Nonlocal conservation laws of PDEs possessing differential coverings
aa r X i v : . [ n li n . S I] S e p NONLOCAL CONSERVATION LAWS OF PDES POSSESSING DIFFERENTIALCOVERINGS
I. KRASIL ′ SHCHIK
To the memory of Alexandre Vinogradov, my teacher
Abstract.
In his 1892 paper [2], L. Bianchi noticed, among other things, that quite simple transfor-mations of the formulas that describe the B¨acklund transformation of the sine-Gordon equation leadto what is called a nonlocal conservation law in modern language. Using the techniques of differentialcoverings [7], we show that this observation is of a quite general nature. We describe the proceduresto construct such conservation laws and present a number of illustrative examples.
Contents
Introduction 11. Preliminaries 22. The main result 53. Examples 7Discussion 10Acknowledgments 10References 10
Introduction
In [2], L. Bianchi, dealing with the celebrated B¨acklund auto-transformation ∂ ( u − w ) ∂x = sin( u + w ) , ∂ ( u + w ) ∂y = sin( u − w ) (1)for the sine-Gordon equation ∂ (2 u ) ∂x∂y = sin(2 u ) (2)in the course of intermediate computations (see [2, p. 10]) notices that the function ψ = ln ∂u∂C , where C is an arbitrary constant on which the solution u may depend, enjoys the relations ∂ψ∂x = cos( u + w ) , ∂ψ∂y = cos( u − w ) . Reformulated in modern language, this means that the 1-form ω = cos( u + v ) dx + cos( u − v ) dy is a nonlocal conservation law for Eq. (1).Actually, Bianchi’s observation is of a very general nature and this is shown below. Date : September 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Nonlocal conservation laws, differential coverings.The work was partially supported by the RFBR Grant 18-29-10013 and IUM-Simons Foundation. I changed the original notation slightly ′ SHCHIK
In Sec. 1, I shortly introduce the basic constructions in nonlocal geometry of PDEs, i.e., the theoryof differential coverings, [7]. Sec. 2 contains an interpretation of the result by L. Bianchi in the mostgeneral setting. In Sec. 3, a number of examples is discussed.Everywhere below we use the notation F ( · ) for the R -algebra of smooth functions, D ( · ) for the Liealgebra of vector fields, and Λ ∗ ( · ) = ⊕ k ≥ Λ k ( · ) for the exterior algebra of differential forms.1. Preliminaries
Following [3], we deal with infinite prolongations E ⊂ J ∞ ( π ) of smooth submanifolds in J k ( π ),where π : E → M is a smooth locally trivial vector bundle over a smooth manifold M , dim M = n ,rank π = m . These E are differential equations for us. Solutions of E are graphs of infinite jets thatlie in E . In particular, E = J ∞ ( π ) is the tautological equation 0 = 0.The bundle π ∞ : E → M is endowed with a natural flat connection C : D ( M ) → D ( E ) called theCartan connection. Flatness of C means that C [ X,Y ] = [ C X , C Y ] for all X , Y ∈ D ( M ). The distributionon E spanned by the fields of the form C X (the Cartan distribution) is Frobenius integrable. We denoteit by C ⊂ D ( E ) as well.A (higher infinitesimal) symmetry of E is a π ∞ -vertical vector field S ∈ D ( E ) such that [ X, C ] ⊂ C .Consider the submodule Λ kh ( E ) generated by the forms π ∗∞ ( θ ), θ ∈ Λ k ( M ). Elements ω ∈ Λ kh ( E )are called horizontal k -forms. Generalizing slightly the action of the Cartan connection, one can applyit to the de Rham differential d : Λ k ( M ) → Λ k +1 ( M ) and obtain the horizontal de Rham complex0 / / F ( E ) / / . . . / / Λ kh ( E ) d h / / Λ k +1 h ( E ) / / . . . / / Λ nh ( E ) / / E . Elements of its ( n − H n − h ( E ) are called conservation laws of E . Wealways assume E to be differentially connected which means that H h ( E ) = R . Coordinates.
Consider a trivialization of π with local coordinates x , . . . , x n in U ⊂ M and u , . . . , u m in the fibers of π | U . Then in π − ∞ ( U ) ⊂ J ∞ ( π ) the adapted coordinates u iσ arise and the Cartan con-nection is determined by the total derivatives C : ∂∂x i D i = ∂∂x i + X j,σ u jσi ∂∂u jσ . Let F = ( F , . . . , F r ), where F j are smooth functions on J k ( π ). The the infinite prolongation of thelocus { z ∈ J k ( π ) | F ( z ) = · · · = F r ( z ) = 0 } ⊂ J k ( π )is defined by the system E = E F = { z ∈ J ∞ ( π ) | D σ ( F j )( z ) = 0 , j = 1 , . . . , r, | σ | ≥ } , where D σ denotes the composition of the total derivatives corresponding to the multi-index σ . Thetotal derivatives, as well as all differential operators in total derivatives, can be restricted to infiniteprolongations and we preserve the same notation for these restrictions. Given an E , we always chooseinternal local coordinates in it for subsequent computations. To restrict an operator to E is to expressthis operator in terms of internal coordinates.Any symmetry of E is an evolutionary vector field E ϕ = X D σ ( ϕ j ) ∂∂u jσ (summation on internal coordinates), where the functions ϕ , . . . , ϕ m ∈ F ( E ) satisfy the system X σ,α ∂F j ∂u ασ D σ ( ϕ α ) = 0 , j = 1 , . . . , r. A horizontal ( n − ω = X i a i dx ∧ · · · ∧ dx i − ∧ dx i +1 ∧ · · · ∧ dx n ONLOCAL CONSERVATION LAWS 3 defines a conservation law of E if X i ( − i +1 D i ( a i ) = 0 . We are interested in nontrivial conservation laws, i.e., such that ω is not exact.Finally, E is differentially connected if the only solutions of the system D ( f ) = · · · = D n ( f ) = 0 , f ∈ F ( E ) , are constants.Consider now a locally trivial bundle τ : ˜ E → E such that there exists a flat connection ˜ C in π ∞ ◦ τ : ˜ E → M . Following [7], we say that τ is a (differential) covering over E if one has τ ∗ ( ˜ C X ) = C X for any vector field X ∈ D ( M ). Objects existing on ˜ E are nonlocal for E : e.g., symmetries of ˜ E are nonlocal symmetries of E , conservation laws of ˜ E are nonlocal conservation laws of E , etc. Aderivation S : F ( E ) → F ( ˜ E ) is called a nonlocal shadow if the diagram F ( E ) C X / / S (cid:15) (cid:15) F ( E ) S (cid:15) (cid:15) F ( ˜ E ) ˜ C X / / F ( ˜ E )is commutative for any X ∈ D ( M ). In particular, any symmetry of the equation E , as well asrestrictions ˜ S (cid:12)(cid:12)(cid:12) F ( E ) of nonlocal symmetries may be considered as shadows. A nonlocal symmetry issaid to be invisible if its shadow ˜ S (cid:12)(cid:12)(cid:12) F ( E ) vanishes.A covering τ is said to be irreducible if ˜ E is differentially connected. Two coverings are equivalentif there exists a diffeomorphism g : ˜ E → ˜ E such that the diagrams˜ E g / / τ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ ˜ E τ (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) E , D ( ˜ E ) g ∗ / / D ( ˜ E ) D ( M ) ˜ C d d ❍❍❍❍❍❍❍❍❍ ˜ C : : ✈✈✈✈✈✈✈✈✈ are commutative. Note also that for any two coverings their Whitney product is naturally defined. Acovering is called linear if τ is a vector bundle and the action of vector fields ˜ C X preserves the subspaceof fiber-wise linear functions in F ( ˜ E ).In the case of 2D equations, there exists a fundamental relation between special type of coveringsover E and conservation laws of the latter. Let τ be a covering of rank l < ∞ . We say that τ is anAbelian covering if there exist l independent conservation laws [ ω i ] ∈ H h ( E ), i = 1 , . . . , l , such that theforms τ ∗ ( ω i ) are exact. Then equivalence classes of such coverings are in one-to-one correspondencewith l -dimensional R -subspaces in H h ( E ). Coordinates.
Choose a trivialization of the covering τ and let w , . . . , w l , . . . be coordinates infibers (the are called nonlocal variables). Then the covering structure is given by the extended totalderivatives ˜ D i = D i + X i , i = 1 , . . . , n, where X i = X α X αi ∂∂w α are τ -vertical vector fields (nonlocal tails) enjoying the condition D i ( X j ) − D j ( X i ) + [ X i , X j ] = 0 , i < j. (3) I. KRASIL ′ SHCHIK
Here D i ( X j ) denotes the action of D i on coefficients of X j . Relations (3) (flatness of ˜ C ) amount tothe fact that the manifold ˜ E endowed with the distribution ˜ C coincides with the infinite prolongationof the overdetermined system ∂w α ∂x i = X αi , which is compatible modulo E .Irreducible coverings are those for which the system of vector fields ˜ D , . . . , ˜ D n has no nontrivialintegrals. If ¯ τ is another covering with the nonlocal tails ¯ X i = P ¯ X βi ∂/∂ ¯ w β , then the Whitney product τ ⊕ ¯ τ of τ and ¯ τ is given by ˜ D i = D i + X α X αi ∂∂w α + X β ¯ X βi ∂∂ ¯ w β . A covering is Abelian if the coefficients X αi are independent of nonlocal variables w j . If n = 2 and ω α = X α dx + X α dx , α = 1 , . . . , l , are conservation laws of E then the corresponding Abeliancovering is given by the system ∂w α ∂x i = X αi , i = 1 , , α = 1 , . . . , l, or ˜ D i = D i + X α X αi ∂∂w α . Vice versa, is such a covering is given, then one can construct the corresponding conservation law.The horizontal de Rham differential on ˜ E is ˜ d h = P i dx i ∧ ˜ D i . A covering is linear if X αi = X β X αi,β w β , (4)where X αi,β ∈ F ( E ). Remark . Denote by X i the F ( E )-valued matrix ( X αi,β ) that appears in (4). Then Eq. (3) may berewritten as D i ( X j ) − D j ( X i ) + [ X i , X j ] = 0 . for linear coverings. Thus, a linear covering defines a zero-curvature reperesentation for E and viceversa.A nonlocal symmetry in τ is a vector field S ϕ,ψ = X ˜ D σ ( ϕ j ) ∂∂u jσ + X ψ α ∂∂w α , where the vector functions ϕ = ( ϕ , . . . , ϕ m ) and ψ = ( ψ , . . . , ψ α , . . . ) on ˜ E satisfy the system ofequations X ∂F j ∂u jσ ˜ D σ ( ϕ j ) = 0 , (5)˜ D i ( ψ α ) = X ∂X αi ∂u jσ ˜ D σ ( ϕ j ) + X ∂X αi ∂w β ψ β . (6)Nonlocal shadows are the derivations ˜ E ϕ = X ˜ D σ ( ϕ j ) ∂∂u jσ , where ϕ satisfies Eq. (5), invisible symmetries are S ϕ, = X ψ α ∂∂w α , ONLOCAL CONSERVATION LAWS 5 where ψ satisfies ˜ D i ( ψ α ) = X ∂X αi ∂w β ψ β . (7)In what follows, we use the notation τ I : ˜ E I → ˜ E for the covering defined by Eq. (7). Remark . Eq. (7) defines a linear covering over ˜ E . Due to Remark 1, we see that for any non-Abeliancovering we obtain in such a way a nonlocal zero-curvature representation with the matrices X i =( ∂X αi /∂w β ). Remark . The covering τ I : ˜ E I → ˜ E is the vertical part of the tangent covering t : T ˜ E → ˜ E , see thedefinition in [6]. 2. The main result
From now on we consider two-dimensional scalar equations with the independent variables x and y .We shall show that any such an equation that admits an irreducible covering possesses a (nonlocal)conservation law. Example 1.
Let us revisit the Bianchi example discussed in the beginning of the paper. Equations (1)define a one-dimensional non-Abelian covering τ : ˜ E = E × R → E over the sine-Gordon equation (2)with the nonlocal variable w . Then the defining equations (7) for invisible symmetries in this coveringare ∂ψ∂x = − cos( u + w ) ψ, ∂ψ∂y = − cos( u − w ) ψ. This is a one-dimensional linear covering over ˜ E which is equivalent to the Abelian covering ∂ ¯ ψ∂x = − cos( u + w ) , ∂ ¯ ψ∂y = − cos( u − w ) , where ¯ ψ = ln ψ . Thus, we obtain the nonlocal conservation law ω = − cos( u + w ) dx − cos( u − w ) dy of the sine-Gordon equation.The next result shows that Bianchi’s observation is of a quite general nature. Proposition 1.
Let τ : ˜ E → E be a one-dimensional non-Abelian covering over E . Then , if τ isirreducible , τ I : ˜ E I → E defines a nontrivial conservation law of the equation ˜ E ( and , consequently , of E too ) .Proof. Consider the total derivatives D I x = ˜ D x + ∂X∂w ψ ∂∂ψ = D x + X ∂∂w + ∂X∂w ψ ∂∂ψD I y = ˜ D y + ∂Y∂w ψ ∂∂ψ = D y + Y ∂∂w + ∂Y∂w ψ ∂∂ψ on E I and assume that a ∈ F ( ˜ E ) is a common nontrivial integral of these fields: D I x ( a ) = D I y ( a ) = 0 , a = const . (8)Choose a point in E I and assume that the formal series a + a ψ + · · · + a j ψ j + . . . , a j ∈ F ( ˜ E ) , (9)converges to a in a neighborhood of this point. Substituting relations (9) to (8) and equating coeffi-cients at the same powers of ψ , we get˜ D x ( a j ) + j ∂X∂w a j = 0 , ˜ D y ( a j ) + j ∂Y∂w a j = 0 , j = 0 , , . . . , I. KRASIL ′ SHCHIK and, since τ is irreducible, this implies that a = k = const and˜ D x ( a j ) a j = j ˜ D x ( a ) a , ˜ D y ( a j ) a j = j ˜ D y ( a ) a . Hence, a j = k j ( a ) j , j >
0. Substituting these relations to (9), we see that a = a ( θ ), where θ = a ψ , a ∈ F ( E ). Then Equation (8) take the form˙ aψ (cid:18) ˜ D x ( a ) + ∂X∂ψ (cid:19) = 0 , ˙ aψ (cid:18) ˜ D y ( a ) + ∂Y∂ψ (cid:19) = 0 , ˙ a = dadθ . Thus ∂X∂w = − ˜ D x ( a ) , ∂Y∂w = − ˜ D y ( a )and the function w + a is a nontrivial integral of ˜ D x and ˜ D y . Contradiction.Finally, repeating the scheme of Example 1, we pass to the equivalent covering by setting ¯ ψ = ln ψ and obtain the nontrivial conservation law ω = ∂X∂w dx + ∂Y∂w dy on E I . (cid:3) Indeed, Bianchi’s result has a further generalization. To formulate the latter, let us say that acovering τ : ˜ E → E is strongly non-Abelian if for any nontrivial conservation law ω of the equation E itslift τ ∗ ( ω ) to the manifold ˜ E is nontrivial as well. Now, a straightforward generalization of Proposition 1is Proposition 2.
Let τ : ˜ E → E be an irreducible covering over a differentially connected equation.Then τ is a strongly non-Abelian covering if and only if the covering τ I is irreducible. We shall now need the following construction. Let τ : ˜ E → E be a linear covering. Consider thefiber-wise projectivization τ P : ˜ E P → E of the vector bundle τ . Denote by p : ˜ E → E P the naturalprojection. Then, obviously, the projection p ∗ ( ˜ C ) is well defined and is an n -dimensional integrabledistribution on E P . Thus, we obtain the following commutative diagram of coverings˜ E p / / τ (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ E P τ P ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ E ,where rank( p ) = 1 and rank( τ P ) = rank( τ ) − Proposition 3.
Let τ : ˜ E → E be an irredicible covering. Then the covering τ P is irreducible as well. Coordinates.
Let rank( τ ) = l > w αx i = l X β =1 X αi,β w β , i = 1 , . . . , n, α = 1 , . . . , l, (10)be the defining equations of the covering τ , see Eq. (4). Choose an affine chart in the fibers of τ P . Tothis end, assume for example that w l = 0 and set¯ w α = w α w l , l = 1 , . . . , l − , in the domain under consideration. Then from Equations (10) it follows that the system¯ w αx i = X αi,l − X li,l ¯ w α + l − X β =1 X αi,β ¯ w β − ¯ w α l − X β =1 X li,β ¯ w β , i = 1 , . . . , n, α = 1 , . . . , l − . locally provides the defining equation for the covering τ P . ONLOCAL CONSERVATION LAWS 7
We are now ready to state and prove the main result.
Theorem 1.
Assume that a differentially connected equation E admits a nontrivial covering τ : ˜ E → E . Then it possesses at least one nontrivial ( nonlocal ) conservation law.Proof. Actually, the proof is a description of a procedure that allows one to construct the desiredconservation law.Note first that we may assume the covering τ to be irreducible. Indeed, otherwise the space ˜ E isfoliated by maximal integral manifolds of the distribution ˜ C . Let l denote the codimension of thegeneric leaf and l = rank( τ ). Then • l > l , because τ is a nontrivial covering; • the integral leaves project to E surjectively, because E is a differentially connected equation.This means that in vicinity of a generic point we can consider τ as an l -parametric family of irreduciblecoverings whose rank is r = l − l >
0. Let us choose one of them and denote it by τ : E → E .If τ is not strongly non-Abelian, then this would mean that E possesses at least one nontrivialconservation law and we have nothing to prove further. Assume now that the covering τ is stronglynon-Abelian. Then due to Proposition 2 the linear covering τ I is irreducible and by Proposition 3its projectivization τ = ( τ I ) P possesses the same property and rank( τ ) = r −
1. Repeating theconstruction, we arrive to the diagram E I p (cid:15) (cid:15) τ I w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ . . . E I r − p (cid:15) (cid:15) τ I r − v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ E E τ o o ( E I ) P = E τ =( τ I ) P o o . . . o o ( E I r − ) P = E r − , τ r − =( τ I r − ) P o o where rank( τ i ) = l − i . Thus, in r − (cid:3) Examples
Let us discuss several illustrative examples.
Example 2.
Consider the Korteweg-de Vries equation in the form u t = uu x + u xxx (11)and the well known Miura transformation [4] u = w x − w . The last formula is a part of the defining equations for the non-Abelian covering w x = u + 16 w ,w t = u xx + 13 wu x + 13 u + 118 w u, the covering equation being w t = w xxx − w w x , i.e., the modified KdV equation. Then the corresponding covering τ I is defined by the system ψ x = 13 wψ,ψ t = 13 (cid:18) u x + 13 wu (cid:19) ψ I. KRASIL ′ SHCHIK that, after relabeling ψ ψ gives us the nonlocal conservation law ω = w dx + (cid:18) u x + 13 wu (cid:19) dt of the KdV equation. Example 3.
The well known Lax pair, see [8], for the KdV equation may be rewritten in terms ofzero-curvature representation D x ( T ) − D t ( X ) + [ X , T ] = 0 , where (2 ×
2) matrices X and T are of the form X = − ( λ + u ) 0 ! , T = − u x (cid:0) u − λ (cid:1) − u xx − u − λu − λ u x ! ,λ ∈ R being a real parameter. As it follows from Remark 1, this amounts to existence of the two-dimensional linear covering τ given by the system w x = 16 w ,w t = − u x w + 19 (cid:18) − λ (cid:19) w ,w x = − ( λ + u ) w ,w t = − (cid:18) u xx + 13 u + 13 λu + 23 λ (cid:19) w + 16 u x w . Let us choose for the affine chart the domain w = 0 and set ψ = w /w . Then the covering τ P isdescribed by the system ψ x = ( λ + u ) ψ + 16 ,ψ t = (cid:18) u xx + 13 u + 13 λu + 23 λ (cid:19) ψ − u x ψ + 19 (cid:18) − λ (cid:19) while τ = ( τ P ) I is given by˜ ψ x = ( λ + u ) ˜ ψ, ˜ ψ t = 2 (cid:18) u xx + 13 u + 13 λu + 23 λ (cid:19) ψ ˜ ψ − u x ˜ ψ. Thus, we obtain the conservation law ω = ( λ + u ) dx + (cid:18) (cid:18) u xx + 13 u + 13 λu + 23 λ (cid:19) ψ − u x (cid:19) dt that depends on the nonlocal variable ψ . Example 4.
Consider the potential KdV equation in the form u t = 3 u x + u xxx Its B¨acklund auto-transformation is associated to the covering τw x = λ − u x −
12 ( w − u ) ,w t = 2 λ − λu x − u x − u xxx + 2 u xx ( w − u ) − ( λ + u x )( w − u ) , where λ ∈ R , see [11]. Then the covering τ I is ψ x = − ( w − u ) ψ,ψ t = 2 (cid:0) u xx ψ − ( λ + u x )( w − u ) (cid:1) ψ, ONLOCAL CONSERVATION LAWS 9 which leads to the nonlocal conservation law ω = − ( w − u ) dx + 2 (cid:0) u xx ψ − ( λ + u x )( w − u ) (cid:1) dt of the potential KdV equation. Example 5.
The Gauss-Mainardi-Codazzi equations read u xy = g − f h sin u , f y = g x + h − g cos u sin u u x , g y = h x − f − g cos u sin u u y , (12)see [10]. This is an under-determined system, and imposing additional conditions on the unknownfunctions u , f , g , and h one obtains equations that describe various types of surfaces in R , cf. [5].System (12) always admits the following C -valued zero-curvature representation D x ( Y ) − D y ( X ) + [ X , Y ] = 0with the matrices X = i2 u x e i u f − g sin u e − i u f − g sin u − u x , Y = i2 iu g − h sin u e − i u g − h sin u The corresponding two-dimensional linear covering τ is defined by the system w x = u x w + e i u f − g sin u w ,w y = e iu g − h sin u w , w x = e − i u f − g sin u w − u x w ,w y = e − i u g − h sin u w . Hence, the covering τ P in the domain w = 0 is ψ x = e i u f − g sin u + 2 u x ψ − e − i u f − g sin u ψ , ψ y = e i u g − h sin u − e − i u g − h sin u ψ . Thus, the covering ( τ P ) I , given by˜ ψ x = 2 (cid:18) u x − e − i u f − g sin u ψ (cid:19) ˜ ψ, ˜ ψ y = − − i u g − h sin u ψ ˜ ψ, defines the nonlocal conservation law ω = (cid:18) u x − e − i u f − g sin u ψ (cid:19) dx − e − i u g − h sin u ψ dy of the Gauss-Mainardi-Codazzi equations. Example 6.
The last example shows that the above described techniques fail for infinite-dimensionalcoverings (such coverings are typical for equations of dimension greater than two).Consider the equation u yy = u tx + u y u xx − u x u xy that arises in the theory of integrable hydrodynamical chains, see [9]. This equation admits thecovering τ with the nonlocal variables w i , i = 0 , , . . . , that enjoy the defining relations w t + u y w x = 0 , w y + u x w x = 0 ,w ix = w i +1 , i ≥ ,w it + D ix ( u y w x ) = 0 , w iy + D ix ( u x w x ) = 0 , i ≥ . see [1]. This is a linear covering, but its projectivization does not lead to construction of conservationlaws. ′ SHCHIK
Discussion
We described a procedure that allows one to associate, in an algorithmic way, with any nontrivialfinite-dimensional covering over a differentially connected equation a nonlocal conservation law. Nev-ertheless, this method fails in the case of infinite-dimensional coverings. It is unclear, at the momentat least, whether this is an immanent property of such coverings or a disadvantage of the method. Ihope to clarify this in future research.
Acknowledgments
I am grateful to Michal Marvan, who attracted my attention to the paper by Luigi Bianchi [2], andto Raffaele Vitolo, who helped me with Italian. I am also grateful to Valentin Lychagin for a fruitfuldiscussion.
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