88) i =1 v ( i +1)00 (cid:15) i (cid:33) = ln (cid:15) + ∞ (cid:88) i =0 G i (cid:15) i , (37)in powers of (cid:15) ; here v (1)00 = 1 u − u , v ( i )00 = 1 u − u i − (cid:88) j =1 v ( j )00 v ( i − j )10 , and v ( i )00 is referred to as the i th order v term. To prove that the CLaws are distinct the following lemmais used. Lemma 4.1.
For each α ∈ N , ∂∂u ( α +1)0 v ( α )00 (cid:54) = 0 and ∂∂u ( α + j )0 v ( α )00 = 0 , j ≥ . (38) Proof.
Proof is by induction; the base case ( α = 1) is immediate. Assume that the lemma holds for α = k −
1, where k ≥
2. For each i ∈ N , ∂∂u ( k + i )0 v ( k )00 = 1 u − u k − (cid:88) j =1 (cid:18) ∂∂u ( k + i )0 v ( j )00 (cid:19) v ( k − j )10 + v ( j )00 ∂∂u ( k + i )0 (cid:16) S m v ( k − j )00 (cid:17) . The first term in the summation is zero for all j , because the highest value j can take is k − i is 1. Similarly, the second term vanishes for all j ≥
2, so only one term is left, namely ∂∂u ( k + i )0 v ( k )00 = (cid:16) v (1)00 (cid:17) S m (cid:18) ∂∂u ( k + i − v ( k − (cid:19) . (39)Using the induction hypothesis, if i = 1 then (39) is nonzero and if i >
1, (39) is zero. (cid:3)
A consequence of this lemma is that, for α ≥ ∂∂u ( α +2)0 (cid:32) v ( α +1)00 v (1)00 (cid:33) = 1 v (1)00 ∂∂u ( α +2)0 (cid:16) v ( α +1)00 (cid:17) = v (1)00 S m (cid:18) ∂∂u ( α +1)0 v ( α )00 (cid:19) (cid:54) = 0 . (40) Kovelevskaya form is convenient for proving that the root characterizes each equivalence class of CLaws. However,for any explicit P∆E, the root (with respect to an appropriate set of initial conditions) can also be calculated withouttransforming to Kovalevskaya form.
As the second factor in (40) doesn’t depend on u , we obtain ∂ ∂u ∂u ( α +2)0 (cid:32) v ( α +1)00 v (1)00 (cid:33) (cid:54) = 0 . (41)Expanding out (37) shows that, for α ≥
1, the highest order v term in G α is v ( α +1)00 /v (1)00 ; by Lemma4.1, this is the only term in G α to depend on u ( α +2)0 . Therefore S n (cid:0) v ( α +1)00 /v (1)00 (cid:1) is the only term inthe CLaw to depend on u ( α +2)1 and so to depend on S α +1 m ∆ (because S m F depends only on points in z except for u – see Figure 4). Thus the root is E ∆ ( S m F α + S n G α ) (cid:12)(cid:12) [∆]= = (cid:40) ∂S m F α ∂ ∆ + α (cid:88) i =0 S − im ∂S n G α ∂S im ∆ + S − ( α +1) m S n (cid:32) ∂∂u ( α +2)0 (cid:32) v ( α +1)00 v (1)00 (cid:33)(cid:33)(cid:41) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [∆]= . The final term is the only one to depend on u − ( α +1) , ; no other term is shifted as far back. From (41), ∂∂u − ( α +1)1 E ∆ ( S m F α + S n G α ) | [∆]= = S − ( α +1) m S n (cid:32) ∂ ∂u ∂u ( α +2)0 (cid:32) v ( α +1)00 v (1)00 (cid:33)(cid:33) (cid:54) = 0 , so this term cannot be a linear combination of the other terms. Therefore, the characteristic does notvanish on solutions of dpKdV and so the CLaw is nontrivial. The roots of the lower-order CLaws withdensities ( F , G ) , . . . , ( F α − , G α − ) do not depend on u − ( α +1)1 , so the CLaw ( F α , G α ) cannot be a linearcombination of these CLaws and their shifts. Thus the CLaws generated by the Gardner transformationare distinct. 5. The converse of Noether’s Theorem
One of the most useful significant applications of the root is to establish that the converse of Noether’sTheorem holds for difference equations. Given a Lagrangian, L , the Euler–Lagrange equation is E ( L ) := (cid:88) ij S − im S − jn (cid:18) ∂L∂u ij (cid:19) = 0 . Each symmetry generator for the Euler–Lagrange equation is the prolongation of X = Q ∂∂u ; (42)the function Q is the characteristic of the symmetry generator. In particular, X generates variationalsymmetries if XL is a total divergence, in which case E ( XL ) ≡ . In this case, there exist functions f and g such that (cid:88) ij (cid:0) S im S jn Q (cid:1) ∂∂u ij L = ( S m − I ) f + ( S n − I ) g. (43)Summation by parts is then used to rewrite (43) as Q · E ( L ) = (cid:88) i,j QS − im S − jn (cid:18) ∂∂u ij L (cid:19) = ( S m − I ) F + ( S n − I ) G, (44)for functions F and G whose precise form is irrelevant. Thus if Q is the characteristic of a variationalsymmetry generator, it is also the characteristic of a CLaw for the Euler–Lagrange equation. Twovariational symmetries are equivalent if they differ by a symmetry whose characteristic vanishes onsolutions of the Euler–Lagrange equation (i.e. a trivial symmetry). Therefore, if the Euler–Lagrangeequation is explicit (in which case the CLaw is trivial if and only only if the root is zero), there is abijective correspondence between equivalence classes of variational symmetries and CLaws. IFFERENCE CONSERVATION LAW CHARACTERISTICS 13 Reconstruction of CLaws from roots
If the characteristic of a CLaw is known then the densities for the CLaw can, in principle, be re-constructed using homotopy operators [11, 19]. For PDEs (given an initialization), the root of a CLawcan be calculated by the same approach as we have used; the root is necessarily a characteristic as aconsequence of the chain rule. By contrast, the root of a CLaw for a P∆E may not be a characteristic.So the key step in reconstructing a CLaw from its root is to find a characteristic which has that root.Our starting-point is the proof of Lemma 3.2. Replacing f by S m F and using the definition (33), weobtain F ( m + 1 , n, u , [ ω ]) = ( ω − g ) (cid:90) λ =0 Q ( m, n, u , [ ω λ ]) dλ + ( S n − I ) H ( m, n, u , [ ω ]) + f ( m + 1 , n, u ) , for some f to be determined. The H term can be set to zero without loss of generality by adding atrivial CLaw of the second kind, so we can assume that F ( m + 1 , n, u , [ ω ]) = ( ω α − g α ) (cid:90) λ =0 Q ( m, n, u , [ ω λ ]) dλ + f ( m + 1 , n, u ) . (45)In general, Q , and as a result (45), contains negative shifts of ω . These can be removed term-by-termby adding trivial CLaws of the second kind, until one obtains an equivalent density S m F that has nonegative shifts of ω .Equation (45) (shifted if necessary, as discussed above) contains all of the ω -dependence of the CLaw.Having obtained this, a characteristic for the CLaw is calculated by replacing ω j in (45) by S jn ∆ + ω j .From (30), Q ( z , [∆]) = (cid:90) λ =0 E ∆ ( F ( m + 1 , n, u , [∆ + ω ])) | ∆ (cid:55)→ λ ∆ dλ, (46)and the CLaw can be written as C := Q ( z , [∆]) · ∆ = Q (cid:0) z , [ u K − ω ( z )] (cid:1)(cid:0) u K − ω ( z ) (cid:1) . Homotopy operators (see [11]) may then be used to find the densities. Alternatively, once the ω depen-dence has been found, the arbitrary function f and the other densities can be constructed directly by avariant of the method that we used for O∆Es (see [10] for details).6.1. Example: reconstruction of CLaws of dpKdV.
To illustrate this, we will reconstruct twoCLaws of dpKdV from their roots. For both examples, we choose g ( n , u ) = 0, so ω λ = λω . (Inpractice, we start with this choice and only change it if the integral is singular.) First, we use the fourthroot in Table 2, which gives F ( m + 1 , n, u , u , ω ) = ω (cid:90) λ =0 − m +1 (cid:8) ( u ) λ ω − λ ω ( u ) + αu − βu (cid:9) dλ + f ( m +1 , n, u , u )= 2 ( − m +1 ω (cid:8) ω (cid:0) ( u ) − ( u ) (cid:1) +2( αu − βu ) (cid:9) + f ( m + 1 , n, u , u ) . (47) This has no negative shifts of ω so, from (45) and (46), a characteristic is Q ( z , ∆) = (cid:90) λ =0 ∂∂ ∆ (cid:110) − m +1 (∆ + ω ) (cid:0) (∆ + ω )( u − u )+2( αu − βu ) (cid:1)(cid:111)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:55)→ λ ∆ dλ, =2 ( − m +1 (cid:0) (∆ + 2 ω )( u − u )+2( αu − βu ) (cid:1) Thus C := Q ∆ = 2 ( − m +1 ( u − ω ) (cid:0) ( u + ω )( u − u )+2( αu − βu ) (cid:1) =2 ( − m ( β + α − u u − u u − u u − u u ) ( u u − u u − u u + u u − β + α ) . (48) This is a total divergence, so applying the homotopy operator from [11] gives the required densities F = 16 ( − m +1 (cid:0) u u − β + 20 u β u + 4 u − αu + 2 u − u β − u u (cid:1) + 16 ( − m +1 (cid:0) u u − − − u βu − − − u u − α − u u α + 2 u − u (cid:1) + 16 ( − m +1 (cid:0) u u − αu u − u − u − u u − − u u − + 5 u u (cid:1) ,G = 16 ( − m (cid:0) − u u α + 2 u − u − u u − − u u (cid:1) + 16 ( − m (cid:0) − u βu − − + 10 u u α + u u − − + 2 u u − β (cid:1) + 16 ( − m (cid:0) u u − β +5 u u − u u − − u u − α − nβ + 12 nα (cid:1) . These densities contain points of the form u − j and u j . To find equivalent densities that are given solelyin terms of z , one must shift the CLaw forwards and then use the dpKdV equation (23) to pull all termsback onto the initial conditions; this leads to even longer expressions!In practice it is much easier to use the direct construction method, which leads to more compactexpressions for the densities. Starting from (47), we know that the densities are of the form F =2 ( − m u (cid:0) u ( u − u )+ 2 ( αu − βu ) (cid:1) + f ( m, n, u , u ) ,G = G ( m, n, u , u ) . Substituting these into the CLaw and evaluating the result on solutions gives0 = C | ∆=0 =2 ( − m +1 (cid:18) u + β − αu − u (cid:19)(cid:18)(cid:18) u + β − αu − u (cid:19) ( u − u )+ 2 ( αu − βu ) (cid:19) − − m u (cid:0) u ( u − u )+ 2 ( αu − βu ) (cid:1) + f ( m + 1 , n, u , u ) − f ( m, n, u , u ) + G ( m, n + 1 , u , u ) − G ( m, n, u , u ) . Differentiating this expression, we obtain0 = ∂ C | ∆=0 ∂u ∂u = ∂ f∂u ∂u . Consequently, f = ˜ f ( m, n, u ) + h ( m, n, u ); however, h can be set to zero by adding a trivial CLaw.Therefore 0 = ∂C | ∆=0 ∂u = 4 ( − m u u + 4 ( − m +1 u α + ∂∂u G ( m, n + 1 , u , u ) , and so G ( m, n, u , u ) = 2 ( − m +1 u u + 4 ( − m u α u + g ( m, n, u ) . Dropping the tilde, 0 = ∂C | ∆=0 ∂u = − ∂f∂u , which yields f = f ( m, n ). The final differentiation is0 = ∂C | ∆=0 ∂u = − ∂g∂u , so g = g ( m, n ). The CLaw now simplifies to( S m − I ) f ( m, n ) + ( S n − I ) g ( m, n ) = 2 ( − m ( α − β ) , a solution of which is f = 0 and g = 2 n ( − m ( α − β ). So the reconstructed densities are F =2 u ( − m (cid:0) u ( u − u ) + 2 α u − β u (cid:1) ,G =2 ( − m (cid:0) − u u + 2 u α u + n ( α − β ) (cid:1) , which are equivalent to the densities found by the homotopy method, as they have the same root. IFFERENCE CONSERVATION LAW CHARACTERISTICS 15
For a more complicated example, consider the densities of the sixth Claw in Table 1. Its root dependson ω − , ω and ω . The characteristic is constructed so that terms depending on ω − do not dependon ω . The term that depends on ω − is h ( m, n, u , ω − , ω ) := ( ω − − ω ) − − ( β − α )( ω − − ω ) − (cid:18) u + β − αω − − ω − u − (cid:19) − . Then let h ( m, n, u , ω , ω ) := ω Q ( m, n, u , [ λω ]) + ( S n − I ) ( ω h ( n , u , [ λω ]))= − ω u − ω u − ω u + ω u − λ ω u + λ ω u − λ ω u − β + λ ω u + α , which is a function that only depends on ω and ω ; this is integrated with respect to λ to obtain S m F | [∆]=0 = (cid:90) h d λ + f ( m + 1 , n, u , u , u )= − ln (cid:18) ( ω − ω )( u − u ) + α − βα − β (cid:19) + f ( m + 1 , n, u , u , u ) . (49)As in the last example, we could calculate the characteristic from (49) and use the homotopy operatorto find densities for the CLaw. Once again, however, the direct construction method is preferable. Byshifting (49) and choosing G to depend on appropriate values of z , the densities have the form F = − ln (cid:18) ( u − u )( u − u ) + α − βα − β (cid:19) + f ( m, n, u , u , u ) ,G = G ( m, n, u , u , u , u ) . The dependence on the dpKdV equation is already determined by (49), so all that remains is to find f and G . The same process (differential elimination followed by integration) is used as before. Skippingthe details, we obtain F = − ln (cid:18) ( u − u )( u − u ) + α − βα − β (cid:19) , G = − ln ( u − u ) , as required. 7. Finding CLaws
The Adjoint of the Linearized Symmetry Operator.
The Gˆateaux derivative of a functional P is the operator defined in [11] by D P ( Q ) = lim (cid:15) → (cid:18) P [ u + (cid:15)Q [ u ]] − P [ u ] (cid:15) (cid:19) = (cid:26) dd(cid:15) P [ u + (cid:15)Q [ u ]] (cid:27) (cid:12)(cid:12)(cid:12) (cid:15) =0 . Explicitly, the Gˆateaux derivative of P is the shift operator with entries D P = (cid:88) ij ∂P∂u ij S im S jn . Therefore its adjoint with respect to the (cid:96) inner product is D ∗ P = (cid:88) i,j (cid:18) S − im S − jn ∂P∂u ij (cid:19) S − im S − jn , and so the Euler operator, E , is defined by E ( P [ u ]) = D ∗ P (1) , just as for PDEs. Using the Leibniz rule, E ( P · Q ) = D ∗ P · Q (1) = D ∗ P ( Q ) + D ∗ Q ( P ) . (50)The action of the vector field X = Q ∂/∂u on the functional P ispr X ( P ) = D P ( Q ) . So the linearized symmetry condition for a given difference equation, ∆ = 0, is0 = D ∆ ( Q ) | [∆]=0 . Table 3.
Solutions of the ALSC for the potential Lotka–Volterra equation Q = ( − n +2) u u − u + n − u + nu u u ( u + u ) Q = u + u u u ( u + u ) − u u − u Q = − mu u − u u ( u + u − ) − ( m +1) u u ( u + u )( u + u ) Q = − u u − u u ( u + u − ) − u u u ( u + u )( u + u ) Q = ( − m + n ( u u + u ( u + u ) ) u u ( u + u ) Q = u u − u ( u + u ) u u ( u + u ) The Euler operator acting on a expression is zero if and only if that expression is a total divergence [11].Therefore Q is a characteristic of a CLaw if and only if0 = E ( Q · ∆) = D ∗ ∆ ( Q ) + D ∗ Q (∆) . Restricting this to solutions of the difference equation gives a necessary condition for Q to be a charac-teristic: 0 = D ∗ ∆ ( Q ) | [∆]=0 . (51)In other words, the characteristics are members of the kernel of the adjoint of the linearized symmetrycondition (ALSC), restricted to solutions. Arriola [4] showed that (51) must be satisfied for first integralsof autonomous ordinary difference equations. In a notable paper that introduces the idea of co-recursionoperators for integrable difference equations [17], Mikhailov et al. define a cosymmetry of a differenceequation as being a member of the kernel of the ALSC. They state (parenthetically) that cosymmetriesare characteristics of CLaws, but do not justify this.In this section, we give an example of a P∆E where not every cosymmetry is a characteristic of aCLaw (see [2, 3] for a discussion of this point for differential equations). First, we find functions Q thatsatisfy (51), using methods similar to those used to find symmetries of difference equations. Then we findadditional constraints on Q by applying the difference Euler operator to Q ∆. Finally, we reconstruct thedensities using homotopy operators or inspection. For brevity, we omit most details of the calculations.7.2. CLaws of the Potential Lotka-Volterra equation.
The potential Lotka–Volterra equation(pLV) u u − u u = 1 . (52)is an integrable equation on the quad-graph. (It belongs to Class 4 of Hietarinta & Viallet’s classificationof quadratic quad-graph equations with polynomial degree growth [8], and is a potential form of thediscrete Lotka-Volterra equation introduced by Hirota & Tsujimoto [9].)Rasin & Hydon’s method for finding symmetries of quad-graph equations [25] is readily adapted tofind solutions of the ALSC, shifted (for convenience) to0 = (cid:8)(cid:0) Q − S n (cid:0) ω , Q (cid:1) − S m (cid:0) ω , Q (cid:1) − S m S n (cid:0) ω , Q (cid:1)(cid:1)(cid:9) (cid:12)(cid:12) [∆]=0 . (53)We search for solutions of (53) which are pulled back onto the initial conditions z = { m, n, u i , u j } . Inparticular, we will look for roots of the form Q = Q ( m, n, u − , u − , u , u , u , u , u ) . (54)The corresponding CLaws depend on m, n, u − , u − , u , u and u only, and are therefore called‘five-point CLaws’.By differential elimination and integration, one obtains the solutions of the ALSC; these are listedin Table 3. However, when one tries to use these roots to construct the corresponding CLaws of thepLV equation, the algorithm fails for Q (see Table 4). It turns out that E ( Q ∆) (cid:54) = 0, and thereforethis cosymmetry does not correspond to the characteristic of a CLaw. This is not surprising, as (51) is IFFERENCE CONSERVATION LAW CHARACTERISTICS 17
Table 4.
Five-point CLaws of the potential Lotka–Volterra equation
The solution Q of the ALSC is not the root of a characteristic. F = ( u − − u )( u − + u − ) u − u − , G = − u − u − F = m ln (cid:16) u u − (cid:17) − ln ( u ) , G = ( m + 1) ln (cid:16) u + u − u (cid:17) − ln ( u + u − ) F = ln (cid:16) u u − (cid:17) , G = ln (cid:16) u + u − u (cid:17) F = ( − m + n u u , G = ( − m + n ( u − u ) u u F = u − u u , G = − u + u u u necessary but not sufficient; nevertheless, to the best of our knowledge, this is the first example of sucha cosymmetry of a P∆E. It raises an interesting question: can an additional constraint be found thatguarantees a solution of the ALSC is a root of a CLaw without the need to work through the (lengthy)process of reconstructing the characteristic? For instance, Mikhailov et al. [17] have used a co-recursionoperator to generate an infinite hierarchy of cosymmetries for the Viallet equation, so a simple test toshow that these produce an infinite hierarchy of CLaws would be useful.8. Acknowledgements
We thank the Natural Environment Research Council for funding this research. We also thank thereferees for their very helpful recommendations.
References [1] V. E. Adler, A. I. Bobenko, and Yu. B. Suris, Classification of integrable equations on quad-graphs. The consistencyapproach, Comm. Math. Phys. 233 (2003), 513–543.[2] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. I.Examples of conservation law classifications, Eur. J. Appl. Math. 13 (2002), 545–566.[3] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. II.General treatment, Eur. J. Appl. Math. 13 (2002), 567–585.[4] L. Arriola, First integrals, invariants and symmetries for autonomous difference equations, Pr. Inst. Mat. Nats. Akad.Nauk Ukr. Mat. 50 (2004), 1253–1260.[5] V. A. Dorodnitsyn, A finite-difference analogue of Noether’s theorem, Dokl. Akad. Nauk 328 (1993), 678–682. (InRussian).[6] V. A. Dorodnitsyn, Noether-type theorems for difference equations, Appl. Numer. Math. 39 (2001), 307–321.[7] T. J. Grant, Characteristics of Conservation Laws for Finite Difference Equations, PhD thesis, Department of Math-ematics, University of Surrey, 2011.[8] J. Hietarinta and C.-M. Viallet, Searching for integrable lattice maps using factorization, J. Phys. A: Math. Theor. 40(2007), 12629–12643.[9] R. Hirota and S. Tsujimoto, Conserved quantities of a class of nonlinear difference-difference equations, J. Phys. Soc.Japan 64 (1995), 31–38.[10] P. E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A 34: Math.Gen. (2001), 10347–10355.[11] P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, Found. Comput. Math. 4 (2004),187–217.[12] P. E. Hydon and E. L. Mansfield, Extensions of Noether’s second theorem: from continuous to discrete systems, Proc.Roy. Soc. Lond. Ser. A 467 (2011), 3206–3221.[13] B. A. Kupershmidt, Discrete Lax Equations and Differential-Difference Calculus, Ast´erisque 123, Soci´et´eMath´ematique de France, Paris, 1985.[14] L. Mart´ınez Alonso, On the Noether map, Lett. Math. Phys. 3 (1979), 419–424.[15] K. Maruno and G. R. W. Quispel, Construction of integrals of higher-order mappings, J. Phys. Soc. JapanDOI:10.1143/JPSJ.75.123001.[16] A. Mikhailov, J. P. Wang, and P. Xenitidis, Recursion operators, conservation laws, and integrability conditions fordifference equations, Theor. Math. Phys. 167 (2011), 421–443.[17] A. V. Mikhailov, J. P. Wang, and P. Xenitidis, Cosymmetries and Nijenhuis recursion operators for difference equations,Nonlinearity 24 (2011), 2079–2097.[18] E. Noether, Invariante Variationsprobleme, Nachr. D. K¨onig. Gesellsch. D. Wiss. Zu G¨ottingen, Math-phys. Klasse(1918), 235257. M. A. Tavel, Invariant variation problems (English translation), Transport Theory Statist. Phys. 1(1971), 186–207.[19] P. J. Olver, Applications of Lie Groups to Differential Equations, second edition, Springer-Verlag, New York, 1993. [20] A. G. Rasin, Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method, J.Phys. A: Math. Theor. DOI:10.1088/1751-8113/43/23/235201.[21] A. G. Rasin and J. Schiff, Infinitely many conservation laws for the discrete KdV equation, J. Phys. A: Math. Theor.DOI:10.1088/1751-8113/42/17/175205.[22] O. G. Rasin and P. E. Hydon, Conservation laws of discrete Korteweg-de Vries equation, Symmetry IntegrabilityGeom. Methods Appl. DOI:10.3842/SIGMA.2005.026.[23] O. G. Rasin and P. E. Hydon, Conservation laws of NQC-type difference equations. J. Phys. A: Math. Gen. 39 (2006),14055–14066.[24] O. G. Rasin and P. E. Hydon, Conservation laws for integrable difference equations, J. Phys. A: Math. Theor. 40(2007), 12763–12773.[25] O. G. Rasin and P. E. Hydon, Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math.119 (2007) 253–269.
Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
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