On symmetries of the Gibbons-Tsarev equation
aa r X i v : . [ n li n . S I] M a y ON SYMMETRIES OF THE GIBBONS–TSAREV EQUATION
H. BARAN, P. BLASCHKE, I.S. KRASIL ′ SHCHIK, AND M. MARVAN
Abstract.
We study the Gibbons–Tsarev equation z yy + z x z xy − z y z xx + 1 = 0 and,using the known Lax pair, we construct infinite series of conservation laws and the algebraof nonlocal symmetries in the covering associated with these conservation laws. We provethat the algebra is isomorphic to the Witt algebra. Finally, we show that the constructedsymmetries are unique in the class of polynomial ones. Introduction
The Gibbons–Tsarev equation considered in this paper was introduced in [6] to clas-sify finite reductions of the infinite Benney system. The Gibbons–Tsarev equation is un-doubtedly integrable [7, 21, 1]. It is known to have infinitely many conservation laws andinfinitely many symmetries ([5, § § Z ( − and Z (1) (constructed in Sections 2 and 4,respectively), they are not of much help, because obtaining them and their commutatorsrequires essentially the same effort as obtaining all symmetries and commutators at once.We present the results as follows. In Section 1, we introduce main notions and the nota-tion. Section 2 deals with the local properties of the Gibbons–Tsarev equation. Coveringsand nonlocal conservation laws are dealt with in Section 3. We introduce an appropriateinfinite system of nonlocal conservation laws in two different but equivalent ways, which isconvenient from the computational point of view. The corresponding infinite-dimensionalcovering is the common ‘ground’ for all the nonlocal symmetries and their commutatorsto be constructed in the sequel. We also consider a one-dimensional covering that allowsto treat the equation as an evolutionary two-component system, which is also convenientfor some proofs. In Section 4, we construct the nonlocal symmetries, starting with their Date : May 24, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Gibbons–Tsarev equation, differential coverings, nonlocal symmetries, nonlo-cal conservation laws, Witt algebra. ′ SHCHIK, AND M. MARVAN shadows. The shadows were found in a way that can be reused in other similar situations.The rest of the section is devoted to the explicit description of full nonlocal symmetriesderived from these shadows and to a proof that the symmetries constitute the Witt alge-bra. Finally, Section 5 is devoted to the proof of uniqueness of the constructed symmetriesin the class of polynomial ones.1.
Preliminaries and notation
We expose here briefly the fundamentals of local [3] and nonlocal [11] geometry ofPDEs. Consider a PDE given by a system of relations { F = 0 } , where F = ( F , . . . , F r )is a vector function in x = ( x , . . . , x n ), u = ( u , . . . , u m ) and finite number of partialderivatives of u with respect to x . To any such a system we put into correspondence alocus E ⊂ J ∞ ( π ) in the space of infinite jets, where π : R m × R n → R n is the trivialbundle and R m , R n are Euclidean spaces with the coordinates u , . . . , u m , x , . . . , x n ,respectively. This locus is defined by all the differential consequences of the system andcalled the infinitely prolonged equation.When the coordinates x i , u j are chosen, the adapted coordinates u jσ arise in J ∞ ( π )and correspond to the partial derivatives ∂ | σ | u j /∂x σ , where σ is a symmetric multi-indexwhose entries are the integers 1 , . . . , n . We always assume that the system is presented inthe passive orthonomic form, see [13], which allows to choose internal coordinates on E .When we say that an object is restricted from J ∞ ( π ) to E , we mean that it is rewrittenin terms of the internal coordinates.The key role in the geometry of PDEs is played by the total derivative operators D x i = ∂∂x i + X j,σ u jσi ∂∂u jσ . These operators can be restricted to any infinitely prolonged equation. Consequently, anydifferential operator in total derivatives (a C -differential operator) is restrictable to E as well. We preserve the same notation for the restrictions if no contradiction arises. Wesay that E is differentially connected if D x i ( f ) = 0, i = 1 , . . . , n implies f = const. Thedistribution spanned by the total derivatives is called the Cartan distribution and denotedby C .A vector field S = X I s jσ ∂∂u jσ on E is a symmetry of E if [ S, D x i ] = 0 for all i . The notation P I means that thesum is taken over the set I of all internal coordinates u jσ . Symmetries form a Lie algebradenoted by sym( E ). To describe symmetries, consider the following construction. Let G = ( G , . . . , G r ) be a function on E . Define its linearisation as the matrix C -differentialoperator ℓ G = (cid:18)X ∂G α ∂u βσ D σ (cid:19) α =1 ,...,rβ =1 ,...,m , where D σ denotes the composition of D x i corresponding to the multi-index σ . We alsouse the notation ℓ E for ℓ F | E . Then the following result is valid: any symmetry is an N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 3 evolutionary vector field E ϕ = X I D σ ( ϕ j ) ∂∂u jσ , where the generating section ϕ = ( ϕ , . . . , ϕ m ) satisfies the equation ℓ E ( ϕ ) = 0. Thecommutator of symmetries induces the Jacobi bracket { ϕ, ϕ ′ } = E ϕ ( ϕ ′ ) − E ϕ ′ ( ϕ ) . We do not distinguish between symmetries and their generating sections below. A sym-metry S is called classical if it is projectable to J ( π ). We say that S is a point symmetryif it is projectable to J ( π ).A conservation law of E is a horizontal ( n − ω = a dx ∧ dx ∧ · · · ∧ dx n + a dx ∧ dx ∧ · · · ∧ dx n + · · · + a n dx ∧ dx ∧ · · · ∧ dx n − closed with respect to the horizontal de Rham differential d h = P ni =1 dx i ∧ D x i , i.e., suchthat P ni =1 ( − i D x i ( a i ) = 0. A conservation law is trivial if ω = d h ρ for some ( n − ρ .The quotient group of all conservation laws modulo trivial ones is denoted by Cl( E ).To compute conservation laws, their generating sections are used. Let ω be a conser-vation law and ¯ ω be its extension to the ambient space J ∞ ( π ) ⊃ E . Then d h (¯ ω ) = ∆( F )for some C -differential operator ∆ and the vector-function ψ = ( ψ , . . . , ψ r ) = ∆ ∗ (1) | E ,where ∆ ∗ denotes the adjoint operator, is the generating section of ω . It possesses two im-portant properties: (a) ψ = 0 if and only if ω is trivial, (b) ℓ ∗ E ( ψ ) = 0. Any solution of thelast equation is called a cosymmetry. The space of cosymmetries is denoted by cosym( E ).Let ˜ E , E be equations. We say that a smooth map τ : ˜ E → E is a morphism if for anypoint ˜ θ ∈ ˜ E one has τ ∗ ( ˜ C ˜ θ ) ⊂ C τ (˜ θ ) . A morphism is a (differential) covering if τ ∗ | ˜ C ˜ θ is aisomorphism for any ˜ θ ∈ ˜ E . Two coverings τ , τ over E are equivalent if there exists anisomorphism f : ˜ E → ˜ E such that τ ◦ f = τ . Assume that E is differentially connected.Then we say that τ is irreducible if ˜ E is differentially connected as well.Take coverings τ and τ and consider the Whitney product τ × τ of the correspondingbundles. It carries a natural structure of a covering, which is called the Whitney productof these coverings.Let τ : ˜ E = E × R s be the trivial bundle. Define the plane ˜ C ˜ θ at a point ˜ θ ∈ ˜ E as theparallel lift of C θ for θ = τ (˜ θ ). This is a covering, and any covering is said to be trivial ifit is equivalent to τ .A one-dimensional covering τ is called Abelian if it is either trivial or there exists anontrivial conservation law ω of E such that its lift τ ∗ ( ω ) becomes trivial on ˜ E . In general,a covering is Abelian if it is equivalent to the Whitney product of the necessary numberof one-dimensional Abelian coverings. Proposition 1 (see [9]) . A finite-dimensional Abelian covering over E is irreducible if andonly if the corresponding system of conservation laws is linearly independent modulo triv-ial ones. Consequently , equivalence classes of irreducible s -dimensional , s < ∞ , Abeliancoverings are in one-to-one correspondence with s -dimensional subspaces in Cl( E ) . We say that a symmetry of the covering equation ˜ E is a nonlocal symmetry of E .Denote by F and ˜ F the algebras of smooth functions on E and ˜ E . Due to τ , one hasthe embedding F ⊂ ˜ F . A derivation F → ˜ F that preserves the Cartan distributions is H. BARAN, P. BLASCHKE, I.S. KRASIL ′ SHCHIK, AND M. MARVAN called a nonlocal shadow. In particular, for any nonlocal symmetry ˜ S its restriction ˜ S | F is a shadow; ˜ S is said to be invisible if its shadow vanishes. Local symmetries of E can betreated as shadows in every covering. A shadow is called reconstructible if there exists anonlocal symmetry such that its shadow is the given one.We also say that a conservation law of ˜ E is a nonlocal conservation law of E .Let us pass to local coordinates. Since the Gibbons–Tsarev equation is two-dimensional,we shall confine ourselves to this case for simplicity. Consider the equation E given by { F ( x, y, u, u x , u y , . . . ) = 0 } and let τ : ˜ E = E × R s → E be a covering (the case s = ∞ isallowed). Let { w α } be coordinates in the fiber (they are called nonlocal variables). Thenthe total derivatives on ˜ E are of the form˜ D x = D x + X α X α ∂∂w α , ˜ D y = D y + X α Y α ∂∂w α ,X α , Y α being smooth functions in all the internal variables and w α . Then τ is a coveringif and only if [ ˜ D x , ˜ D y ] = 0, or, equivalently, D x ( Y α ) − D y ( X α ) + X β (cid:18) X β ∂Y α ∂w β − Y β ∂X α ∂w β (cid:19) = 0 , α = 1 , . . . , s. Equivalently, the system w αx = X α , w αy = Y α , (1)is compatible modulo E . If the functions X α , Y α do not depend on the nonlocal variables,then the covering is Abelian.Any nonlocal symmetry in τ is defined by its generating section Φ = ( ϕ, . . . , ψ α , . . . ),where ϕ = ( ϕ , . . . , ϕ m ) and ψ α are functions on ˜ E satisfying˜ D x ( ψ α ) = ˜ ℓ X α ( ϕ ) + X β ∂X α ∂w β ψ β , ˜ D y ( ψ α ) = ˜ ℓ Y α ( ϕ ) + X β ∂Y α ∂w β ψ β , ˜ ℓ E ( ϕ ) = 0 , (2)where the ‘tilde’ over a C -differential operator denotes its natural lift from E to ˜ E . Nonlo-cal shadows are given by functions ϕ that satisfy Equation (2), while invisible symmetriesare sections Φ with ϕ = 0 and ψ α satisfying˜ D x ( ψ α ) = X β ∂X α ∂w β ψ β , ˜ D y ( ψ α ) = X β ∂Y α ∂w β ψ β . Assume that the right-hand sides of (1) depend on a parameter λ (which is called thespectral parameter). A parameter is non-removable (essential) if the coverings τ λ are pair-wise inequivalent (cf. [12]). Having a family of coverings with an essential parameter, onecan expand the functions X α , Y α in formal series in λ . If substitution of ψ α = P i ∈ Z ψ αi λ i to this expansion is well defined, one obtains an infinite-dimensional covering with thenonlocal variables ψ αi . In the case when this covering is Abelian we get an infinite familyof conservation laws (perhaps, trivial or dependent). A classical example of this procedureis the construction of the infinite series of conservation laws for the Korteweg–de Vriesequation, [16]. N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 5
But even if the covering at hand does not depend on a parameter, there exists a standardway to insert such a parameter formally (the so-called reversion procedure, see [17]).Namely, assume for simplicity that τ is one-dimensional and is given by w x = X ( x, y, w, u, u x , u y , . . . ) , w y = Y ( x, y, w, u, u x , u y , . . . ) . Then v x = − X ( x, y, λ, u, u x , u y , . . . ) v λ , v y = − Y ( x, y, λ, u, u x , u y , . . . ) v λ is a covering as well, see [10] for the geometric interpretation. We use this constructionbelow to construct infinite series of nonlocal conservation laws for the Gibbons–Tsarevequation. 2. Local symmetries and conservation laws
Consider the Gibbons–Tsarev equation [6] in the form z yy + z x z xy − z y z xx + 1 = 0 (3)(obtained from [6, eq. (15)] by the exchange x ↔ y and z ↔ − z ). For a monomial X = x i y j , let us use the notation z X = ∂ i + j z∂x i ∂y j . In particular, z X = z when i = j = 0. For internal coordinates on E we choose x , y , z X such that X = x k or X = x k y , k ≥
0, while z yyX = D X ( z y z xx − z x z xy − X .If not stated otherwise, sums are taken over all internal coordinates. The total deriva-tives on E are D x = ∂∂x + X z xX ∂∂z X , D y = ∂∂y + X z yX ∂∂z X , (summation over all internal coordinates z X ). It is straightforward to check that that (3)is a differentially connected equation.2.1. Weights.
The Gibbons–Tsarev equation becomes homogeneous if we assign theweights | x | = 3, | y | = 2, | z | = 4 (due to the scaling symmetry, see Subsection 2.2 below)and | z x k | = | z | − k | x | = 4 − k, (cid:12)(cid:12) z x k y (cid:12)(cid:12) = | z | − k | x | − | y | = 2 − k. To any monomial in x , y , z x k , and z x k y we assign the weight that equals the sum of weightsof its factors. The total derivatives preserve the space of polynomials and, as operators,have the weights | D x | = − | D y | = − H. BARAN, P. BLASCHKE, I.S. KRASIL ′ SHCHIK, AND M. MARVAN
Local symmetries.
Let S = E Z be a symmetry of E . Then the defining equationfor the generating sections of symmetries is ℓ E ( Z ) ≡ D y ( Z ) + z x D x D y ( Z ) − z y D x ( Z ) + z xy D x ( Z ) − z xx D y ( Z ) = 0 . (4)Solving (4) for functions Z of small jet order, we found that Equation (3) possesses fivelocal symmetries Z ( − = 1 , z -translation ,Z ( − = z x , x -translation ,Z ( − = z y , y -translation , (5) Z ( − = yz x − x, generalized Galilean boost ,Z (0) = 3 xz x + 2 yz y − z, scaling . All these symmetries are point ones. In Section 5, it will be shown that this is the com-plete set of local symmetries. The vector field S ( i ) = E Z ( i ) , as an operator, has theweight (cid:12)(cid:12) S ( i ) (cid:12)(cid:12) = i .All commutators of the symmetries S ( − , . . . , S (0) vanish except for[ S (0) , S ( − ] = 2 S ( − , [ S (0) , S ( − ] = S ( − , [ S (0) , S ( − ] = S ( − , [ S (0) , S ( − ] = S ( − , [ S ( − , S ( − ] = − S ( − , [ S ( − , S ( − ] = − S ( − . Remark . Note that changing the basis by S (0)
7→ − S (0) we arrive to the commutatorrelations [ S ( i ) , S ( j ) ] = ( j − i ) S ( i + j ) , where formally S ( α ) = 0 for α < −
4. In whatfollows, we use the latter choice of the basic symmetries.It will be shown in Section 4 that this set of five symmetries can be extended to ahierarchy of nonlocal symmetries infinite in both positive and negative directions.2.3.
Cosymmetries.
The defining equation for cosymmetries of (3) is ℓ ∗ E ( R ) ≡ D y ( R ) + z x D x D y ( R ) − z y D x ( R ) − z xy D x ( R ) + 2 z xx D x ( R ) = 0 . Solutions of lower order include six local cosymmetries of the first order R (0) = 1 , R (1) = 2 z x , R (2) = 3 z x + 2 z y + 3 y, R (3) = 4 z x + 6 z x z y + 8 yz x + 2 x, R (4) = 5 z x + 12 z x z y + 15 yz x + 3 z y + 6 xz x + 10 yz y + z + y , R (5) = 6 z x + 20 z x z y + 24 yz x + 12 z x z y + 12 xz x + 36 yz x z y + 4( z + 6 y ) z x + 8 xz y + 12 xy (compare with [20, p. 156]) and a single one of the third order R ( − = z xxx . N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 7
We have verified by direct computation that Equation (3) has no local generating sectionof order 2 and 4.2.4.
Conservation laws.
All the above listed cosymmetries are the generating sectionsof conservation laws ρ ( i ) = P ( i ) d x + Q ( i ) d y , where P (0) = z x + z y + y,Q (0) = z x z y ,P (1) = z x + 2 z x z y − x,Q (1) = z x z y + z y − z,P (2) = z x + 3 z x z y + 3 yz x + z y + 3 yz y − z,Q (2) = z x z y + 2 z x z y + 3 yz x z y − xy,P (3) = z x + 4 z x z y + 4 yz x + 3 z x z y + 2 xz x + 8 yz x z y − zz x + 2 xz y − xy,Q (3) = z x z y + 3 z x z y + 4 yz x z y + z y + 2 xz x z y + 4 yz y − zz y − yz − x ,P (4) = z x + 5 z x z y + 5 yz x + 6 z x z y + 3 xz x + 15 yz x z y + z y + ( z + y ) z x + 6 xz x z y + 5 yz y + ( z + y ) z y − yz − x ,Q (4) = z x z y + 4 z x z y + 5 yz x z y + 3 z x z y + 3 xz x z y + 10 yz x z y + ( z + y ) z x z y + 3 xz y − x (4 z + 5 y ) ,P (5) = z x + 6 z x z y + 6 yz x + 10 z x z y + 4 xz x + 24 yz x z y + 4 z x z y + 4( z + 3 y ) z x + 12 xz x z y + 18 yz x z y + 12 xyz x + 4( z + 6 y ) z x z y + 4 xz y + 12 xyz y − xz,Q (5) = z x z y + 5 z x z y + 6 yz x z y + 6 z x z y + 4 xz x z y + 18 yz x z y + z y + 4( z + 3 y ) z x z y + 8 xz x z y + 6 yz y + 12 xyz x z y + 4( z + 3 y ) z y − z − y z − x y and P ( − = − z x z xx − z xy z xx ,Q ( − = − z y z xx − z xy . Note that (cid:12)(cid:12) ρ ( i ) (cid:12)(cid:12) = i + 5, i = − , , . . . , nonlocal conservation laws for theGibbons–Tsarev equation (3).3. Coverings and the infinite series of nonlocal conservation laws
Using two known coverings [6, 7] of the Gibbons–Tsarev equation, we construct herean infinite series of (nonlocal) conservation laws that later (Section 4) will be used to con-struct the corresponding infinite-dimensional Abelian covering and describe the algebraof nonlocal symmetries in this covering. It will also be shown that the obtained infinitedimensional coverings are equivalent.
H. BARAN, P. BLASCHKE, I.S. KRASIL ′ SHCHIK, AND M. MARVAN
Coverings.
Consider the nonlinear non-Abelian covering τ z : ˜ E → E over Equa-tion (3) given by ϕ x = 1 z y + z x ϕ − ϕ , ϕ y = − z x − ϕz y + z x ϕ − ϕ . (6)The covering introduced by Gibbons and Tsarev in [7] can be rewritten in this way.To simplify the subsequent computations, let us introduce new variables u and v suchthat z x = u + v, z y = − uv. (7)Due to the compatibility condition( u + v ) y + ( uv ) x = 0 (8)and by Equation (3) we deduce that the new variables enjoy the system of evolutionequations u y + vu x = 1 v − u , v y + uv x = 1 u − v (9)Denote this equation by E . The equation is homogeneous with respect to the weights | x | = 3, | y | = 2, | u | = | v | = 1. Due to (8), the form ( u + v ) d x − uv d y is a conservation lawof the equation E while (7) defines the covering E → E associated with this conservationlaw.The covering τ z defined by (6) generates the covering τ uv : ˜ E → E given by the relations ϕ x = − ϕ − u )( ϕ − v ) , ϕ y = u + v − ϕ ( ϕ − u )( ϕ − v ) (10)and the diagram of coverings ˜ E τ z (cid:15) (cid:15) ˜ τ / / ˜ E τ uv (cid:15) (cid:15) E τ / / E is commutative.3.2. Nonlocal conservation laws.
We construct an infinite hierarchy of nonlocal con-servation laws for the Gibbons-Tsarev equation using two different but related ways.3.2.1.
The first way.
Consider an arbitrary gauge symmetry ϕ ψ ( ϕ ) of the covering τ uv .For the sake of convenience, relabel the variable ϕ to λ . Then, applying the reversionprocedure described in Section 1 to the covering (10), one obtains ψ x = 1( λ − u )( λ − v ) · ψ λ , ψ y = λ − ( u + v )( λ − u )( λ − v ) · ψ λ . (11)Now, we consider λ as a formal parameter and expand ψ in the Laurent series ψ = ψ ( − λ + ψ (0) + ψ (1) λ + · · · + ψ ( k ) λ k + . . . (12)One also has the obvious expansions1 λ − u = 1 λ X i ≥ u i λ i , λ − v = 1 λ X i ≥ v i λ i N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 9 which imply 1( λ − u )( λ − v ) = 1 λ (cid:16) σ λ + · · · + σ k λ k + . . . (cid:17) , where σ k = X i + j = k u i v j . (13) Remark . Note that since the quantities σ k are symmetric in the variables u and v , theycan be rewritten as polynomials in z x = u + v and z y = − uv . See formula (24) below.Now, from the expansion (12) one obtains ψ x = ψ ( − x λ + ψ (0) x + ψ (1) x λ + · · · + ψ ( k ) x λ k + . . . ,ψ y = ψ ( − y λ + ψ (0) y + ψ (1) y λ + · · · + ψ ( k ) y λ k + . . . , and ψ λ = ψ ( − − ψ (1) λ − ψ (2) λ − · · · − k ψ ( k ) λ k +1 + . . . Substituting all the above expansions to Equations (11), one obtains ψ ( − x λ + ψ (0) x + ψ (1) x λ + · · · + ψ ( k ) x λ k + . . . = 1 λ (cid:16) σ λ + · · · + σ k λ k + . . . (cid:17) (cid:18) ψ ( − − ψ (1) λ − ψ (2) λ − · · · − kψ ( k ) λ k +1 + . . . (cid:19) ,ψ ( − y λ + ψ (0) y + ψ (1) y λ + · · · + ψ ( k ) y λ k + . . . = (cid:18) λ − σ λ (cid:19) (cid:16) σ λ + · · · + σ k λ k + . . . (cid:17) (cid:18) ψ ( − − ψ (1) λ − ψ (2) λ − · · · − kψ ( k ) λ k +1 + . . . (cid:19) . Denote by A + A λ + · · · + A k λ k + . . . the result of multiplication of the last two factors in the previous expressions, i.e., A = ψ ( − , A = σ ψ ( − , A = σ ψ ( − − ψ (1) , and A k = σ k ψ ( − − σ k − ψ (1) − σ k − ψ (2) − · · · − ( k − σ ψ ( k − − ( k − ψ ( k − , k ≥ . Consequently, ψ ( − x λ + ψ (0) x + ψ (1) x λ + · · · + ψ ( k ) x λ k + · · · = 1 λ (cid:18) A + A λ + · · · + A k λ k + . . . (cid:19) ,ψ ( − y λ + ψ (0) y + ψ (1) y λ + · · · + ψ ( k ) y λ k + · · · = (cid:18) λ − σ λ (cid:19) (cid:18) A + A λ + · · · + A k λ k + . . . (cid:19) and thus ψ ( − x = 0 , ψ (0) x = 0 , ψ (1) x = 0 , ψ ( − y = 0 , ψ (0) y = 0 , ψ (1) y = A ′ SHCHIK, AND M. MARVAN and ψ ( k ) x = A k − , ψ ( k ) y = A k − − σ A k − (14)for k ≥
2. Without loss of generality we can set ψ ( − = 1 and skip the variable ψ (0) , sincethe coefficients A k are independent of it. Then, using the obtained expressions for A and A , we obtain ψ (1) x = 0, ψ (1) y = 1, ψ (2) x = 1, ψ (2) y = 0 and set ψ (1) = y, ψ (2) = x, (15)without loss of generality as well. Thus, we have A = 1 , A = σ , A = σ − y, A = σ − σ y − x and A k = σ k − σ k − y − σ k − x − σ k − ψ (3) − · · · − ( k − σ ψ ( k − − ( k − ψ ( k − for k > σ σ k − σ k +1 = uvσ k − , we obtain from (14) ψ (3) x = σ , ψ (3) y = − uv − y ; ψ (4) x = σ − y, ψ (4) y = − uvσ − x ; ψ (5) x = σ − σ y − x, ψ (5) y = − uv ( σ − y ) − ψ (3) ; ψ (6) x = σ − σ y − σ x − ψ (3) , ψ (6) y = − uv ( σ − σ y − x ) − ψ (4) ; ψ (7) x = σ − σ y − σ x − σ ψ (3) − ψ (4) , ψ (7) y = − uv ( σ − σ y − σ x − ψ (3) ) − ψ (5) and ψ ( k ) x = σ k − − σ k − y − σ k − x − k − X i =3 iσ k − i − ψ ( i ) , (16) ψ ( k ) y = − uv ( σ k − − σ k − y − σ k − x − k − X i =3 iσ k − i − ψ ( i ) ) − ( k − ψ ( k − . (17)for k ≥
7. Denote by X ( k ) and Y ( k ) the right-hand sides of the obtained equations, i.e., ψ ( k ) x = X ( k ) , ψ ( k ) y = Y ( k ) , k ≥ . (18)Obviously, we have (cid:12)(cid:12) X ( k ) (cid:12)(cid:12) = k − (cid:12)(cid:12) Y ( k ) (cid:12)(cid:12) = k − (cid:12)(cid:12) ψ ( k ) (cid:12)(cid:12) = k + 1.Let us now return back to the equation E given by (9) and consider the spaces E = E × R (3) , . . . , E k = E k − × R ( k +1) , . . . , where R ( k ) is R with the distinguished coordinate ψ ( k ) , k ≥
3. Consider also the naturalprojections τ k,k − : E k → E k − , τ k : E k → E . Let E ∗ be the inverse limit of the infinite sequence E . . . o o E k − o o E kτ k,k − o o . . . o o N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 11 and τ ∗ : E ∗ → E be the corresponding projection. Endow the spaces E k with the vectorfields D ( k ) x = D x + k +1 X i =3 X ( i ) ∂∂ψi , D ( k ) y = D y + k +1 X i =3 Y ( i ) ∂∂ψi , where D x and D y are the total derivatives on E . Similarly, we define the fields D ( ∗ ) x and D ( ∗ ) y on E ∗ . Proposition 2.
For all k , including the case k = ∗ , one has [ D ( k ) x , D ( k ) y ] = 0 .Proof. This is an immediate consequence of the fact that (10) is a covering over E . (cid:3) Hence, all the maps τ k carry covering structures; these coverings are irreducible: Proposition 3.
Let f ∈ F ( E k ) be a function such that D ( k ) x ( f ) = D ( k ) y ( f ) = 0 . Then f =const .Proof. Let x, y, . . . , u i = ∂ i u∂x i , v i = ∂ i v∂x i , . . . be coordinates on E and D x = ∂∂x + X i ≥ (cid:18) u i +1 ∂∂u i + v i +1 ∂∂v i (cid:19) ,D y = ∂∂y + X i ≥ (cid:18) D ix (cid:18) v − u + vu (cid:19) ∂∂u i + D ix (cid:18) u − v + uv (cid:19) ∂∂v i (cid:19) be the total derivatives in these coordinates. Consider a function f = f ( x, y, u, v, . . . , u i , v j , ψ (3) , . . . , ψ ( k ) )on E k and assume that D x ( f ) + X (3) ∂f∂ψ (3) + · · · + X ( k ) ∂f∂ψ ( k ) = D y ( f ) + Y (3) ∂f∂ψ (3) + · · · + Y ( k ) ∂f∂ψ ( k ) = 0 . (19)Since the coefficients X (3) , Y (3) , . . . , X ( k ) , Y ( k ) are independent of the variables u α , v β forall α and β >
0, from the above formulas for D x an D y it follows that f cannot dependon these variables either as well as on u and v and thus Equation (19) reads now ∂f∂x + X (3) ∂f∂ψ (3) + · · · + X ( k ) ∂f∂ψ ( k ) = ∂f∂y + Y (3) ∂f∂ψ (3) + · · · + Y ( k ) ∂f∂ψ ( k ) = 0 . But X ( α ) and Y ( β ) are polynomials in u and v of degrees α − β −
1, respectively,and this finishes the proof. (cid:3)
Obviously, every map τ k,k − : E k → E k − is also a covering; moreover, it is an Abeliancovering associated to the conservation law ω ( k ) = X ( k ) d x + Y ( k ) d y ∈ Cl( E k − )and (cid:12)(cid:12) ω ( k ) (cid:12)(cid:12) = k + 1. Proposition 4.
The conservation law ω ( k ) is nontrivial on E k − . ′ SHCHIK, AND M. MARVAN
Proof.
This readily follows from general properties of coverings (see Section 1) and Propo-sition 3. (cid:3)
Remark . By the very construction, the equation E is equivalent to the Gibbons–Tsarevequation (3). Moreover, it can be checked that the conservation laws ω (4) , . . . , ω (9) areequivalent to the conservation laws ρ (0) , . . . , ρ (5) , respectively, described in Subsection 2.4. Remark . Of course, the initial choice (15) for the values of ψ ( − , ψ (1) , and ψ (2) is notunique. Nevertheless, one can easily show that other admissible values lead to equivalentresults.3.2.2. The second method.
Consider now the covering (10) and assume that ϕ = ϕ ( − λ + ϕ (0) + ϕ (1) λ + · · · + ϕ ( k ) λ k + . . . (20)Then, rewriting (11) in the form( ϕ − u )( ϕ − v ) ϕ x = − , ( ϕ − u )( ϕ − v ) ϕ y = u + v − ϕ and substituting expansion (20), one obtains the following defining system for the coeffi-cients ϕ ( i ) : B − ϕ ( − x = 0 , B − ϕ ( − y = 0 ,B − ϕ (0) x + B − ϕ ( − x = 0 , B − ϕ (0) y + B − ϕ ( − y = 0 ,B − ϕ (1) x + B − ϕ (0) x + B ϕ ( − x = 0 , B − ϕ (1) y + B − ϕ (0) y + B ϕ ( − y = − ϕ ( − ,B − ϕ (2) x + B − ϕ (1) x + B ϕ (0) x + B ϕ ( − x B − ϕ (2) y + B − ϕ (1) y + B ϕ (0) y + B ϕ ( − y = − , = u + v − ϕ (0) ,B − ϕ (3) x + B − ϕ (2) x + B ϕ (1) x + B ϕ (0) x B − ϕ (3) y + B − ϕ (2) y + B ϕ (1) y + B ϕ (0) y + B ϕ ( − x = 0 , + B ϕ ( − y = − ϕ (1) ,. . . . . .B − ϕ ( k +2) x + B − ϕ ( k +1) x + · · · + B k +1 ϕ ( − x B − ϕ ( k +2) y + B − ϕ ( k +1) y + · · · + B k +1 ϕ ( − y = 0 , = − ϕ ( k ) ,. . . . . . where ( ϕ − u )( ϕ − v ) = B − λ + B − λ + B + B λ + · · · + B k λ k + . . . is the expansion of the product ( ϕ − u )( ϕ − v ), i.e., B − = (cid:0) ϕ ( − (cid:1) ,B − = ϕ ( − (cid:0) ϕ (0) − u − v (cid:1) ,B = 2 ϕ ( − ϕ (1) + (cid:0) ϕ (0) − u (cid:1) (cid:0) ϕ (0) − v (cid:1) ,B = 2 ϕ ( − ϕ (2) + (cid:0) ϕ (0) − u − v (cid:1) ϕ (1) ,B = 2 ϕ ( − ϕ (3) + (cid:0) ϕ (0) − u − v (cid:1) ϕ (2) + (cid:0) ϕ (1) (cid:1) , N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 13 B = 2 ϕ ( − ϕ (4) + (cid:0) ϕ (0) − u − v (cid:1) ϕ (3) + 2 ϕ (1) ϕ (2) ,. . .B k = 2 ϕ ( − ϕ (2 k +1) + (cid:0) ϕ (0) − u − v (cid:1) ϕ (2 k ) + 2 ϕ (1) ϕ (2 k − + · · · + 2 ϕ ( k − ϕ ( k +1) + (cid:0) ϕ ( k ) (cid:1) ,B k +1 = 2 ϕ ( − ϕ (2 k +2) + (cid:0) ϕ (0) − u − v (cid:1) ϕ (2 k +1) + 2 ϕ (1) ϕ (2 k ) + · · · + 2 ϕ ( k ) ϕ ( k +1) ,. . . Analysing the first four equations of the defining system, we see that the followingchoice of coefficients is possible: ϕ ( − = 1 , ϕ (0) = 0 , ϕ (1) = − y, ϕ (2) = − x. (21)Then B − = 1, while B − = − ( u + v ) ,B = − y + uv,B = − x + y ( u + v ) ,B = 2 ϕ (3) + x ( u + v ) + y ,B = 2 ϕ (4) − ( u + v ) ϕ (3) + 2 xy,B = 2 ϕ (5) − ( u + v ) ϕ (4) − yϕ (3) + 2 x ,B = 2 ϕ (6) − ( u + v ) ϕ (5) − yϕ (4) − xϕ (3) ,B = 2 ϕ (7) − ( u + v ) ϕ (6) − yϕ (5) − xϕ (4) + (cid:0) ϕ (3) (cid:1) ,B = 2 ϕ (8) − ( u + v ) ϕ (7) − yϕ (6) − xϕ (5) + 2 ϕ (3) ϕ (4) ,. . .B k = 2 ϕ (2 k +1) − ( u + v ) ϕ (2 k ) − yϕ (2 k − − xϕ (2 k − + 2 ϕ (3) ϕ (2 k − + . . . · · · + 2 ϕ ( k − ϕ ( k +1) + (cid:0) ϕ ( k ) (cid:1) ,B k +1 = 2 ϕ (2 k +2) − ( u + v ) ϕ (2 k +1) − yϕ (2 k ) − xϕ (2 k − + 2 ϕ (3) ϕ (2 k − + . . . · · · + 2 ϕ ( k ) ϕ ( k +1) ,. . . Hence, the initial defining system transforms to ϕ (3) x = − ( u + v ) , ϕ (3) y = uv − y,ϕ (4) x = − y − u − uv − v , ϕ (4) y = − x + uv ( u + v ) , while for k > ϕ ( k ) x = B k − − B k − ϕ (3) x − · · · − B − ϕ ( k − x ,ϕ ( k ) y = B k − − B k − ϕ (3) y − · · · − B − ϕ ( k − y − ϕ ( k − . (22)Denote by ¯ X ( k ) and ¯ Y ( k ) the right-hand sides of equations (22), i.e., ϕ ( k ) x = ¯ X ( k ) , ϕ ( k ) y = ¯ Y ( k ) , k ≥ . (23)We have (cid:12)(cid:12) ϕ ( k ) (cid:12)(cid:12) = k + 1. ′ SHCHIK, AND M. MARVAN
Now, exactly as in Subsection 3.2.1, we introduce the spaces ¯ E k = ¯ E k − × ¯ R ( k +1) , k = 2 , . . . , where ¯ R ( k ) = R with the coordinate ϕ ( k ) , the projections¯ τ k,k − : ¯ E k → ¯ E k − , ¯ τ k : ¯ E k → E and ¯ τ ∗ : ¯ E ∗ → E as the inverse limit. We endow these spaces with the vector fields¯ D ( k ) x = D x + k +1 X i =3 ¯ X ( i ) ∂∂ϕ ( i ) , ¯ D ( k ) y = D y + k +1 X i =3 ¯ Y ( i ) ∂∂ϕ ( i ) . Similarly, we define ¯ D ( ∗ ) x and ¯ D ( ∗ ) y . Proposition 5.
For all k ≥ and k = ∗ one has [ ¯ D ( k ) x , ¯ D ( k ) y ] = 0, i.e. , all the maps ¯ τ k and ¯ τ k,k − are coverings. All these coverings are irreducible. Consider the forms ¯ ω ( k ) = ¯ X ( k ) d x + ¯ Y ( k ) d y. One has (cid:12)(cid:12) ¯ ω ( k ) (cid:12)(cid:12) = k + 1 and Proposition 6.
For every k ≥ the form ¯ ω ( k ) is a nontrivial conservation law of theequation ¯ E k − .Remark . As before, the choice (21) of initial values for ϕ ( − , . . . , ϕ (2) is not unique, butall admissible choices lead to equivalent results.Finally, the following statement is valid: Proposition 7.
The pairs of coverings τ k,k − and ¯ τ k,k − , τ k and ¯ τ k , τ ∗ and ¯ τ ∗ are equiv-alent. We provide the proof in the next subsection.3.3.
Proof of Proposition 7.
Let us turn back to the Gibbons–Tsarev equation (3).For reader’s convenience, we summarise the results of the previous section in terms of thevariables x, y, z . We recall that ψ (0) = 0 , ψ (1) = y, ψ (2) = x, ψ (3) = z − y , while ψ ( k ) , k >
3, are genuine nonlocal variables of the Gibbons–Tsarev equation, satis-fying ψ ( k ) x = σ k − − k − X i =1 iσ k − i − ψ ( i ) , ψ ( k ) y = z y ψ ( k − x − ( k − ψ ( k − . In terms of z , we have σ k = X ≤ j ≤ k − j (cid:18) k − jj (cid:19) z k − jx z jy , k > . (24)To prove formula (24), we consider the formal power series in an auxiliary variable λ withcoefficients taken from the two sides of formula (24) and show that they coincide. Usingthe left-hand side, we have, according to formula (13), X k ≥ σ k λ k = X i,j ≥ u i v j λ i + j = X i,j ≥ ( uλ ) i ( vλ ) j = 11 − uλ · − vλ . N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 15
Using the right-hand side, where we substitute for z x , z y from formulas (7), we obtain thesame series: X k ≥ X ≤ j ≤ k − j (cid:18) k − jj (cid:19) z k − jx z jy λ k = X i ≥ X ≤ j ≤ i (cid:18) ij (cid:19) z i − jx z jy λ i + j = X i ≥ X ≤ j ≤ i (cid:18) ij (cid:19) ( z x λ ) i − j ( z y λ ) j = X i ≥ ( z x λ + z y λ ) i = X i ≥ (cid:0) ( u + v ) λ − uvλ (cid:1) i = X i ≥ (cid:0) − (1 − uλ )(1 − vλ ) (cid:1) i = 1(1 − uλ )(1 − vλ ) . Thus, formula (24) is proved.The first method of the previous section uses the expansion (12), i.e., ψ ( λ ) = λ + ψ (1) λ + · · · + ψ ( k ) λ k + · · · , (25)where ψ satisfies the linear system (11), which we rewrite in terms of z x , z y : ψ x = 1 λ − z x λ − z y · ψ ′ , ψ y = λ − z x λ − z x λ − z y · ψ ′ , (26)where the ‘prime’ denotes the λ -derivative. The second method uses the expansion (20),i.e., ϕ ( λ ) = 1 λ + ϕ (1) λ + · · · + ϕ ( k ) λ k + · · · , (27)where ϕ satisfies the nonlinear system (6), i.e., ϕ x = − ϕ − z x ϕ − z y , ϕ y = − ϕ − z x ϕ − z x ϕ − z y . (28)Recall that composition b ◦ a of formal series b ( µ ) = P j ≥ s b j µ j and a ( λ ) = P i ≥ r a i λ i ,i.e., b ( a ( λ )) = X j ≥ s b j (cid:16)X i ≥ r a i λ i (cid:17) j = X j ≥ s b j (cid:16)X i ≥ r a i r λ i (cid:17) . . . (cid:16)X i j ≥ r a i j λ i j (cid:17) = X j ≥ s b j X i ≥ r,...,i j ≥ r a i . . . a i j λ i + ··· + i j is a formal series if and only if the coefficients at powers of λ are finite sums. This iscertainly the case when P j ≥ s b j µ j is a polynomial or when r ≥
1, i.e., when P i ≥ r a i λ i isa power series without the constant term.Computing 1 ϕ ( λ ) = λ − ϕ (1) λ k +2 + · · · ′ SHCHIK, AND M. MARVAN we see that 1 /ϕ ( λ ) is a power series without the constant term and therefore, the com-position series ψ ◦ ϕ , i.e., ψ ( ϕ ( λ )) = ϕ ( λ ) + ψ (1) ϕ ( λ ) + · · · + ψ ( k ) ϕ ( λ ) k + · · · , is well defined. Proposition 8.
Let ψ ( λ ) and ϕ ( λ ) be the formal expansions (25) and (27) , respectively.Then each pair of the conditions (1) equation (26);(2) equation (28);(3) ψ ( ϕ ) = c ( λ ) , where c ( λ ) is a constant ( possibly depending on λ ), implies the remaining condition.Proof. Assume that (26) and (28) hold. Substituting ϕ for λ in (26), an easy computationyields ( ψ ( ϕ )) x = ψ x ( ϕ ) + ψ ′ ( ϕ ) ϕ x = 0 , ( ψ ( ϕ )) y = ψ y ( ϕ ) + ψ ′ ( ϕ ) ϕ y = 0by virtue of (28). Then ψ ( ϕ ) is a constant with respect to x and y , since the covering (28)is differentially connected.Conversely, assume that ψ ( ϕ ) = c ( λ ), where c ( λ ) does not depend on x and y . Then ψ ( ϕ ) x = ψ ( ϕ ) y = 0 and ψ x ( ϕ ) = − ψ ′ ( ϕ ) ϕ x , ψ y ( ϕ ) = − ψ ′ ( ϕ ) ϕ y , which yields the equivalence of Equations (26) and (28). (cid:3) Under the substitution λ → /λ , the expansion (25) acquires the form ψ (cid:16) λ (cid:17) = 1 λ + ψ (1) λ + · · · + ψ ( k ) λ k + · · · , i.e., ϕ ( λ ) and ψ (1 /λ ) are Laurent series of the lowest degree −
1. Consequently, 1 /ϕ ( λ )and 1 /ψ (1 /λ ) are power series without a constant term. So, they are composable witheach other.There is a preferable choice of the constant c ( λ ) in Proposition 8. Proposition 9.
The expansions ϕ , ψ can be chosen so that ψ ( ϕ ( λ )) = 1 /λ, i.e. , the power series /ψ (1 /λ ) and /ϕ ( λ ) are compositionally inverse one to another.Proof. According to Proposition 8, we are free to choose c ( λ ) = 1 /λ , i.e., ψ ( ϕ ( λ )) = 1 /λ .Substituting 1 /ϕ ( λ ) for λ in 1 /ψ (1 /λ ), we obtain 1 /ψ ( ϕ ( λ )) = 1 /c ( λ ) = λ . Hence thestatement. (cid:3) With this choice of c ( λ ), the k -tuples of coefficients ψ (1) , . . . , ψ ( k ) and ϕ (1) , . . . , ϕ ( k ) determine each other uniquely, thereby providing the induction step in the proof of the N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 17 equalities E k = ¯ E k . It is, however, necessary to check that the condition c ( λ ) = 1 /λ iscompatible with the choices ψ (1) = y, ϕ (1) = − y,ψ (2) = x, ϕ (2) = − x,ψ (3) = z − y , ϕ (3) = − z − y . (29)made in Section 3. To this end, we compute1 λ = ψ ( ϕ ) = ϕ ( λ ) + ψ (1) ϕ ( λ ) + · · · + ψ ( k ) ϕ ( λ ) k + · · · = 1 λ + ( ψ (1) + ϕ (1) ) λ + ( ψ (2) + ϕ (2) ) λ + ( ψ (3) + ϕ (3) − ψ (1) ϕ (1) ) λ + ( ψ (4) + ϕ (4) − ψ (1) ϕ (2) − ψ (2) ϕ (1) ) λ + · · · One easily sees that the coefficients at λ i , i = 1, 2, 3, vanish under the above mentionedchoices and we obtain the recurrent formulas ψ ( k ) = − X m ≥ ( − m X i + ··· + i m = k +1 k (cid:18) km (cid:19) ϕ ( i − · · · ϕ ( i m − ,ϕ ( k ) = − X m ≥ X i + ··· + i m = k +1 k (cid:18) km (cid:19) ψ ( i − · · · ψ ( i m − that provide the needed equivalence of coverings.4. Nonlocal symmetries
It is straightforward to compute the first-degree nonlocal shadows depending on anynumber of nonlocal variables. It may seem to be insignificant whether we use ψ ( i ) or ϕ ( i ) ,but the formulas to follow turn out to be simpler if the latter choice is made. Thus, wegive here an explicit description of nonlocal symmetries in the covering ¯ τ ∗ and prove thatthey form the Witt algebra. As the first step, we obtain the shadows.4.1. The hierarchy of symmetry shadows.
Consider the covering ¯ τ ∗ with the nonlocalvariables ϕ ( i ) and present the total derivatives in the form˜ D x = D x + X i ¯ X ( i ) ∂∂ϕ ( i ) , ˜ D y = D y + X i ¯ Y ( i ) ∂∂ϕ ( i ) , where ¯ X ( i ) and ¯ Y ( i ) are the right-hand sides in (22).Now, using the expansion (27), let us introduce a new set of nonlocal variables ϕ λ i =d i ϕ/ d λ i and consider the product ¯ E λ = E × J ( λ ; ϕ ), where J ( λ ; ϕ ) is the space with the ′ SHCHIK, AND M. MARVAN coordinates λ and ϕ λ i , and the covering ¯ τ λ : ¯ E λ → E . In what follows we abbreviate the‘index’ λ n as Λ. We equip ¯ E λ with the total derivatives˜ D x = D x + X Λ ϕ x Λ ∂∂ϕ Λ , ˜ D y = D y + X Λ ϕ y Λ ∂∂ϕ Λ , ˜ D λ = dd λ + X Λ ϕ λ Λ ∂∂ϕ Λ , (30)where the coefficients ϕ x Λ and ϕ y Λ can be computed by means of Equations (28). Then ˜ E endowed with the vector fields (30) is equivalent to the system consisting of the Gibbons–Tsarev equation (3), the condition z λ = 0 , (31)and the pair (28) over the extended set of independent variables x , y , λ . Proposition 10.
Denote Z = ( ϕ − z x ϕ − z y ) ϕ λ , (32) Under the expansion (27), Z is a formal Laurent series of the form Z = ∞ X n = − Z ( n ) λ n − . (33) Then Z ( n ) are shadows of symmetries of the Gibbons–Tsarev equation in the covering ¯ τ ∗ .Proof. It is a routine computation to insert (32) into the linearisation˜ ℓ E ( Z ) ≡ ˜ D y ( Z ) + z x ˜ D x ˜ D y ( Z ) − z y ˜ D x ( Z ) + z xy ˜ D x ( Z ) − z xx ˜ D y ( Z ) (34)and check that ˜ ℓ E ( Z ) = 0 modulo equations (3), (28) and (31). If Z is replaced with itsexpansion (33), we obtain 0 = ˜ ℓ E ( Z ) = ∞ X n = − ˜ ℓ E ( Z ( n ) ) λ n − . Since ˜ ℓ E ( Z ( n ) ) do not depend on λ , they have to vanish modulo equation (3) and expandedsystem (28), i.e., equations (23). Hence the statement. (cid:3) Remark . It is easy to compute functions Z such that ˜ ℓ E ( Z ) = 0 modulo equations (3),(28) and (31) (cf. the proof of Proposition 10). Besides the expression (32), another suchfunction is Z = ϕ λ , which, however, generates just the invisible symmetries (see Sect. 1).Moreover, if some Z satisfies ˜ ℓ E ( Z ) = 0, then so does f ( λ ) Z for any function f ( λ ). Thisdoes not extend the linear space of generated shadows Z ( i ) , however. Remark . Although the condition z λ = 0 is necessary for Z given by (32) to be a shadowof the Gibbons–Tsarev equation, the same Z does not satisfy ˜ D λ Z = 0 and, therefore, isnot a shadow of the system consisting of the Gibbons–Tsarev equation and the equation z λ = 0.Proposition 10 says that Z is the generating section for an infinite hierarchy of shadowsof the Gibbons–Tsarev equation. These shadows are easy to obtain explicitly. Let P ( • ) denote summation where indices run through all integers from − N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 19
Proposition 11.
Let A ( k,n )2 = X ( • ) i + ··· + i k +2 = n i i ϕ ( i ) · · · ϕ ( i k +2 ) , k ≥ . (35) Then Z ( n ) = A (1 ,n )2 z x + A (0 ,n )2 z y − A (2 ,n )2 . Proof.
Considering the expansion (27), we have ϕ k ϕ ′ = (cid:16)X ( • ) i ϕ ( i ) λ i (cid:17) · · · (cid:16)X ( • ) i k ϕ ( i k ) λ i k (cid:17) × λ (cid:16)X ( • ) i k +1 i k +1 ϕ ( i k +1 ) λ i k +1 (cid:17)(cid:16)X ( • ) i k +2 i k +2 ϕ ( i k +2 ) λ i k +2 (cid:17) = X n X ( • ) i + ··· + i k +2 = n i k +1 i k +2 ϕ ( i ) · · · ϕ ( i k +2 ) λ n − = X n A ( k,n )2 λ n − . Inserting into Z given by formula (32), we obtain the result immediately. (cid:3) The hierarchy of full symmetries.
Here the shadows Z ( n ) obtained in the previ-ous section will be extended to full symmetries of the covering ¯ τ ∗ . To this end, consider anonlocal symmetry in the form a ∂∂x + b ∂∂y + c ∂∂z + X i> f ( i ) ∂∂ϕ ( i ) + · · · , where a , b , c , f ( i ) are functions on ¯ E ∗ . Then the corresponding vertical field, obtained bysubtracting a ˜ D x + b ˜ D y , is S = X Ξ ˜ D Ξ ( c − az x − bz y ) ∂∂z Ξ + X i> ( f ( i ) − aϕ ( i ) x − bϕ ( i ) y ) ∂∂ϕ ( i ) , (36)where ϕ ( i ) x = ¯ X ( i ) , ϕ ( i ) y = ¯ Y ( i ) are given by recurrent relations (22) and z Ξ are internalcoordinates in E (see Section 2). Then S is a symmetry of ¯ E ∗ if and only if S ( z yy + z x z xy − z y z xx + 1) = 0 , S ( ϕ ( i ) x − ¯ X ( i ) ) = 0 , S ( ϕ ( i ) y − ¯ Y ( i ) ) = 0 (37)modulo equations (3) and (23). Using formulas (29), variables x, y, z can be expressed interms of ϕ (1) , ϕ (2) , ϕ (3) . Consequently, we can rewrite (36) and (37) in terms of ϕ ( i ) and z Ξ , | Ξ | >
0, alone.
Proposition 12.
In terms of coordinates ϕ ( i ) , i > and z Ξ , a vertical evolutionary fieldin the covering ¯ τ ∗ can be written as S = X i> Φ ( i ) ∂∂ϕ ( i ) + X | Ξ | > ˜ D Ξ Z ∂∂z Ξ ,Z = ( z y − ϕ (1) ) f (1) + z x f (2) − f (3) , Φ ( i ) = f ( i ) + f (2) ¯ X ( i ) + f (1) ¯ Y ( i ) . (38) ′ SHCHIK, AND M. MARVAN
The field S is a symmetry if and only if ˜ ℓ E Z = 0 and ˜ D x Φ ( i ) − S ¯ X ( i ) = 0 , ˜ D y Φ ( i ) − S ¯ Y ( i ) = 0 . (39) Proof.
Formulas (38) are obtained by direct computation, while (39) follows from (37)immediately. (cid:3)
Let us now pass from the covering ¯ τ ∗ with the nonlocal variables ϕ ( i ) to the covering¯ E λ → E obtained from the covering (28) by means of the expansion (27), i.e., ϕ = 1 λ + X i ϕ ( i ) λ i . Then ϕ λ = − /λ + P i iϕ ( i ) λ i − , ϕ λλ = 2 /λ + P i i ( i − ϕ ( i ) λ i − , etc. Hence, ∂ϕ∂ϕ ( i ) = λ i , ∂ϕ λ ∂ϕ ( i ) = iλ i − = d λ i d λ , ∂ϕ λλ ∂ϕ ( i ) = i ( i − λ i − = d λ i d λ , . . . and, therefore, ∂∂ϕ ( i ) = X Λ ∂ Λ λ i ∂∂ϕ Λ , where, as above, Λ stands for λ n , n ≥
0, and ∂ λ n = d n /dλ n . Alternatively speaking,the vector field ∂/∂ϕ ( i ) , when rewritten in the coordinates λ, ϕ Λ , is the prolongation of λ i ∂/∂ϕ . For completeness, we note that the vector field written as ∂/∂λ in the coordinates λ, ϕ (1) , ϕ (2) , ϕ (3) , . . . becomes ∂∂λ + ϕ λ ∂∂ϕ + ϕ λλ ∂∂ϕ λ + · · · = ∂∂λ + X Λ ϕ λ Λ ∂∂ϕ Λ = D λ in the coordinates λ, ϕ Λ . Proposition 13.
In terms of the coordinates ϕ Λ and z Ξ , | Ξ | > a vertical infinitelyprolonged field in the covering ¯ τ λ can be written as S = X Λ ∂ Λ Φ ∂∂ϕ Λ + X | Ξ | > ˜ D Ξ Z ∂∂z Ξ , (40) where Z = ( z y − ϕ (1) ) f (1) + z x f (2) − f (3) , Φ = f − f (2) + f (1) ( ϕ − z x ) ϕ − z x ϕ − z y ,f = X i> f ( i ) λ i . (41) The field S is a symmetry if and only if ˜ ℓ E ( Z ) = 0, see (34), and ˜ D x Φ + ϕ ˜ D x Z + ˜ D y Z − (2 ϕ − z x )Φ( ϕ − z x ϕ − z y ) = 0 , ˜ D y Φ + z y ˜ D x Z + ( ϕ − z x ) ˜ D y Z − (( ϕ − z x ) + z y )Φ( ϕ − z x ϕ − z y ) = 0 . (42) N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 21
Proof.
Formula (38) can be rewritten as S = X i> Φ ( i ) ∂∂ϕ ( i ) + X | Ξ | > ˜ D Ξ Z ∂∂z Ξ = X Λ ∂ Λ (cid:16)X i> Φ ( i ) λ i (cid:17) ∂∂ϕ Λ + X | Ξ | > ˜ D Ξ Z ∂∂z Ξ , where Z = ( z y − ϕ (1) ) f (1) + z x f (2) − f (3) , and X i> Φ ( i ) λ i = X i> f ( i ) λ i + f (2) X i> ¯ X ( i ) λ i + f (1) X i> ¯ Y ( i ) λ i = f + f (2) X i> ϕ ( i ) x λ i + f (1) X i> ϕ ( i ) y λ i = f + f (2) ϕ x + f (1) ϕ y = f − f (2) + f (1) ( ϕ − z x ) ϕ − z x ϕ − z y , where f = P i> f ( i ) λ i . Hence formulas (41).Now, S is a symmetry if and only if˜ D x Φ + S (cid:18) ϕ − z x ϕ − z y (cid:19) = 0 , ˜ D y Φ + S (cid:18) ϕ − z x ϕ − z x ϕ − z y (cid:19) = 0 . These are formulas (42). (cid:3)
The last proposition suggests the following construction. Let f = X i> f ( i ) λ i , where the coefficients f ( i ) are independent of λ . Then we set S f = X i> f ( i ) ∂∂ϕ ( i ) . Transforming to the coordinates λ, ϕ, . . . , ϕ Λ , . . . , we obtain S f = X Λ ∂ Λ f ∂∂ϕ Λ which is the usual prolongation of a vertical generator f ∂/∂ϕ . Obviously,[ S f , S g ] = S { f,g } , { f, g } = S f g − S g f. (43)Using this notation, the symmetries we are looking for can be written as S = S Φ + X | Ξ | > ˜ D Ξ Z ∂∂z Ξ , ′ SHCHIK, AND M. MARVAN where Z = ( z y − ϕ (1) ) f (1) + z x f (2) − f (3) , Φ = f − f (2) + f (1) ( ϕ − z x ) ϕ − z x ϕ − z y ,f = X i> f ( i ) λ i , and should satisfy ˜ ℓ E ( Z ) = 0, see (34), as well as Equations (42).Now, using the formal series f = λ n ϕ λ = − λ n − + X i ≥ iϕ ( i ) λ n + i − (44)and the field S f we shall show that all the shadows described in Subsection 4.1 are liftedto a nonlocal symmetry in ¯ τ ∗ . The proof depends on the integer n . The case n ≥ . The series λ n ϕ λ is of the form required by the definition of S f . In allthese cases, conditions (34) and (42) are easily checked by straightforward computation,which is omitted.For n = 3, we have f = λ ϕ λ = − λ + P i ≥ iϕ ( i ) λ i , i.e., f (1) = − f (2) = 0, f (3) = ϕ (1) .In this case, Z = − z y = − Z ( − , i.e., we obtain the lift S λ ϕ λ = − ∂∂ϕ (1) + X i ≥ iϕ ( i ) ∂∂ϕ (2+ i ) of the y -translation.For n = 4, we have f = λ ϕ λ = − λ + P i ≥ iϕ ( i ) λ i , i.e., f (1) = f (3) = 0, f (2) = − Z = − z x = − Z ( − , i.e., we obtain the lift S λ ϕ λ = − ∂∂ϕ (2) + X i ≥ iϕ ( i ) ∂∂ϕ (3+ i ) of the x -translation.If n = 5, then f = λ ϕ λ = − λ + P i ≥ iϕ ( i ) λ i , i.e., f (1) = f (2) = 0, f (3) = − Z = 1 = Z ( − , i.e., we recover the first classical symmetry and obtained itslift S ( − = S λ ϕ λ = − ∂∂ϕ (3) + X i ≥ iϕ ( i ) ∂∂ϕ (4+ i ) . If n ≥
6, then the coefficients f (1) , f (2) , f (3) are zero. Obviously, Z = 0 and we obtainthe invisible symmetries S (1 − n ) = S λ n ϕ λ = − ∂∂ϕ ( n − + X i ≥ iϕ ( i ) ∂∂ϕ ( n + i − , n ≥ . (45) The case n < . In this case, the series (44) contains non-positive terms and so wecannot construct the corresponding field S f directly. To overcome this problem, we dothe following. N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 23
For any formal series T = P i c i t i we use the notation [ t n ] T = c n and define the operators P ϕ f = m X k =0 [ ϕ k ] f · ϕ k , P ϕ f = f − P ϕ f = f − m X k =0 [ ϕ k ] f · ϕ k . Then P ϕ f is a positive series in λ and a polynomial in ϕ .Consider now the family f n = P ϕ ( λ n ϕ λ ), n ≤
2. The first members are f = P ϕ ( λ ϕ λ ) = λ ϕ λ + 1 , f = P ϕ ( λϕ λ ) = λϕ λ + ϕ, f = P ϕ ( ϕ λ ) = ϕ λ + ϕ − ϕ (1) , etc. Proposition 14.
For any n ≤ all the vector fields S of the form (40) with f = f n = P ϕ ( ϕ λ λ n ) (46) are nonlocal symmetries in ¯ τ ∗ .Example. To illustrate how the construction works, let us discuss the case of n = 1 inmore detail. We have λϕ λ = − λ + ϕ (1) λ + 2 ϕ (2) λ + 3 ϕ (3) λ + . . . and f = P λϕ λ = λϕ λ + ϕ = 2 ϕ (1) λ + 3 ϕ (2) λ + 4 ϕ (3) λ + . . . Consequently, f ( i )1 = ( i + 1) ϕ ( i ) . In particular, f (1) = 2 ϕ (1) = − y , f (2) = 3 ϕ (2) = − x , f (3) = 4 ϕ (3) = − z − y . Substituting into formulas (41), we get Z = ( z y − ϕ (1) ) f (1) + z x f (2) − f (3) = − yz y − xz x + 4 z = − Z (0) , Φ = λϕ λ + ϕ + 3 x + 2 y ( ϕ − z x ) ϕ − z x ϕ − z y . Thus, we have the lift of the scaling symmetry.To prove Proposition 14, we introduce the generating function f = f ( λ, ξ ) = ∞ X n = − ξ n − f − n . (47)If we show that f satisfies equations (42), then, by linearity, (42) will be satisfied for all f n . To turn this observation into a proof, we need an analytic expression for f and alsofor the first three coefficients f ( i ) of the expansion f = P i f ( i ) ( ξ ) λ i . Lemma 1.
The generating function (47) admits the representation f = (cid:18) λξ (cid:19) ϕ λ ( λ ) λ − ξ + ϕ ξ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) . In addition , f (1) = − ϕ ξ ( ξ ) , f (2) = − ϕ ξ ( ξ ) ϕ ( ξ ) , f (3) = ϕ ξ ( ξ ) (cid:0) ϕ (1) − ϕ ( ξ ) (cid:1) . Proof.
We break the proof into several steps.
Step 1.
We show that (cid:2) ϕ ( ξ ) k (cid:3) ϕ ξ ( ξ ) ξ n − = − (cid:2) ξ n − (cid:3) ϕ ξ ( ξ ) ϕ ( ξ ) k +1 . ′ SHCHIK, AND M. MARVAN
Recall that the ‘formal residue’ of the Laurent series g ( ξ ) = P ∞ k = −∞ g k ξ k is defined byRes g = (cid:2) ξ − (cid:3) g ( ξ ) = g − . It is straightforward to check that it has the following properties:Res αg + βh = α Res g + β Res h, α, β = const , Res g ′ = 0 , [ ξ n ] g = Res gξ n +1 , Res ( g ( h )) h ′ = Res h ′ h · Res g, where the ‘prime’ denotes the ξ -derivative and g ( h ) is the composition of formal series.The last property is valid for all Laurent series g of the form g = g − n ξ − n + g − n ξ − n + · · · + g + g ξ + g ξ + · · · , i.e., whose principal part is finite, and for h of the form h = ξ m ( h + h ξ + h ξ + · · · ) or h = ξ − m ( h + h ξ − + h ξ − + · · · ), where m > g = Res ( g ◦ h ) h ′ Res ( h ′ /h ) , with g ( ω ) = ψ ( ω ) and h ( ξ ) = ϕ ( ξ ), where ψ is the compositional inverse of ϕ , i.e. ψ ( ϕ ( λ )) = ϕ ( ψ ( λ )) = λ . In other words ω = ϕ ( ξ ), ξ = ψ ( ω ). Since ϕ and ψ are composi-tionally mutually inverse, one has ϕ ′ ( ψ ( ω )) = 1 ψ ′ ( ω ) , ψ ′ ( ϕ ( ξ )) = 1 ϕ ′ ( ξ ) , and using the obvious identity Res ( ϕ ′ ( ξ ) /ϕ ( ξ )) = −
1, we obtain (cid:2) ϕ ( ξ ) k (cid:3) ϕ ′ ( ξ ) ξ n − = (cid:2) ω k (cid:3) ψ ′ ( ω ) ψ ( ω ) n − = Res 1 ψ ′ ( ω ) ψ ( ω ) n − ω k +1 = − Res ϕ ′ ( ξ ) ψ ′ ( ϕ ( ξ )) ξ n − ϕ ( ξ ) k +1 = − Res ϕ ′ ( ξ ) ξ n − ϕ ( ξ ) k +1 = − (cid:2) ξ n − (cid:3) ϕ ′ ( ξ ) ϕ ( ξ ) k +1 . Step 2.
Substituting this result into the definition of f n we obtain f − n = ϕ ′ ( λ ) λ n − + n +1 X k =0 ϕ ( λ ) k (cid:2) ξ n − (cid:3) ϕ ′ ( ξ ) ϕ ( ξ ) k +1 , (48)where, as a matter of fact, we can extend the upper summation bound to infinity since (cid:2) ξ n − (cid:3) ϕ ′ ( ξ ) ϕ ( ξ ) k +1 = 0 , k > n + 1 . Thus, we obtain f − n = ϕ ′ ( λ ) λ n − + (cid:2) ξ n − (cid:3) ∞ X k =0 ϕ ( λ ) k ϕ ′ ( ξ ) ϕ ( ξ ) k +1 = ϕ ′ ( λ ) λ n − + (cid:2) ξ n − (cid:3) ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) . (49) N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 25
Step 3.
To get the closed formula sought for the generating function f ( λ, ξ ) = ∞ X n = − ξ n − f − n , it remains to use the identities ∞ X n = − ξ n − ϕ ′ ( λ ) λ n − = (cid:18) λξ (cid:19) ϕ ′ ( λ ) λ − ξ , ∞ X n = − ξ n − (cid:2) ξ n − (cid:3) ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) = ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) . Step 4.
Computation of the first three coefficients f (1) = [ λ ] f ( λ, ξ ) , f (2) = (cid:2) λ (cid:3) f ( λ, ξ ) , f (3) = (cid:2) λ (cid:3) f ( λ, ξ )is straightforward. (cid:3) Proof of Proposition 14.
Let us verify equations (42) for the generating function f = f ( λ, ξ ).In this case, we have Z = − (cid:0) z y + z x ϕ ( ξ ) − ϕ ( ξ ) (cid:1) ϕ ξ ( ξ ) . This is a shadow of symmetries of the Gibbons–Tsarev equation as proved in Proposi-tion 10, so Equation (34) is satisfied. We haveΦ = (cid:18) λξ (cid:19) ϕ λ ( λ ) λ − ξ + ϕ ξ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) ϕ ( ξ ) − z x ϕ ( ξ ) − z y ϕ ( λ ) − z x ϕ ( λ ) − z y . These expressions can be put directly into equations (42). The proof that equations(42) indeed hold is a matter of direct computation with the help of the identities ∂ϕ ( t ) ∂x = − ϕ ( t ) − z x ϕ ( t ) − z y ,∂ϕ ( t ) ∂y = − ϕ ( t ) − z x ϕ ( t ) − z x ϕ ( t ) − z y ,∂ϕ t ( t ) ∂x = (2 ϕ ( t ) − z x ) ϕ t ( t )( ϕ ( t ) − z x ϕ ( t ) − z y ) ,∂ϕ t ( t ) ∂y = − (( ϕ ( t ) − z x ) + z y ) ϕ t ( t )( ϕ ( t ) − z x ϕ ( t ) − z y ) ,z yy = z y z xx − z x z xy − , where t is either λ or ξ . (cid:3) Collecting together the above facts, we obtain the main result of this section:
Theorem 1.
In the notation of Proposition 13 , the vector fields S with f = f n (46) aresymmetries for all n ∈ Z . Remarkably, we are able to obtain explicit formulas for the symmetries. Denote by A ( k,n ) r = [ λ n − r ] ϕ k ϕ ′ r , ′ SHCHIK, AND M. MARVAN the coefficient at λ n − r in the product ϕ k ϕ ′ r , where k, r are arbitrary integers, cf. (35). For r = 0 we have A ( k,n )0 = X ( • ) j + ··· + j k = n ϕ ( j ) ϕ ( j ) · · · ϕ ( j k ) , k > P ( • ) ... means summation where indices run through all integersfrom − Proposition 15.
Vector fields S ( n ) = S f − n admit the explicit formula S ( n ) = X m ≥ (cid:18) ( n + m ) ϕ ( n + m ) + n +1 X k =0 A ( k,m )0 A ( − k − ,n )2 (cid:19) ∂∂ϕ ( m ) . (50) Proof.
This is a direct consequence of the representation of f − n , see (48), and the defini-tion of S f . (cid:3) Remark . Alternatively, we can also write S ( n ) = − X m ≥ m X k =0 A ( − k − ,m )0 A ( k,n )2 ∂∂ϕ ( m ) . The Lie algebra structure.
The main result of this part is
Theorem 2.
The vector fields S ( n ) = S f − n satisfy [ S ( n ) , S ( m ) ] = ( m − n ) S ( n + m ) , i.e. , constitute a basis of the Witt algebra. For the proof we are going to make use of the following lemma:
Lemma 2.
Let g = g ( ϕ, ϕ ′ , λ ) , h = h ( ϕ, ϕ ′ , λ ) , be two formal series in λ of the lowest order . Then [ S g , S h ] = S { g,h } , (51) where { g, h } = h ϕ g − g ϕ h + h ϕ ′ D λ g − g ϕ ′ D λ h, where, as before, D λ = ∂/∂λ + P Λ ϕ λ Λ ∂/∂ Λ .Proof. Obviously from (27), one has ∂ϕ∂ϕ ( m ) = λ m , ∂ϕ ′ ∂ϕ ( m ) = mλ m − for all integers m >
0. Therefore, ∂ ϕ m g ( λ ) = g ϕ λ m + g ϕ ′ D λ λ m . Hence S h g ( λ ) = ∞ X m =1 [ ξ m ] h ( ξ ) ∂ ϕ m g ( λ ) = ∞ X m =1 [ ξ m ] h ( ξ ) ( g ϕ λ m + g ϕ ′ D λ λ m ) = g ϕ h ( λ ) + g ϕ ′ D λ h ( λ ) . N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 27
The last equality stems from the fact that h is of the lowest order one. Finally,[ S g , S h ] = ∞ X m =1 [ λ m ] ( S g h ( λ ) − S h g ( λ )) ∂ ϕ m . (cid:3) Proof of Theorem 2.
We have to prove the identity[ S f n , S f m ] = ( n − m ) S f n + m . (52)Recall that S g = ∞ X m =1 [ λ m ] g ( λ ) ∂ ϕ m , where g is a series in λ of the lowest order one.For n >
1, the quantity f n = λ n +1 ϕ ′ is of the lowest order one and depends on ϕ ′ only,hence is admissible for the commutation rule above. It is easy to show that for n, m > (cid:8) λ n +1 ϕ ′ , λ m +1 ϕ ′ (cid:9) = λ m +1 D λ λ n +1 ϕ ′ − λ n +1 D λ λ m +1 ϕ ′ = ( n − m ) λ m + n +1 ϕ ′ . In other words [ S f n , S f m ] = S ( n − m ) λ m + n +1 ϕ ′ = ( n − m ) S f m + n . For n ≤
1, dependence of f − n on ϕ ( i ) is different, thus another approach is needed. Weare going to use the representation (49) for f − n ( λ ), that is, f − n ( λ ) = ϕ ′ ( λ ) λ n − + [ ξ n − ] ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) . Note that f − n ( λ ) does not depend on ξ .Let us compute S f − n ( λ ) f − m ( λ ) = S f − n ( λ ) ϕ ′ ( λ ) λ m − + [ ξ m − ] S f − n ( λ ) ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ )= 1 λ m − f ′ − n ( λ ) + [ ξ m − ] f − n ( λ ) ϕ ′ ( ξ ) ( ϕ ( ξ ) − ϕ ( λ )) − [ ξ m − ] f − n ( ξ ) ϕ ′ ( ξ ) ( ϕ ( ξ ) − ϕ ( λ )) + [ ξ m − ] f ′ − n ( ξ ) 2 ϕ ′ ( ξ ) ϕ ( ξ ) − ϕ ( λ ) , where the ′ in f ′ − n denotes the total derivative. Collecting terms that contain ϕ ′ ( λ ), ϕ ′′ ( λ )and the rest that depends only on ϕ ( λ ), we can rewrite the last result as follows: S f − n ( λ ) f − m ( λ ) = − ( n − ϕ ′ ( λ ) λ m + n − + ϕ ′′ ( λ ) λ m + n − + [ ξ n − ] ϕ ′ ( ξ ) ( ϕ ( ξ ) − ϕ ( λ )) ϕ ′ ( λ ) λ m − + [ ξ m − ] ϕ ′ ( ξ ) ( ϕ ( ξ ) − ϕ ( λ )) ϕ ′ ( λ ) λ n − + ∞ X k =0 c k ϕ ( λ ) k , where c k are some coefficients depending on ϕ ( i ) . ′ SHCHIK, AND M. MARVAN
Notice that all terms except the first one and the last one are symmetric with respectto swapping n ↔ m . Hence, { f − n ( λ ) , f − m ( λ ) } = S f − n ( λ ) f − m ( λ ) − S f − m ( λ ) f − n ( λ )= ( m − n ) ϕ ′ ( λ ) λ m + n − + ∞ X k =0 c k ϕ ( λ ) k , where, again, c k are some coefficients depending on ϕ ( i ) , i ≥ g, h that are of lowest order 1, we have[ ϕ ( λ ) j ] { g ( λ ) , h ( λ ) } = 0 , j ≥ , since the operator S g does not act on λ and thus cannot produce non-positive powersof λ . In our case, both f − m ( λ ) and f − n ( λ ) are of lowest order 1 and thus0 = [ ϕ ( λ ) j ] { f − n ( λ ) , f − m ( λ ) } = [ ϕ ( λ ) j ] ( m − n ) ϕ ′ ( λ ) λ m + n − + ∞ X k =0 c k ϕ ( λ ) k ! = ( m − n )[ ϕ ( λ ) j ] ϕ ′ ( λ ) λ m + n − + c j . Hence, c j = − ( m − n )[ ϕ ( λ ) j ] ϕ ′ ( λ ) λ m + n − and { f − n ( λ ) , f − m ( λ ) } = ( m − n ) ϕ ′ ( λ ) λ m + n − − ( m − n ) ∞ X k =0 ϕ ( λ ) k (cid:2) ϕ ( ξ ) k (cid:3) ϕ ′ ( ξ ) ξ m + n − = ( m − n ) f − m − n ( λ ) . Finally, (cid:2) S f − n , S f − m (cid:3) = S { f − n ,f − m } = ( − n − ( − m )) S f − m − n , as claimed. (cid:3) Uniqueness of symmetries
We prove here some uniqueness results for the nonlocal symmetries of the Gibbons–Tsarev equation (3) in the covering τ ∗ defined in Section 3.2.1. For technical reasons, itis more convenient for us to deal with System (9), i.e., u y + vu x = 1 v − u , v y + uv x = 1 u − v . Due to (7), the relation between shadows of (3) and those of (9) is established by U = uD x ( Z ) + D y ( Z ) u − v , V = vD x ( Z ) + D y ( Z ) v − u , (53)where Z is a shadow for the Gibbons–Tsarev equation, while ( U, V ) is a shadow for thesystem in u and v . In particular, for the symmetries S ( − , . . . , S (0) with the generatingsections given by (5) one has Z ( − W ( − = ( u x , v x ) , N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 29 Z ( − W ( − = ( u y , v y ) ,Z ( − W ( − = (1 − yu x , − yv x ) ,Z (0) W (0) = (3 xu x + 3 yu y − u, xv x + 3 yv y − v ) , while the symmetry S ( − becomes invisible.In what follows, we, for convenience, use the notation u y = f ( u, v ) − vu x , v y = g ( u, v ) − uv x , (54)where f and g may be considered as functions in u and v such that the partial deriva-tives f u , f v , g u , g v do not vanish. We choose the functions x, t, u i = ∂ i u∂x i , v i = ∂ i v∂x i , i ≥ , for the internal coordinates on E , and then the total derivatives are D x = ∂∂x + X i ≥ (cid:18) u i +1 ∂∂u i + v i +1 ∂∂v i (cid:19) ,D y = ∂∂y + X i ≥ (cid:18) D ix ( f − vu ) ∂∂u i + D ix ( g − uv ) ∂∂v i (cid:19) . Then ℓ E = (cid:18) vD x + D y − f u u − f v v − g u uD x + D y − g v (cid:19) and the defining equations for symmetries of (54) are D y ( U ) = f u U − vD x ( U ) + ( f v − u ) V,D y ( V ) = g v V − uD x ( V ) + ( g u − v ) U. (55)5.1. Uniqueness of polynomial shadows.
Consider now the covering τ ∗ : E ∗ → E with the nonlocal variables ψ (3) , . . . , ψ ( k ) , . . . defined in Subsection 3.2.1. We say that afunction F on E ∗ is of order k if at least one of the partial derivatives F u k or F u k does notvanish, while F u i = F u i = 0 for all i > k .Let us estimate the higher order terms of τ ∗ -shadows. The defining equations for τ ∗ -shadows is obtained from (55) by changing the total derivatives D x and D y to D ( ∗ ) x = D x + X i ≥ X ( i ) ∂∂ψ ( i ) , D ( ∗ ) y = D y + X i ≥ Y ( i ) ∂∂ψ ( i ) , i.e., they are of the form D ( ∗ ) y ( U ) = f u U − vD ( ∗ ) x ( U ) + ( f v − u ) V, (56) D ( ∗ ) y ( V ) = g v V − uD ( ∗ ) x ( V ) + ( g u − v ) U. (57)Note that the coefficients X ( i ) and Y ( i ) are of order zero.We shall need the following ‘asymptotics’ below: (cid:0) D ( ∗ ) x (cid:1) p ( f − vu ) = − vu p +1 + ( f u − pv ) u p + ( f v − u ) v p + O ( p − , (cid:0) D ( ∗ ) y (cid:1) p ( g − uv ) = − uv p +1 + ( g v − pu ) v p + ( g u − v ) u p + O ( p −
1) (58)for an arbitrary p >
1. Here and in what follows O ( α ) denotes terms of order ≤ α . ′ SHCHIK, AND M. MARVAN
Proposition 16.
Equation (54) admits no τ ∗ -shadow of order > .Proof. Let us assume that the components U and V of the shadow under considerationare of order k and, using (58), differentiate Equation (56) with respect to v k +1 . The resultis − uU v k = − vU v k . In a similar way, applying ∂/∂u k +1 to (57), we get − vV u k = − uV u k .Consequently, U = U ( . . . , u k − , v k − , u k ) , V = V ( . . . , u k − , v k − , v k ) , (59)where ‘dots’ stand for the variables of order ≤ k − ∂/∂u k and ∂/∂v k to Equations (56) and (57): ∂ (56) ∂u k : ∂D ( ∗ ) y ∂u k ( U ) + D ( ∗ ) y ( U u k ) = f u U u k − v (cid:18) ∂D ( ∗ ) x ∂u k ( U ) + D ( ∗ ) x ( U u k ) (cid:19) + ( f v − u ) U u k ,∂ (56) ∂v k : ∂D ( ∗ ) y ∂v k ( U ) + D ( ∗ ) y ( U v k ) = f u U v k − v (cid:18) ∂D ( ∗ ) x ∂v k ( U ) + D ( ∗ ) x ( U v k ) (cid:19) + ( f v − u ) U v k ,∂ (57) ∂u k : ∂D ( ∗ ) y ∂u k ( V ) + D ( ∗ ) y ( V u k ) = g v V u k − u (cid:18) ∂D ( ∗ ) x ∂u k ( V ) + D ( ∗ ) x ( V u k ) (cid:19) + ( g u − v ) V u k ,∂ (57) ∂v k : ∂D ( ∗ ) y ∂v k ( V ) + D ( ∗ ) y ( V v k ) = g v V v k − u (cid:18) ∂D ( ∗ ) x ∂v k ( V ) + D ( ∗ ) x ( V v k ) (cid:19) + ( g u − v ) V v k (the partial derivatives above are applied to the coefficients of the corresponding opera-tors). Using now (59), we see that the above equalities amount to ∂D ( ∗ ) y ∂u k ( U ) + D ( ∗ ) y ( U u k ) = f u U u k − v (cid:18) ∂D ( ∗ ) x ∂u k ( U ) + D ( ∗ ) x ( U u k ) (cid:19) ,∂D ( ∗ ) y ∂v k ( V ) + D ( ∗ ) y ( V v k ) = g v V v k − u (cid:18) ∂D ( ∗ ) x ∂v k ( V ) + D ( ∗ ) x ( V v k ) (cid:19) and ∂D ( ∗ ) y ∂v k ( U ) = − v ∂D ( ∗ ) x ∂v k ( U ) + ( f v − u ) V v k ,∂D ( ∗ ) y ∂u k ( V ) = − u ∂D ( ∗ ) x ∂u k ( V ) + ( g u − v ) U u k . Now, by (58), we have ∂D ( ∗ ) x ∂u k = ∂∂u k − , ∂D ( ∗ ) y ∂u k = ( f u − kv ) ∂∂u k + ( g u − v ) ∂∂v k − v ∂∂u k − ,∂D ( ∗ ) x ∂v k = ∂∂v k − , ∂D ( ∗ ) y ∂v k = ( f u − u ) ∂∂u k + ( g v − ku ) ∂∂v k − u ∂∂v k − . and, using (59) again, we arrive to − kv U u k + D ( ∗ ) y ( U u k ) = − vD ( ∗ ) x ( U u k ) , − ku V v k + D ( ∗ ) y ( V v k ) = − uD ( ∗ ) x ( V v k ) N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 31 and ( f v − u ) U u k + ( v − u ) U v k − = ( f v − u ) V v k , (60)( g u − v ) V v k + ( u − v ) V u k − = ( g u − v ) U u k . (61)Then, differentiating Equation (60) with respect to v k and Equation (61) with respectto u k , we obtain ( f v − u ) V v k v k = ( g u − v ) U u k u k = 0 , i.e., U = au k + b, V = bv k + d, (62)where the functions a , b , c , and d are of order k −
1. Let us substitute the obtainedexpressions (62) to the defining system (56)–(57): D ( ∗ ) y ( a ) u k + aD ( ∗ ) y ( u k ) + D ( ∗ ) y ( b ) = f u ( au k + b ) − v (cid:0) D ( ∗ ) x ( a ) u k + au k +1 + D ( ∗ ) ( b ) (cid:1) (63)+ ( f v − u )( cv k + d ) D ( ∗ ) y ( c ) v k + cD ( ∗ ) y ( v k ) + D ( ∗ ) y ( d ) = g v ( cv k + d ) − u (cid:0) D ( ∗ ) x ( c ) v k + cv k +1 + D ( ∗ ) ( d ) (cid:1) (64)+ ( g u − u )( au k + b ) . Using the estimates (58) and comparing the terms containing u k +1 , v k +1 and u k , v k , wesee that the terms with u k +1 , v k +1 and u k , v k are cancelling, whilein Eq. (63) at u k v k : − ua v k − = − va v k − , in Eq. (64) at u k v k : − vc u k − = − uc u k − , in Eq. (63) at u k : − kv a = 0 , (65)in Eq. (64) at v k : − ku c = 0 , (66)in Eq. (63) at v k : ( f v − u )( a − c ) = ( u − v ) b v k − , in Eq. (64) at u k : ( g u − v )( c − a ) = ( v − u ) d u k − . In particular, from Equations (65) and (66) we see that the coefficients a and c vanishand thus, by virtue of (62), the functions U and V are of order k −
1. We repeat theprocedure until the order of the shadows at hand becomes equal to 1. (cid:3)
Using Proposition 16, we shall now prove that the symmetries S ( i ) = E Z ( i ) , i = − , − , . . . , exhaust all the polynomial symmetries in the covering τ ∗ . Theorem 3.
Any τ ∗ -nonlocal symmetry of the Gibbons–Tsarev equation of weight k , polynomial in all variables , coincides with S ( k ) up to a constant factor , k ≥ − .Proof. Using Equations (53), we pass from symmetries of the Gibbons–Tsarev equation (3)to those of System (9). The proof is accomplished by induction on the weight.For small weights ( | S | = − , . . . ,
0) this fact can be checked by direct computationsdue to Proposition 16.Let us fix a k > k the statement is true. Toproceed with the proof, we need a number of auxiliary facts. The first two of them canbe observed from the results of Section 4. ′ SHCHIK, AND M. MARVAN
Fact . For a symmetry S ( i ) = E Z ( i ) , one has the following ‘asymptotics’ in ψ s: Z (1) = − ψ (4) + local termsand, for i > Z ( i ) = − ( i + 2) ψ ( i +3) + 2 i + 32 z x ψ ( i +2) + Υ( i + 1) , i ≥ , where Υ( α ) denotes the terms independent of ψ ( β ) for β > α . This means, by (53), thatthe corresponding generating section for the system is of the form W (1) = 52 ψ (3) W ( − + local termsand, for i > U ( i ) = 2 i + 32 ψ ( i +2) U ( − + ( i + 1) ψ ( i +1) U ( − + Υ( i ) ,V ( i ) = 2 i + 32 ψ ( i +2) V ( − + ( i + 1) ψ ( i +1) V ( − + Υ( i ) , or W ( i ) = 2 i + 32 ψ ( i +2) W ( − + ( i + 1) ψ ( i +1) W ( − + Υ( i ) , (67)where W ( − = ( u , v ) , W ( − = (cid:18) v − u − vu , u − v − uv (cid:19) , are the generating sections of the infinitesimal x - and y -translations, respectively. Fact . The shadow W ( − = (1 − yu , − yv ) of the generalised Galilean boost extendsto E ∗ as follows S ( − = X l ≥ (cid:18) D lx (1 − yu ) ∂∂u l + D lx (1 − yv ) ∂∂v l (cid:19) + (cid:0) x − yX (3) (cid:1) ∂∂ψ (3) + X j> (cid:0) ( j − ψ ( j − − yX ( j ) (cid:1) ∂∂ψ ( j ) . Since the last expression can be rewritten in the form S ( − = ∂∂u + ∂∂v + y ∂∂x − yD ( ∗ ) x + 2 x ∂∂ψ (3) + X j> ( j − ψ ( j − ∂∂ψ ( j ) , while ℓ W ( − = − yD ( ∗ ) x − yD ( ∗ ) x ! , we obtain { W ( − , W } = ∂U∂u + ∂U∂v + y ∂U∂x + 2 x ∂U∂ψ (3) + X j> ( j − ψ ( j − ∂U∂ψ ( j ) ∂V∂u + ∂V∂v + y ∂V∂x + 2 x ∂V∂ψ (3) + X j> ( j − ψ ( j − ∂V∂ψ ( j ) (68) N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 33 for any W = ( U, V ), and, in particular, { W ( − , W ( i ) } = ( i + 1) W ( i − + Υ( i − , i ≥ . Fact . A straightforward, but important consequence of (68) is that the adjoint ac-tion W
7→ { W ( − , W } is a derivation, i.e., { W ( − , hW } = h { W ( − , W } + X W ( − ( h ) W, h ∈ F ( E ∗ ) , where X W ( − = ∂∂u + ∂∂v + y ∂∂x + 2 x ∂∂ψ (3) + X j> ( j − ψ ( j − ∂∂ψ ( j ) is a vector field on E . Fact . Let W = ( U, V ) be a solution of System (56)-(57) of weight k . Let also l bethe minimal integer such that ∂U/∂ψ ( j ) and ∂V /∂ψ ( j ) vanish for all j > l . Recall (seeSubsection 3.2.1) that the ‘nonlocal tails’ of the total derivatives D ( ∗ ) x and D ( ∗ ) y on E ∗ areof the form X = X j ≥ X ( j ) ∂∂ψ ( j ) , Y = X j ≥ Y ( j ) ∂∂ψ ( j ) , where X ( j ) = − ( j − ψ ( j − + Υ( j − , Y ( j ) = − ( j − ψ ( j − + Υ( j − . This implies that if W = ( U, V ) and l is chosen as above, then ∂W∂ψ ( l ) = (cid:18) ∂U∂ψ ( l ) , ∂V∂ψ ( l ) (cid:19) , ∂W∂ψ ( l − = (cid:18) ∂U∂ψ ( l − , ∂V∂ψ ( l − (cid:19) are shadows as well (of weights k − l − k − l , respectively).Let us now return to the main course of the proof. Since k − l < k , then due to theinduction hypothesis we have ∂W∂ψ ( l ) = αW ( k − l − = α (cid:18) k − l + 12 ψ ( k − l +1) W ( − + ( k − l ) ψ ( k − l ) W ( − (cid:19) + Υ( k + l − , (69)where α ∈ R is a nonvanishing constant. Note also that due to the definition of l onehas l ≥ k − l + 1, or 2 l ≥ k + 1 . (70)We now consider two cases: Inequality (70) is either strict or an equality. The case l > k + 1 . In this case, Equation (69) implies W = αW ( k − l − ψ ( l ) + Υ( l − . (71)Let us apply the operator { W ( − , ·} to both sides of (71): { W ( − , W } = α { W ( − , W ( k − l − } ψ ( l ) + αW ( k − l − X W ( − ( ψ ( l ) ) + Υ( l − α { W ( − , W ( k − l − } ψ ( l ) + α ( l − W ( k − l − ψ ( l − + Υ( l − . ′ SHCHIK, AND M. MARVAN
But (cid:12)(cid:12) { W ( − , W } (cid:12)(cid:12) = k − { W ( − , W } = βW ( k − + Υ( k − β (cid:18) k + 12 W ( − ψ ( k +1) + Υ( k ) (cid:19) = α ( k − l ) W ( k − l − ψ ( l ) + α ( l − W ( k − l − ψ ( l − + Υ( l − . The last equality can hold only when l = k + 1 , l = − k + 12 β. Consider the shadow ˜ W = W − βW ( k ) . There are two possibilities: (a) ˜ W = 0 and thenthe proof is finished; (b) ˜ W = 0 and then there should exist the minimal integer ˜ l < l such that ∂ ˜ W /∂ψ (˜ l ) = 0. The only possibility is ˜ l = ( k + 1) / The case l = k + 1 . Now Equation (69) reads ∂W∂ψ ( l ) = α l − ψ ( l ) W ( − + Υ( l − , or W = α l − (cid:0) ψ ( l ) (cid:1) W ( − + terms linear in ψ ( l ) + Υ( l − . Let us apply the operator { W ( − , ·} to the last equation. Then in the left-hand sidewe obtain a shadow of weight k − l − ψ (2 l ) W ( − + Υ(2 l − (cid:3) Uniqueness of invisible symmetries.
Consider a symmetry S = X i ≥ (cid:18)(cid:0) D ( ∗ ) x (cid:1) i ( U ) ∂∂u i + (cid:0) D ( ∗ ) x (cid:1) i ( V ) ∂∂v i (cid:19) + X α ≥ Ψ ( α ) ∂∂ψ ( α ) of E ∗ . Let us say that S is invisible of depth k if U = V = Ψ (3) = · · · = Ψ ( k − = 0, i.e., S = X α ≥ k Ψ ( α ) ∂∂ψ ( α ) . The defining equations for such symmetries are [ S , D ( ∗ ) x ] = [ S , D ( ∗ ) y ] = 0, or D ( ∗ ) x (Ψ ( α ) ) = α − X β = k Ψ ( β ) ∂X ( α ) ∂ψ ( β ) , D ( ∗ ) y (Ψ ( α ) ) = α − X β = k Ψ ( β ) ∂Y ( α ) ∂ψ ( β ) (72)for all α ≥ k , where, as before, D ( ∗ ) x = D x + X α ≥ X ( α ) ∂∂ψ ( α ) , D ( ∗ ) y = D y + X α ≥ Y ( α ) ∂∂ψ ( α ) and X ( α ) , Y ( α ) are the right-hand sides of (16) and (17), respectively. N SYMMETRIES OF THE GIBBONS–TSAREV EQUATION 35
Theorem 4.
Any nontrivial invisible symmetry of depth k is of the form ∂∂ψ ( k ) + γ ∂∂ψ ( k +1) + X α ≥ k +2 Ψ ( α ) ∂∂ψ ( α ) , where γ = const .Proof. Indeed, the right-hand sides of Equations (72) vanish for α = k and α = k + 1, i.e., D ( ∗ ) x (Ψ ( k ) ) = D ( ∗ ) x (Ψ ( k +1) ) = 0 , D ( ∗ ) y (Ψ ( k ) ) = D ( ∗ ) y (Ψ ( k +1) ) = 0 . But, by Proposition 3, the equation E ∗ is differentially connected and thus Ψ ( k ) and Ψ ( k +1) are constants. (cid:3) Remark . Actually, one can say more about the structure of the coefficients Ψ ( α ) (seeEquation (45)), but for our cause the above said is sufficient. Acknowledgments
This research was undertaken within the framework of the OPVK programme, projectCZ.1.07/2.300/20.0002. The work of I.S. Krasil ′ shchik was partially supported by theRFBR Grant 18-29-10013 and the Simons-IUM fellowship. The work of P. Blaschke andM. Marvan was partially supported by GA ˇCR under project P201/12/G028.The symbolic computations were performed with the aid of the Jets software [2]. Theauthors are grateful to Jenya Ferapontov who draw our attention to the problem. We alsowant to express our gratitude to Maxim Pavlov, Vladimir Sokolov, and Sergey Tsarev forfruitful discussions.
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InverseProblems (2000) 605–618. Mathematical Institute in Opava, Silesian University in Opava, Na Rybn´ıˇcku 626/1,746 01 Opava, Czech Republic
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Mathematical Institute in Opava, Silesian University in Opava, Na Rybn´ıˇcku 626/1,746 01 Opava, Czech Republic
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V.A. Trapeznikov Institute of Control Sciences RAS, Profsoyuznaya 65, 117342 Moscow,Russia & Independent University of Moscow, 119002, Bolshoy Vlasyevskiy Pereulok 11,Moscow
E-mail address : [email protected] Mathematical Institute in Opava, Silesian University in Opava, Na Rybn´ıˇcku 626/1,746 01 Opava, Czech Republic
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