Nonlocal conservation laws of PDEs possessing differential coverings

In his 1892 paper [L. Bianchi, Sulla trasformazione di B\"{a}cklund per le superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12], L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the B\"{a}cklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings [I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and B\"{a}cklund transformations, Acta Appl. Math. v. 15 (1989) 1-2, 161--209], we show that this observation is of a quite general nature. We describe the procedures to construct such conservation laws and present a number of illustrative examples.

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.
In Sec. 1, I shortly introduce the basic constructions in nonlocal geometry of PDEs, i.e., the theory of differential coverings, [7]. Sec. 2 contains an interpretation of the result by L. Bianchi in the most general 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 Lie algebra of vector fields, and Λ * (·) = ⊕ k≥0 Λ k (·) for the exterior algebra of differential forms.

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 that lie 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 : The distribution on E spanned by the fields of the form C X (the Cartan distribution) is Frobenius integrable. We denote it 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 k h (E ) generated by the forms π * ∞ (θ), θ ∈ Λ k (M ). Elements ω ∈ Λ k h (E ) are called horizontal k-forms. Generalizing slightly the action of the Cartan connection, one can apply it to the de Rham differential d : Λ k (M ) → Λ k+1 (M ) and obtain the horizontal de Rham complex Elements of its (n − 1)st cohomology group H n−1 h (E ) are called conservation laws of E . We always assume E to be differentially connected which means that H 0 h (E ) = R. Coordinates. Consider a trivialization of π with local coordinates x 1 , . . . , x n in U ⊂ M and u 1 , . . . , u m in the fibers of π| U . Then in π −1 ∞ (U ) ⊂ J ∞ (π) the adapted coordinates u i σ arise and the Cartan connection is determined by the total derivatives Let F = (F 1 , . . . , F r ), where F j are smooth functions on J k (π). The the infinite prolongation of the locus { z ∈ J k (π) | F 1 (z) = · · · = F r (z) = 0 } ⊂ J k (π) is defined by the system where D σ denotes the composition of the total derivatives corresponding to the multi-index σ. The total derivatives, as well as all differential operators in total derivatives, can be restricted to infinite prolongations and we preserve the same notation for these restrictions. Given an E , we always choose internal local coordinates in it for subsequent computations. To restrict an operator to E is to express this operator in terms of internal coordinates.
Any symmetry of E is an evolutionary vector field A horizontal (n − 1)-form 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 are constants.
Consider now a locally trivial bundle τ :Ẽ → E such that there exists a flat connectionC in π ∞ • τ :Ẽ → 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 onẼ are nonlocal for E : e.g., symmetries ofẼ are nonlocal symmetries of E , conservation laws ofẼ are nonlocal conservation laws of E , etc. A derivation S : is commutative for any X ∈ D(M ). In particular, any symmetry of the equation E , as well as restrictionsS of nonlocal symmetries may be considered as shadows. A nonlocal symmetry is said to be invisible if its shadowS

vanishes.
A covering τ is said to be irreducible ifẼ is differentially connected. Two coverings are equivalent if there exists a diffeomorphism g :Ẽ 1 →Ẽ 2 such that the diagrams are commutative. Note also that for any two coverings their Whitney product is naturally defined. A covering is called linear if τ is a vector bundle and the action of vector fieldsC X preserves the subspace of fiber-wise linear functions in F (Ẽ ).
In the case of 2D equations, there exists a fundamental relation between special type of coverings over E and conservation laws of the latter. Let τ be a covering of rank l < ∞. We say that τ is an Abelian covering if there exist l independent conservation laws [ . . , l, such that the forms τ * (ω i ) are exact. Then equivalence classes of such coverings are in one-to-one correspondence with l-dimensional R-subspaces in H 1 h (E ). Coordinates. Choose a trivialization of the covering τ and let w 1 , . . . , w l , . . . be coordinates in fibers (the are called nonlocal variables). Then the covering structure is given by the extended total derivativesD are τ -vertical vector fields (nonlocal tails) enjoying the condition Here D i (X j ) denotes the action of D i on coefficients of X j . Relations (3) (flatness ofC ) amount to the fact that the manifoldẼ endowed with the distributionC coincides with the infinite prolongation of the overdetermined system Irreducible coverings are those for which the system of vector fieldsD 1 , . . . ,D n has no nontrivial integrals. Ifτ is another covering with the nonlocal tailsX i = X β i ∂/∂w β , then the Whitney product τ ⊕τ of τ andτ is given byD A covering is Abelian if the coefficients X α i are independent of nonlocal variables w j . If n = 2 and ω α = X α 1 dx 1 + X α 2 dx 2 , α = 1, . . . , l, are conservation laws of E then the corresponding Abelian covering is given by the system Vice versa, is such a covering is given, then one can construct the corresponding conservation law.
where X α i,β ∈ F (E ). Remark 1. Denote by X i the F (E )-valued matrix (X α i,β ) that appears in (4). Then Eq. (3) may be rewritten as for linear coverings. Thus, a linear covering defines a zero-curvature reperesentation for E and vice versa.
Remark 2. Eq. (7) defines a linear covering overẼ . Due to Remark 1, we see that for any non-Abelian covering we obtain in such a way a nonlocal zero-curvature representation with the matrices X i = (∂X α i /∂w β ). Remark 3. The covering τ I :Ẽ I →Ẽ is the vertical part of the tangent covering t : TẼ →Ẽ , see the definition in [6].

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 :Ẽ = E × R → E over the sine-Gordon equation (2) with the nonlocal variable w. Then the defining equations (7) for invisible symmetries in this covering are This is a one-dimensional linear covering overẼ which is equivalent to the Abelian covering Proof. Consider the total derivatives on E I and assume that a ∈ F (Ẽ ) is a common nontrivial integral of these fields: Choose a point in E I and assume that the formal series a 0 + a 1 ψ + · · · + a j ψ j + . . . , a j ∈ F (Ẽ ), converges to a in a neighborhood of this point. Substituting relations (9) to (8) and equating coefficients at the same powers of ψ, we get and, since τ is irreducible, this implies that a 0 = k 0 = const and Hence, a j = k j (a 1 ) j , j > 0. Substituting these relations to (9), we see that a = a(θ), where θ = a 1 ψ, a 1 ∈ F (E ). Then Equation (8) take the forṁ and the function w + a 1 is a nontrivial integral ofD x andD y . Contradiction. Finally, repeating the scheme of Example 1, we pass to the equivalent covering by settingψ = ln ψ and obtain the nontrivial conservation law Indeed, Bianchi's result has a further generalization. To formulate the latter, let us say that a covering τ :Ẽ → E is strongly non-Abelian if for any nontrivial conservation law ω of the equation E its lift τ * (ω) to the manifoldẼ is nontrivial as well. Now, a straightforward generalization of Proposition 1 is Proposition 2. Let τ :Ẽ → 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 :Ẽ → E be a linear covering. Consider the fiber-wise projectivization τ P :Ẽ P → E of the vector bundle τ . Denote by p :Ẽ → E P the natural projection. Then, obviously, the projection p * (C ) is well defined and is an n-dimensional integrable distribution on E P . Thus, we obtain the following commutative diagram of coverings where rank(p) = 1 and rank(τ P ) = rank(τ ) − 1.
Proposition 3. Let τ :Ẽ → E be an irredicible covering. Then the covering τ P is irreducible as well.
Coordinates. Let rank(τ ) = l > 1 and be the defining equations of the covering τ , see Eq. (4). Choose an affine chart in the fibers of τ P . To this end, assume for example that w l = 0 and set in the domain under consideration. Then from Equations (10) it follows that the system locally provides the defining equation for the covering τ P .
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 :Ẽ → 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 desired conservation law. Note first that we may assume the covering τ to be irreducible. Indeed, otherwise the spaceẼ is foliated by maximal integral manifolds of the distributionC . Let l 0 denote the codimension of the generic leaf and l = rank(τ ). Then • l > l 0 , 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 0 -parametric family of irreducible coverings whose rank is r = l − l 0 > 0. Let us choose one of them and denote it by τ 0 : If τ 0 is not strongly non-Abelian, then this would mean that E possesses at least one nontrivial conservation law and we have nothing to prove further. Assume now that the covering τ 0 is strongly non-Abelian. Then due to Proposition 2 the linear covering τ I 0 is irreducible and by Proposition 3 its projectivization τ 1 = (τ I 0 ) P possesses the same property and rank(τ 1 ) = r − 1. Repeating the construction, we arrive to the diagram where rank(τ i ) = l − i. Thus, in r − 1 steps at most we shall arrive to a one-dimensional irreducible covering and find ourselves in the situation of Proposition 1 and this finishes the proof.

Examples
Let us discuss several illustrative examples.

Example 2. Consider the Korteweg-de Vries equation in the form
and the well known Miura transformation [4] The last formula is a part of the defining equations for the non-Abelian covering the covering equation being i.e., the modified KdV equation. Then the corresponding covering τ I is defined by the system wu ψ that, after relabeling ψ → 3 ln ψ gives us the nonlocal conservation law which leads to the nonlocal conservation law of the potential KdV equation.
Example 5. The Gauss-Mainardi-Codazzi equations read see [10]. This is an under-determined system, and imposing additional conditions on the unknown functions u, f , g, and h one obtains equations that describe various types of surfaces in R 2 , cf. [5]. System (12) always admits the following C-valued zero-curvature representation with the matrices The corresponding two-dimensional linear covering τ is defined by the system Hence, the covering τ P in the domain w 2 = 0 is Thus, the covering (τ P ) I , given bỹ ψ x = 2 u x − e −iu f − g sin u ψ ψ ,ψ y = −2 e −iu g − h sin u ψψ, defines the nonlocal conservation law ω = u x − e −iu f − g sin u ψ dx − e −iu 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-dimensional coverings (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 the covering τ with the nonlocal variables w i , i = 0, 1, . . . , that enjoy the defining relations w 0 t + u y w 1 x = 0, w 0 y + u x w 1 x = 0, w i x = w i+1 , i ≥ 0, see [1]. This is a linear covering, but its projectivization does not lead to construction of conservation laws.

Discussion
We described a procedure that allows one to associate, in an algorithmic way, with any nontrivial finite-dimensional covering over a differentially connected equation a nonlocal conservation law. Nevertheless, this method fails in the case of infinite-dimensional coverings. It is unclear, at the moment at least, whether this is an immanent property of such coverings or a disadvantage of the method. I hope to clarify this in future research.