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Article

# Nonlocal Conservation Laws of PDEs Possessing Differential Coverings †

by
Iosif Krasil’shchik
V.A. Trapeznikov Institute of Control Sciences RAS, Profsoyuznaya 65, 117342 Moscow, Russia
To the memory of Alexandre Vinogradov, my teacher.
Symmetry 2020, 12(11), 1760; https://doi.org/10.3390/sym12111760
Submission received: 22 September 2020 / Revised: 20 October 2020 / Accepted: 20 October 2020 / Published: 23 October 2020
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)

## Abstract

:
In his 1892 paper, L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings, 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.
MSC:
37K10

## 1. Introduction

In [1], L. Bianchi, dealing with the celebrated Bäcklund auto-transformation (I changed the original notation slightly)
$∂ ( u − w ) ∂ x = sin ( u + w ) , ∂ ( u + w ) ∂ y = sin ( u − w )$
for the sine-Gordon equation
$∂ 2 ( 2 u ) ∂ x ∂ y = sin ( 2 u )$
in the course of intermediate computations (see ([1], 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
$ω = cos ( u + w ) d x + cos ( u − w ) d y$
is a nonlocal conservation law for Equation (1).
It became clear much later, some 100 years after the publication of [1], that nonlocal conservation laws are important invariants of PDEs and are used in numerous applications, e.g.,: numerical methods [2,3], sociological models [4,5], integrable systems [6], electrodynamics [7,8], mechanics [9,10,11], etc.
Actually, Bianchi’s observation is of a very general nature and this is shown below.
In Section 2, I shortly introduce the basic constructions in nonlocal geometry of PDEs, i.e., the theory of differential coverings, [12]. Section 3 contains an interpretation of the result by L. Bianchi in the most general setting. In Section 4, 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.

## 2. Preliminaries

Following [13], 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 $C : D ( M ) → D ( E )$ called the Cartan connection. Flatness of $C$ means that $C [ X , Y ] = [ C X , C Y ]$ for all X, $Y ∈ D ( M )$. 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 $Λ h k ( E )$ generated by the forms $π ∞ * ( θ )$, $θ ∈ Λ k ( M )$. Elements $ω ∈ Λ h k ( E )$ 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
$0 ⟶ F ( E ) ⟶ ⋯ ⟶ Λ h k ( E ) ⟶ d h Λ h k + 1 ( E ) ⟶ ⋯ ⟶ Λ h n ( E ) ⟶ 0$
on $E$. Elements of its $( n − 1 )$st cohomology group $H h n − 1 ( E )$ are called conservation laws of $E$. We always assume $E$ to be differentially connected which means that $H h 0 ( E ) = R$.
Remark 1.
The concept of a differentially connected equation reflects Vinogradov’s correspondence principle [14], (p. 195): when ‘secondary dimension’ (dimension of the Cartan distribution) $Dim → 0$, the objects of PDE geometry degenerate to their counterparts in geometry of finite-dimensional manifolds. Following this principle, we informally have
$lim Dim → 0 H h i ( E ) = H dR i ( M ) .$
Since $H dR 0 ( M )$ is responsible for topological connectedness of M, the group $H h 0 ( E )$ stands for differential one.
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 $u σ i$ arise and the Cartan connection is determined by the total derivatives
$C : ∂ ∂ x i ↦ D i = ∂ ∂ x i + ∑ j , σ u σ i j ∂ ∂ u σ j .$
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
$E = E F = { z ∈ J ∞ ( π ) ∣ D σ ( F j ) ( z ) = 0 , j = 1 , ⋯ , r , σ ≥ 0 } ,$
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
$E φ = ∑ D σ ( φ j ) ∂ ∂ u σ j$
(summation on internal coordinates), where the functions $φ 1 , ⋯ , φ m ∈ F ( E )$ satisfy the system
$∑ σ , α ∂ F j ∂ u σ α D σ ( φ α ) = 0 , j = 1 , ⋯ , r .$
A horizontal $( n − 1 )$-form
$ω = ∑ i a i d x 1 ∧ ⋯ ∧ d x i − 1 ∧ d x i + 1 ∧ ⋯ ∧ d x n$
defines a conservation law of $E$ if
$∑ i ( − 1 ) i + 1 D i ( a i ) = 0 .$
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
$D 1 ( f ) = ⋯ = D n ( f ) = 0 , f ∈ F ( E ) ,$
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 [12], 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. A derivation $S : F ( E ) → F ( E ˜ )$ is called a nonlocal shadow if the diagram
is commutative for any $X ∈ D ( M )$. In particular, any symmetry of the equation $E$, as well as restrictions $S ˜ F ( E )$ of nonlocal symmetries may be considered as shadows. A nonlocal symmetry is said to be invisible if its shadow $S ˜ F ( E )$ vanishes.
A covering $τ$ is said to be irreducible if $E ˜$ is differentially connected. Two coverings are equivalent if there exists a diffeomorphism $g : E ˜ 1 → E ˜ 2$ 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 fields $C ˜ X$ preserves the subspace of fiber-wise linear functions in $F ( E ˜ )$.
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 $[ ω i ] ∈ H h 1 ( E )$, $i = 1 , ⋯ , l$, 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 $H h 1 ( E )$.
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 derivatives
$D ˜ i = D i + X i , i = 1 , ⋯ , n ,$
where
$X i = ∑ α X i α ∂ ∂ w α$
are $τ$-vertical vector fields (nonlocal tails) enjoying the condition
$D i ( X j ) − D j ( X i ) + [ X i , X j ] = 0 , i < j .$
Here $D i ( X j )$ denotes the action of $D i$ on coefficients of $X j$. Relations (3) (flatness of $C ˜$) amount to the fact that the manifold $E ˜$ endowed with the distribution $C ˜$ coincides with the infinite prolongation of the overdetermined system
$∂ w α ∂ x i = X i α ,$
which is compatible modulo $E$.
Irreducible coverings are those for which the system of vector fields $D ˜ 1 , ⋯ , D ˜ n$ has no nontrivial integrals. If $τ ¯$ is another covering with the nonlocal tails $X ¯ i = ∑ X ¯ i β ∂ / ∂ w ¯ β$, then the Whitney product $τ ⊕ τ ¯$ of $τ$ and $τ ¯$ is given by
$D ˜ i = D i + ∑ α X i α ∂ ∂ w α + ∑ β X ¯ i β ∂ ∂ w ¯ β .$
A covering is Abelian if the coefficients $X i α$ are independent of nonlocal variables $w j$. If $n = 2$ and $ω α = X 1 α d x 1 + X 2 α d x 2$, $α = 1 , ⋯ , l$, are conservation laws of $E$ then the corresponding Abelian covering is given by the system
$∂ w α ∂ x i = X i α , i = 1 , 2 , α = 1 , ⋯ , l ,$
or
$D ˜ i = D i + ∑ α X i α ∂ ∂ w α .$
Vice versa, if such a covering is given, then one can construct the corresponding conservation law.
The horizontal de Rham differential on $E ˜$ is $d ˜ h = ∑ i d x i ∧ D ˜ i$. A covering is linear if
$X i α = ∑ β X i , β α w β ,$
where $X i , β α ∈ F ( E )$.
Remark 2.
Denote by $X i$ the $F ( E )$-valued matrix $( X i , β α )$ that appears in (4). Then Equation (3) may be rewritten 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 representation for $E$ and vice versa.
A nonlocal symmetry in $τ$ is a vector field
$S φ , ψ = ∑ D ˜ σ ( φ j ) ∂ ∂ u σ j + ∑ ψ α ∂ ∂ w α ,$
where the vector functions $φ = ( φ 1 , ⋯ , φ m )$ and $ψ = ( ψ 1 , ⋯ , ψ α , ⋯ )$ on $E ˜$ satisfy the system of equations
$∑ ∂ F j ∂ u σ j D ˜ σ ( φ j ) = 0 ,$
$D ˜ i ( ψ α ) = ∑ ∂ X i α ∂ u σ j D ˜ σ ( φ j ) + ∑ ∂ X i α ∂ w β ψ β .$
$E ˜ φ = ∑ D ˜ σ ( φ j ) ∂ ∂ u σ j ,$
where $φ$ satisfies Equation (5), invisible symmetries are
$S 0 , ψ = ∑ ψ α ∂ ∂ w α ,$
where $ψ$ satisfies
$D ˜ i ( ψ α ) = ∑ ∂ X i α ∂ w β ψ β .$
In what follows, we use the notation $τ I : E ˜ I → E ˜$ for the covering defined by Equation (7).
Remark 3.
Equation (7) defines a linear covering over $E ˜$. Due to Remark 2, we see that for any non-Abelian covering we obtain in such a way a nonlocal zero-curvature representation with the matrices $X i = ( ∂ X i α / ∂ w β )$.
Remark 4.
The covering $τ I : E ˜ I → E ˜$ is the vertical part of the tangent covering$t : T E ˜ → E ˜$, see the definition in [15].

## 3. 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. Equation (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 Equation (7) for invisible symmetries in this covering are
$∂ ψ ∂ x = − cos ( u + w ) ψ , ∂ ψ ∂ y = − cos ( u − w ) ψ .$
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
$ω = − cos ( u + w ) d x − cos ( u − w ) d y$
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 is irreducible, $τ I : E ˜ I → E ˜$ defines a nontrivial conservation law of the equation $E ˜$ (and, consequently, of $E$ too).
Proof.
Consider the total derivatives
$D x I = D ˜ x + ∂ X ∂ w ψ ∂ ∂ ψ = D x + X ∂ ∂ w + ∂ X ∂ w ψ ∂ ∂ ψ D y I = D ˜ y + ∂ Y ∂ w ψ ∂ ∂ ψ = D y + Y ∂ ∂ w + ∂ Y ∂ w ψ ∂ ∂ ψ$
on $E I$ and assume that $a ∈ F ( E ˜ )$ is a common nontrivial integral of these fields:
$D x I ( a ) = D y I ( a ) = 0 , a ≠ const .$
Choose a point in $E I$ and assume that the formal series
$a 0 + a 1 ψ + ⋯ + a j ψ j + ⋯ , a j ∈ F ( E ˜ ) ,$
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
$D ˜ x ( a j ) + j ∂ X ∂ w a j = 0 , D ˜ y ( a j ) + j ∂ Y ∂ w a j = 0 , j = 0 , 1 , ⋯ ,$
and, since $τ$ is irreducible, this implies that $a 0 = k 0 = const$ and
$D ˜ x ( a j ) a j = j D ˜ x ( a 1 ) a 1 , D ˜ y ( a j ) a j = j D ˜ y ( a 1 ) a 1 .$
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 form
$a ˙ ψ D ˜ x ( a 1 ) + ∂ X ∂ w = 0 , a ˙ ψ D ˜ y ( a 1 ) + ∂ Y ∂ w = 0 , a ˙ = d a d θ .$
Thus
$∂ X ∂ w = − D ˜ x ( a 1 ) , ∂ Y ∂ w = − D ˜ y ( a 1 )$
and the function $w + a 1$ 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
$ω = ∂ X ∂ w d x + ∂ Y ∂ w d y$
on $E I$. □
Indeed, Bianchi’s result has a further generalization. To formulate the latter, let us say that a covering $τ : E ˜ → E$ is strongly non-Abelian if for any nontrivial conservation law $ω$ of the equation $E$ its lift $τ * ( ω )$ to the manifold $E ˜$ is nontrivial as well. Now, a straightforward generalization of Proposition 1 is
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 the fiber-wise projectivization $τ P : E ˜ P → E$ of the vector bundle $τ$. Denote by $p : E ˜ → E P$ 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 $τ : E ˜ → E$ be an irredicible covering. Then the covering $τ P$ is irreducible as well.
Coordinates. Let $rank ( τ ) = l > 1$ and
$w x i α = ∑ β = 1 l X i , β α w β , i = 1 , ⋯ , n , α = 1 , ⋯ , l ,$
be the defining equations of the covering $τ$, see Equation (4). Choose an affine chart in the fibers of $τ P$. To this end, assume for example that $w l ≠ 0$ and set
$w ¯ α = w α w l , l = 1 , ⋯ , l − 1 ,$
in the domain under consideration. Then from Equation (10) it follows that the system
$w ¯ x i α = X i , l α − X i , l l w ¯ α + ∑ β = 1 l − 1 X i , β α w ¯ β − w ¯ α ∑ β = 1 l − 1 X i , β l w ¯ β , i = 1 , ⋯ , n , α = 1 , ⋯ , l − 1 .$
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 two-dimensional equation $E$ admits a nontrivial covering $τ : E ˜ → E$ of finite rank. 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 space $E ˜$ is foliated by maximal integral manifolds of the distribution $C ˜$. 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 $τ 0 : E 0 → E$.
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 $τ 0 I$ is irreducible and by Proposition 3 its projectivization $τ 1 = ( τ 0 I ) P$ 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. □

## 4. Examples

Let us discuss several illustrative examples.
Example 2.
Consider the Korteweg-de Vries equation in the form
$u t = u u x + u x x x$
and the well known Miura transformation [16]
$u = w x − 1 6 w 2 .$
The last formula is a part of the defining equations for the non-Abelian covering
$w x = u + 1 6 w 2 , w t = u x x + 1 3 w u x + 1 3 u 2 + 1 18 w 2 u ,$
the covering equation being
$w t = w x x x − 1 6 w 2 w x ,$
i.e., the modified KdV equation. Then the corresponding covering $τ I$ is defined by the system
$ψ x = 1 3 w ψ , ψ t = 1 3 u x + 1 3 w u ψ$
that, after relabeling $ψ ↦ 3 ln ψ$ gives us the nonlocal conservation law
$ω = w d x + u x + 1 3 w u d t$
of the KdV equation.
Example 3.
The well known Lax pair, see [17], for the KdV equation may be rewritten in terms of zero-curvature representation
$D x ( T ) − D t ( X ) + [ X , T ] = 0 .$
The $( 2 × 2 )$ matrices $X$ and $T$ become much simpler if we present the equation in the form
$u t = 6 u u x − u x x x .$
In this case, they are
$X = 0 1 u − λ 0 , T = − u x 2 ( u + 2 λ ) 2 u 2 − u x x + 2 λ u − 4 λ 2 u x ,$
$λ ∈ R$ being a real parameter. As it follows from Remark 2, this amounts to existence of the two-dimensional linear covering τ given by the system
$w 1 , x = w 2 , w 1 , t = − u x w 1 + 2 ( u + 2 λ ) w 2 , w 2 , x = ( u − λ ) w 1 , w 2 , t = ( 2 u 2 − u x x + 2 λ u − 4 λ 2 ) w 1 + u x w 2 .$
Let us choose for the affine chart the domain $w 2 ≠ 0$ and set $ψ = w 1 / w 2$. Then the covering $τ P$ is described by the system
$ψ x = 1 − ( u − λ ) ψ , ψ t = 2 ( u + 2 λ ) − 2 u x ψ − ( 2 u 2 − u x x + 2 λ u − 4 λ 2 ) ψ 2 ,$
while $τ 1 = ( τ P ) I$ is given by
$ψ ˜ x = ( λ − u ) ψ ˜ , ψ ˜ t = − 2 u x + ( 2 u 2 − u x x + 2 λ u − 4 λ 2 ) ψ ψ ˜ .$
Thus, we obtain the conservation law
$ω = ( λ − u ) d x − 2 u x + ( 2 u 2 − u x x + 2 λ u − 4 λ 2 ) ψ d t$
that depends on the nonlocal variable ψ.
Example 4.
Consider the potential KdV equation in the form
$u t = 3 u x 2 + u x x x$
Its Bäcklund auto-transformation is associated to the covering τ
$w x = λ − u x − 1 2 ( w − u ) 2 , w t = 2 λ 2 − 2 λ u x − u x 2 − u x x x + 2 u x x ( w − u ) − ( λ + u x ) ( w − u ) 2 ,$
where $λ ∈ R$, see [18]. Then the covering $τ I$ is
$ψ x = − ( w − u ) ψ , ψ t = 2 u x x ψ − ( λ + u x ) ( w − u ) ψ ,$
which leads to the nonlocal conservation law
$ω = − ( w − u ) d x + 2 u x x ψ − ( λ + u x ) ( w − u ) d t$
of the potential KdV equation.
Example 5.
$u x y = g − f h sin u , f y = g x + h − g cos u sin u u x , g y = h x − f − g cos u sin u u y ,$
see [19]. 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. [20]. System (12) always admits the following $C$-valued zero-curvature representation
$D x ( Y ) − D y ( X ) + [ X , Y ] = 0$
with the matrices
$X = i 2 u x e i u f − g sin u e − i u f − g sin u − u x , Y = i 2 0 e i u g − h sin u e − i u g − h sin u 0$
The corresponding two-dimensional linear covering τ is defined by the system
$w x 1 = u x w 1 + e i u f − g sin u w 2 , w y 1 = e i u g − h sin u w 2 , w x 2 = e − i u f − g sin u w 1 − u x w 2 , w y 2 = e − i u g − h sin u w 1 .$
Hence, the covering $τ P$ in the domain $w 2 ≠ 0$ is
$ψ x = e i u f − g sin u + 2 u x ψ − e − i u f − g sin u ψ 2 , ψ y = e i u g − h sin u − e − i u g − h sin u ψ 2 .$
Thus, the covering $( τ P ) I$, given by
$ψ ˜ x = 2 u x − e − i u f − g sin u ψ ψ ˜ , ψ ˜ y = − 2 e − i u g − h sin u ψ ψ ˜ ,$
defines the nonlocal conservation law
$ω = u x − e − i u f − g sin u ψ d x − e − i u g − h sin u ψ d y$
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
$u y y = u t x + u y u x x − u x u x y$
that arises in the theory of integrable hydrodynamical chains, see [21]. This equation admits the covering τ with the nonlocal variables $w i$, $i = 0 , 1 , ⋯$, that enjoy the defining relations
$w t 0 + u y w x 1 = 0 , w y 0 + u x w x 1 = 0 , w x i = w i + 1 , i ≥ 0 , w t i + D x i ( u y w x 1 ) = 0 , w y i + D x i ( u x w x 1 ) = 0 , i ≥ 1 .$
see [22]. This is a linear covering, but its projectivization does not lead to construction of conservation laws.

## 5. 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.

## Funding

The work was partially supported by the RFBR Grant 18-29-10013 and IUM-Simons Foundation.

## Acknowledgments

I am grateful to Michal Marvan, who attracted my attention to the paper by Luigi Bianchi [1], and to Raffaele Vitolo, who helped me with Italian. I am also grateful to Valentin Lychagin for a fruitful discussion.

## Conflicts of Interest

The authors declare no conflict of interest.

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Krasil’shchik, I. Nonlocal Conservation Laws of PDEs Possessing Differential Coverings. Symmetry 2020, 12, 1760. https://doi.org/10.3390/sym12111760

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Krasil’shchik I. Nonlocal Conservation Laws of PDEs Possessing Differential Coverings. Symmetry. 2020; 12(11):1760. https://doi.org/10.3390/sym12111760

Chicago/Turabian Style

Krasil’shchik, Iosif. 2020. "Nonlocal Conservation Laws of PDEs Possessing Differential Coverings" Symmetry 12, no. 11: 1760. https://doi.org/10.3390/sym12111760

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