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Recursion Operators in the Cotangent Covering of the Rddym Equation

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Recursion Operators in the Cotangent Covering of the Rddym Equation RECURSION OPERATORS IN THE COTANGENT COVERING OF THE RDDYM EQUATION I.S. KRASIL′SHCHIK AND A.M. VERBOVETSKY Abstract. We describe a general method of constructing nonlocal recursion operators for symmetries of PDEs. As an example, the cotangent equation to the 3D rdDym equation uyt = uxuxy − uyuxx for which two mutually inverse operators are found. The exposition includes a rigorous criterion to check the hereditary property. Contents Introduction 1 1. Preliminaries and notation 3 Coordinates 4 2. The main result 5 Symmetries 6 Tangent equation 7 Conservation laws 7 Shadows 8 Recursion operators 8 Actions 9 Lifts of shadows 9 3. Conclusions 10 Acknowledgments 10 References 10 Introduction The rdDym equation (rth dispersionless Dym equation) uyt = uxuxy − uyuxx (1) arXiv:2106.06763v1 [nlin.SI] 12 Jun 2021 belongs to the class of linearly degenerate equations [8] and arose in the Mart´ınez Alonso-Shabat hierarchy [16]. It is Lax integrable and its symmetry properties were studied in [1, 3]. Recursion operators for symmetries of Equation (1) was found by O. Morozov in [19]. In [22], V. Ovsienko considered (in slightly different notation) the system vyt = 2(vyuxx − vxuxy)+ uxvxy − uyvxx − 2(uyuxx +2uxuxy), (2) uyt = uxuxy − uyuxx, 2010 Mathematics Subject Classification. 35B06. Key words and phrases. Partial differential equations, integrable linearly degenerate equations, nonlocal symmetries, recursion operators. The work ISK was partially supported by the RFBR Grant 18-29-10013 and RSF Grant 21-71-20034. 1 ′ 2 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY which is a two-component extension of the rdDym equation (another two-component generalization of (1) was studied in [20]). Treating (2) as the Euler equation on the Virasoro algebra, its bi-Hamiltonian structure was established. System (2) may be also considered in the following way. Recall (see [12, 13]) that to any equation E = {F [u]=0} in unknowns u its cotangent equation T∗E ∗ F [u]=0, ℓF [u,p]=0 ∗ is naturally associated, ℓF is the operator adjoint to the linearisation of F (here and below [u], [u,p], etc. denotes the variables together with their derivatives up to certain order). Then (2) may be understood as the cotangent equation of (1), with v instead of p, ‘deformed’ by the Lagrangian term δL with L = u u2 . δu y x System (2) is an Euler-Lagrange equations. Lagrangian deformations of some linearly degenerate equations were studied in [2]. In particular, the system vyt = 2(vyuxx − vxuxy)+ uxvxy − uyvxx − 2(2kuxyux + (kuy + l)uxx), (3) uyt = uxuxy − uyuxx, which is considered below, is the deformation of the cotangent to rdDym equation 2 R with the deforming Lagrangian L = (kuy + l)ux, k, l ∈ . System (2) is obtained from (3) when k = 1, l = 0. We find below recursion operators for symmetries of (3). Remark 1 (courtesy O.I. Morozov). Actually, both parameters k and l are fake in a sense: the change of variables v 7→ v − 2kxux − ku + ly kills them and transforms System (3) to the pure cotangent equation of (1). In particular, the Ovsienko system (2) also reduces to the cotangent equation. Though the computations in the general case are not much harder that in a particular one, the result look more complicated. For that reason we set k = l = 0 everywhere below, i.e., the equation we study below will be of the form vyt = 2(vyuxx − vxuxy)+ uxvxy − uyvxx, (4) uyt = uxuxy − uyuxx. There exist several approaches to construct recursion operators (see, for exam- ple, [21, 25]). Our method is based on the technology described in [13] and orig- inating from [11]. It is based on geometrical interpretation of recursion operators for symmetries as B¨acklund auto-transformations of the equation tangent to the given one (cf. [17]) and essentially uses the theory of differential coverings [14]. Remark 2. Recursion operators as B¨acklund auto-transformations of the tangent equations initially aroused in M. Marvan’s paper [17], though implicitly the tangent equation as an instrument for computation of recursion operators appears in [9]. Informally, tangent and cotangent bundles to PDEs are discussed in the paper [15] by B. Kupershmidt. A detailed discussion of the tangent covering and its role in the theory of Hamiltonian (Poisson) structures can be found in [12] and especially in [10]. The necessary theoretical matters needed for computations are shortly described in Section 1, while Section 2 contains the main results. RO FOR THE 2-COMPONENT RDDYM 3 1. Preliminaries and notation The exposition here is based on [5, 13, 14]. Let π : E → M be a vector bundle over a smooth manifold M, dim M = n, ∞ rank π = m, and π∞ : J (π) → M be the bundle of its infinite jets. Consider an infinitely prolonged differential equation E ∈ J ∞(π) and assume that E = {F =0}, ∗ E where F ∈ Γ(π∞(ξ)), where ξ : N → M is another vector bundle. Any is naturally endowed with an n-dimensional Frobenius integrable distribution C (the Cartan distribution) that defines its geometry. The Cartan distribution is π∞-horizontal and, consequently, defines a flat connection in π∞ : E → M. ˜ ˜ A morphism of equations f : E → E is a smooth map, such that f∗(Cθ) ⊂ Cf(θ) for any θ ∈ E˜. A morphism is called a (differential) covering if it is a submersion and the restriction f | of its differential to any Cartan plane is an isomorphism. Two ∗ Cθ coverings fi : Ei → E, i = 1, 2, are called equivalent if there exists a diffeomorphism g : E1 → E2 that preserves the Cartan distributions and such that f1 = f2 ◦ g. Denote by F = F(E) the algebra of smooth functions on E. A (higher infinites- imal) symmetry of E is a π∞-vertical derivation S : F → F (a vector field on E) that preserves the Cartan distribution. There exists a one-to-one correspondence between symmetries and solutions of the equation ℓE(ϕ)=0, where ℓE is the restriction of the linearisation ℓF : κ → P to E and κ denotes the ∗ E module of sections Γ(π∞(π)). A conservation law of is a π∞-horizontal form n−1 E ω ∈ Λh ( ) closed with respect to the horizontal de Rham differential dh. Trivial n−2 conservation laws are those of the form ω = dhρ, ρ ∈ Λ (E). Given a covering τ : E˜ → E, symmetries and conservation laws of E˜ are called nonlocal for E. Let S : F(E) → F(E˜) be a π∞-vertical derivation. It is called a τ-shadow if C˜X ◦ S = S ◦ CX for any vector field X on M. We say that a symmetry S˜ is a lift of the shadow S if S˜ = S. E Let an equation E be given by {F =0}. Then the system TE F [u]=0, ℓF [u, q]=0 is called the tangent equation of E, while the projection t: [u, q] 7→ [u] is the tangent covering over E. It must be stressed that q is considered as an odd variable of degree 1. Sections of t that preserve the Cartan distributions are identified with symmetries of E and consequently B¨acklund auto-transformations Rτ,τ ′ of the form W ❇ (5) ⑤⑤ ❇❇ ′ τ ⑤⑤ ❇❇τ ⑤⑤ ❇❇ ⑤~ ⑤ ❇ TE TE, where τ, τ ′ are coverings, are naturally interpreted as recursion operators for sym- metries of E. To construct these operators, we use a scheme consisting of two steps. In the first step, we are looking for two-component conservation laws ω of TE linear on the fibers of the projection t. Provided such a law was found, we consider the covering τ = τω associated with it and, as the second step, try to find τω-shadows (also linear on fibers both of t and τω). If such a shadow S exists, it delivers the ′ R covering τ = τS and the desired recursion operator τω ,τS . ′ 4 I.S.KRASIL SHCHIK AND A.M. VERBOVETSKY Finally, few words about the type of equations that are the subject of our study. E ∗ ˆ κ Assume = {F = 0}, F ∈ P , and consider the operator ℓF : P → ˆ formally n T∗E adjoint to ℓF , where •ˆ = hom(•, Λh). Then the system ∗ ℓF [u,p]=0, F [u] = 0 (6) is said to be the cotangent equation of E, while the projection t∗ : T∗E → E, (u,p) 7→ (u), is called the cotangent covering. Similar to the case of the tangent equation, the variable p is odd. System (6) is always an Euler-Lagrange equation with the Lagrangian density L = pj F j and we are interested in equations ∗ ℓFP[u,p]+ G[u,p]=0, F [u]=0, (‘Lagrangian deformations’ of (6)), such that (a) they are Lagrangian as well and (b) they possess a nontrivial recursion operator. Equation (2) is exactly of this TE T∗E ∗ type. Note that for Euler-Lagrange equations and coincide, since ℓF = ℓF in this case. Coordinates. Let U ⊂ M be a coordinate neighborhood with local coordinates x = (x1,...,xn) and E ⊃ π−1(U) ≃ U × Rm be a trivialization of π with fiber- 1 m j ∞ wise coordinates u = (u ,...,u ). Then adapted coordinates uσ arise on J (π), j |σ| j σ where uσ correspond to the partial derivative ∂ u /∂x . The Cartan distribution is spanned by the total derivatives ∂ j ∂ Di = i + uσi j , ∂x ∂uσ i X while the Cartan connection takes ∂/∂x to Di. Equation E is given by the system l j E of relations F (x,...,uσ,... ) = 0, l =1,...,r. To restrict necessary objects to , one needs to choose internal coordinates on E (cf. [18]) and express these objects in terms of internal coordinates. In particular, the Cartan distribution on E is got by rewriting the operators Di in internal coordinates.
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