On the algebra of nonlocal symmetries for the 4D Mart\'ınez Alonso-Shabat equation
aa r X i v : . [ n li n . S I] A ug ON THE ALGEBRA OF NONLOCAL SYMMETRIES FOR THE 4DMART´INEZ ALONSO-SHABAT EQUATION
I.S. KRASIL ′ SHCHIK AND P. VOJ ˇC ´AK
Abstract.
We consider the 4D Mart´ınez Alonso-Shabat equation E u ty = u z u xy − u y u xz (also referred to as the universal hierarchy equation) and using its known Lax pair constructtwo infinite-dimensional differential coverings over E . In these coverings, we give a completedescription of the Lie algebras of nonlocal symmetries. In particular, our results generalizethe ones obtained in [12] and contain the constructed there infinite hierarchy of commutingsymmetries as a subalgebra in a much bigger Lie algebra. Contents
Introduction 11. Preliminaries 22. The equation and its coverings 33. Algebras of nonlocal symmetries 7Acknowledgments 13References 13
Introduction
To the best of our knowledge, the equation u ty = u z u xy − u y u xz (1)was introduced in the work [9] by L. Mart´ınez Alonso, A.B. Shabat, where the authors studiedmulti-dimensional systems whose reductions lead to the known (1 + 1)-integrable equations (seealso [8] for additional motivations). By this reason we call Equation (1) the Mart´ınez Alonso-Shabat equation, or shortly the 4D
MASh equation . The equation arises also in classification ofintegrable 4D systems, see [4].A differential covering (Lax pair) with a non-removable parameter was constructed in [11], aswell as a recursion operator for symmetries of the 4D MASh equation. Using this covering, theauthors of [12] found a hierarchy of nonlocal symmetries and proved its commutativity.We study Equation (1) using the approach successfully applied to integrable linearly degen-erate 3D systems in [1] and [5]. Expanding defining equations of the Lax pair in formal seriesof the spectral parameter, we construct two differential coverings (which we call the negativeand positive ones) and describe the algebras of nonlocal symmetries in these coverings. As thereader will see, the structure of these algebras is quite complicated. The commutative hierarchyfound in [12] appears as a subalgebra in one of them. We also analyze the action of the recursionoperator from [11] on our symmetries.The structure of the paper is as follows: in Section 1 we present very briefly necessary factsfrom the geometrical theory of PDEs [3] and differential coverings [7]. Section 2 contains theconstruction of the positive and negative coverings and defining equtions for symmetries in them.In Section 3, the symmetry algebras are described.
Mathematics Subject Classification.
Key words and phrases.
4D Mart´ınez Alonso-Shabat equation, universal hierarchy equation, Lax pairs, differ-ential coverings, nonlocal symmetries. ′ SHCHIK AND P. VOJˇC ´AK Preliminaries
Let us very shortly recall the necessary theoretical background. All the details may be found,e.g., in [3] and [7]. A particular implementation of all the general constructions will be presentedin Section 2.
Equations.
From the geometrical viewpoint, a differential equation is a submanifold ina jet space. More precisely, this means the following. Let π : E → M be a locally trivialvector bundle over a smooth manifold, dim M = n , rank π = m , and π ∞ : J ∞ ( π ) → M bethe corresponding bundle of infinite jets. For us, a differential equation (imposed on sectionsof π ) is a submanifold E ⊂ J ∞ ( π ) obtained by the prolongation procedure from a submanifoldin the space of finite jets. We use the same notation π ∞ for the restriction π ∞ | E : E → M .The structure of equation on E is defined by the Cartan connection C , which takes vectorfields X ∈ D ( M ) to vector fields C X ∈ D ( E ) on E . The connection is flat, i.e., C [ X,Y ] = [ C X , C Y ].The corresponding integrable π ∞ -horizontal distribution is called the Cartan distribution on E and its maximal ( n -dimensional) integral manifolds are identified with solutions of E . Local symmetries.
An (infinitesimal higher local) symmetry of E is a π ∞ -vertical vectorfield S ∈ D ( E ) on E such that the commutator [ S, C X ] lies in the Cartan distribution forany X ∈ D ( M ). Symmetries form a Lie R -algebra denoted by sym E .To describe sym E , consider another vector bundle ξ : G → M , rank ξ = r , and assume that E = { F = 0 } is given as the set of zeros of some section F ∈ P = Γ( π ∗∞ ( ξ )), where Γ( · ) denotesthe C ∞ ( M )-module of sections. Consider also the module κ = Γ( π ∗∞ ( π )) and the linearizationoperator ℓ E = ℓ F | E : κ → P. Then one has sym E = ker ℓ E . Thus, to any symmetry S ∈ sym E there corresponds a section ϕ ∈ κ , its generating section, orcharacteristic, and we use the notation S = E ϕ in this case. The commutator of symmetriesgenerates a bracket in the R -space of generating sections defined by [ E ϕ , E ψ ] = E { ϕ,ψ } . Thebracket {· , ·} is called the (higher) Jacobi bracket. Differential coverings.
Let τ : ˜ E → E be a locally trivial bundle. It is called a (differential)covering over E if there exists a flat connection ˜ C in the bundle ˜ π ∞ = π ∞ ◦ τ : ˜ E → M such that τ ∗ (˜ C X ) = C X for any field X ∈ D ( M ). The manifold ˜ E , locally at least, is always an equation insome bundle over M and is called the covering equation. The number s = rank τ is the coveringdimension and it may be infinite. Coordinates in fibers of τ are called nonlocal variables.Let τ i : E i → E , i = 1, 2, be two coverings. Then their Whitney product τ ⊕ τ carries anatural structure of a covering called the Whitney product of τ and τ and all the arrows inthe diagram E × E E τ ∗ ( τ ) / / τ ∗ ( τ ) (cid:15) (cid:15) τ ⊕ τ & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E τ (cid:15) (cid:15) E τ / / E are coverings.A B¨acklund transformation between equations E and E is a diagram of the form˜ E τ (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) τ (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ E E , D MASH EQUATION 3 where τ and τ are coverings. When E = E then we speak about B¨acklund auto-transformation.If the equations E i , i = 1, 2, are given by the systems { F i ( u i ) = 0 } then the system { ˜ F ( u , u ) =0 } that corresponds to ˜ E possesses the following property: if u is a solution of E and ( u , u )is a solution of ˜ E then u solves E and vice versa. Nonlocal symmetries and shadows.
A nonlocal symmetry of E in the covering τ is asymmetry of ˜ E . These symmetries form the algebra sym τ E = sym ˜ E . Thus, to find nonlocalsymmetries, we need to solve the equation ℓ ˜ E ( ˜ ϕ ) = 0.Denote by F and by ˜ F the algebras of smooth functions on E and ˜ E , respectively. Theprojection τ leads to the embedding τ ∗ : F → ˜ F . We say that an R -linear derivation Y : F → ˜ F is a shadow in τ if the diagram F C X / / Y (cid:15) (cid:15) F Y (cid:15) (cid:15) ˜ F ˜ C X / / ˜ F is commutative for any X ∈ D ( M ). In particular, for any nonlocal symmetry ˜ S : ˜ F → ˜ F therestriction ˜ S (cid:12)(cid:12)(cid:12) F : F → ˜ F is a shadow. We say that ˜ S is invisible if its shadow vanishes. Notethat any local symmetry S may be regarded as a τ -shadow if one takes the composition τ ∗ ◦ S .A nonlocal symmetry ˜ S is a lift of a shadow Y if ˜ S (cid:12)(cid:12)(cid:12) F = Y . A lift, if it exists, is defined up toinvisible symmetries. The defining equation for shadows is˜ ℓ E ( ˜ ϕ ) = 0 , where ˜ ℓ E is the natural extension of the linearization operator from E to ˜ E . Recursion operators (see [6, 10] ). Let an equation E be given by { F ( u ) = 0 } . Then itstangent equation is TE : F ( u ) = 0 ℓ F ( p ) = 0 , where p = ( p , . . . , p m ) is a new unknown of the same dimension as u . The projection t : TE → E , ( u, p ) u , is called the tangent covering of E . Properties of TE are closely related withsymmetries of E : sections of t which take the Cartan distribution on E to that on TE are inone-to-one correspondence with symmetries.Let R be a B¨acklund auto-transformation of TE . Then it relates shadows of symmetries of E with each other, i.e., may be understood as recursion operator.2. The equation and its coverings
Here we present the necessary formulas for the computations to be done in Section 3.
Internal coordinates and the total derivatives.
The manifold E corresponding to Equa-tion (1) lies in J ∞ ( π ), where π : R × R → R is the trivial bundle. We denote the coordinatesin the base by x , y , z , t , while u denotes a coordinate in the fiber. Then internal coordinates u x i z j , u x i z j y k , u x i z j t l , i, j ≥ , k, l > , on E arise. Then the Cartan connection is completely determined by its values on the basisvector fields ∂/∂x , ∂/∂y , ∂/∂z , ∂/∂t . The result is the corresponding total derivatives on E : D x = ∂∂x + X i,j ≥ ,k,l> (cid:18) u x i +1 z j ∂∂u x i z j + u x i +1 z j y k ∂∂u x i z j y k + u x i +1 z j t l ∂∂u x i z j t l (cid:19) ,D y = ∂∂y + X i,j ≥ ,k,l> (cid:18) u x i z j y ∂∂u x i z j + u x i z j y k +1 ∂∂u x i z j y k + D ix D jy D l − t ( u z u xy − u y u xz ) ∂∂u x i z j t l (cid:19) , I.S. KRASIL ′ SHCHIK AND P. VOJˇC ´AK D z = ∂∂z + X i,j ≥ ,k,l> (cid:18) u x i z j +1 ∂∂u x i z j + u x i z j +1 y k ∂∂u x i z j y k + u x i z j +1 t l ∂∂u x i z j t l (cid:19) ,D t = ∂∂t + X i,j ≥ ,k,l> (cid:18) u x i z j t ∂∂u x i z j + D ix D jy D k − y ( u z u xy − u y u xz ) ∂∂u x i z j y k + u x i z j t l +1 ∂∂u x i z j t l (cid:19) . The Cartan distribution on E is spanned by these fields. The defining equations for local symmetries.
The linearization of Equation (1) has theform D y D t ( ϕ ) = u xy D z ( ϕ ) − u xz D y ( ϕ ) + u z D x D y ( ϕ ) − u y D x D z ( ϕ ) , (2)where ϕ is a function that depends on a finite number of internal coordinates. The vector fieldon E that corresponds to a solution ϕ is E ϕ = X i,j ≥ ,k,l> (cid:18) D ix D jz ( ϕ ) ∂∂u x i z j + D ix D jz D ky ( ϕ ) ∂∂u x i z j y k + D ix D jz D lt ( ϕ ) ∂∂u x i z j t l (cid:19) , (3)but we shall mainly deal with the generating functions ϕ rather than with the fields E ϕ them-selves.Note that it can be easily shown that Equation (1) admits point symmetries only, i.e., solutionsof (2) may depend only on the variables x , y , z , t , u , u x , u y , u z , and u t . The τ + - and τ − coverings All our subsequent nonlocal constructions are based on thecovering w t = u z w x − λ − w z , w y = λu y w x , (4)where 0 = λ ∈ R and w is the nonlocal variable, see [11]. It is readily checked that thecompatibility conditions for the overdetermined system (4) amount to Equation (1). We denotethe covering (4) by τ λ . Remark . At first glance, the covering τ λ is one-dimensional. This is not the case, actually,because x - and z - derivatives of w are not defined in (4). To make the definition complete, wemust introduce infinite number of nonlocal variables w α,β , α , β = 0 , , , . . . , w , = w and set w α,βx = w α +1 ,β , w α,βz = w α,β +1 w α,βt = ( u z w x − λ − w z ) x α z β , w α,βy = ( λu y w x ) x α z β . So, (4) defines an infinite-dimensional covering.Assume now that w = w ( λ ) and consider the expansion w = P i ∈ Z λ i w i . Substituting thelatter into (4), we get w i,t = u z w i,x − w i +1 ,z , w i,y = u y w i − ,x , i ∈ Z . (5)Thus, we obtain an infinite-dimensional covering over E , but the problem is that this is ‘badinfinity’ which has ‘neither beginning nor end’. To overcome this inconvenience, we divide (5)in two parts assuming that w i = 0 for i > w i = 0 for i < τ − ) and positive ( τ + )ones, respectively. After suitable relabellings, the defining equations for these coverings acquirethe form τ − : E − → E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = y,r i,t = u z u − y r i − ,y − r i − ,z ,r i,x = u − y r i − ,y , i ≥ , (6)and τ + : E + → E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − = x, q = u,q i,y = u y q i − ,x ,q i,z = u z q i − ,x − q i − ,t , i ≥ . (7) D MASH EQUATION 5
So, q , q . . . are the nonlocal variables in τ + and r , r , . . . are those in τ − . Remark . Strictly speaking, we must enrich (6) with infinite number of formal variables thatwould define y - and z -derivatives of r . In a similar way, additional variables that define x -and t -derivatives of q are needed (cf. Remark 1). To be more precise, in τ − , we consider thevariables r α,βi , α , β = 0 , , . . . , such that r , i = r i and r α,βi,y = r α +1 ,βi , r α,βi,z = r α,β +1 i ,r α,βi,t = ( u z u − y r i − ,y − r i − ,z ) y α z β , r α,βi,x = ( u − y r i − ,y ) y α z β Similarly, we introduce q α,βi in τ + and set q , i = q i , q α,βi,x = q α +1 ,βi , q α,βi,t = q α,β +1 i ,q α,βi,y = ( u y q i − ,x ) x α t β , q α,βi,z = ( u z q i − ,x − q i − ,t ) x α t β . But, as we shall see below, this formalization does not influence the subsequent computations.
The defining equations for nonlocal symmetries.
Let us begin with writing down thetotal derivatives in the negative and positive coverings. In τ − , due to (6) and Remark 2, onehas D − x = D x + X − , D − y = D y + Y − , D − z = D z + Z − , D − t = D t + T − , where X − = D x + ∞ X i =1 ∞ X α,β =0 ( u − y r i − ,y ) y α z β ∂∂r α,βi ,Y − = D y + ∞ X i =1 ∞ X α,β =0 r α +1 ,βi ∂∂r α,βi ,Z − = D z + ∞ X i =1 ∞ X α,β =0 r α,β +1 i ∂∂r α,βi ,T − = D t + ∞ X i =1 ∞ X α,β =0 ( u z u − y r i − ,y − r i − ,z ) ∂∂r α,βi . The total derivatives in τ + are D + x = D x + X + , D + y = D y + Y + , D + z = D z + Z + , D + t = D t + T + , where X + = D x + ∞ X i =1 ∞ X α,β =0 q α +1 ,βi ∂∂q α,βi ,Y + = D y + ∞ X i =1 ∞ X α,β =0 ( u y q i − ,x ) x α t β ∂∂q α,βi ,Z + = D z + ∞ X i =1 ∞ X α,β =0 ( u z q i − ,x − q i − ,t ) x α t β ∂∂q α,βi ,T + = D t + ∞ X i =1 ∞ X α,β =0 q α,β +1 i ∂∂q α,βi . Finally, the total derivatives in the Whitney product τ ± = τ − ⊕ τ + of τ − and τ + read D ± x = D x + X − + X + , D ± y = D y + Y − + Y + , D ± z = D z + Z − + Z + , D ± t = D t + T − + T + and the lift of ℓ E to τ ± will be denoted by ℓ ± E with the obvious meaning of the notation. I.S. KRASIL ′ SHCHIK AND P. VOJˇC ´AK
To proceed, let us agree on notation. Denote by E ± ϕ the field on τ ± obtained from the field E ϕ presented in (3) by changing the total derivatives D • to D ±• , where • denotes x , y , z or t . Wealso obtain operators ℓ ± E from ℓ E in the same way.In this notation, any τ − -nonlocal symmetry is of the form S = E − ϕ + ∞ X i =1 ∞ X α,β =0 ϕ α,βi ∂∂r α,βi , where ϕ , ϕ α,βi are functions on on τ − . Then ϕ α,βi = ( D − y ) α ( D − z ) β ( ϕ i ), ϕ i = ϕ α,βi , and ℓ − E ( ϕ ) = 0 ,D − t ( ϕ i ) = u − y ( u y D − z ( ϕ ) − u z D − y ( ϕ )) r i − ,y + u z u − y D − y ( ϕ i − ) − D − z ( ϕ i − ) , (8) D − x ( ϕ i ) = − u − y D − y ( ϕ ) r i − ,y + u − y D − y ( ϕ i − ) . Hence, any such a symmetry S = S Φ is completely determined by the vector-function Φ =( ϕ, ϕ , . . . ) and the formula [ S Φ , S Ψ ] = S { Φ , Ψ } defines a bracket on the space of these functions.Nonlocal shadows are just the functions ϕ that satisfy the first of Equations (8), while invisiblesymmetries are Φ = (0 , ϕ , . . . ) with D − t ( ϕ i ) = u z u − y D − y ( ϕ i − ) − D − z ( ϕ i − ) ,D − x ( ϕ i ) = u − y D − y ( ϕ i − ) . (9)Of course, the scheme is almost the same in τ + . Any symmetry is S = E + ϕ + ∞ X i =1 ∞ X α,β =0 ϕ α,βi ∂∂q α,βi , where ϕ , ϕ α,βi are functions on on τ + and ϕ α,βi = ( D + x ) α ( D + t ) β ( ϕ i ), ϕ i = ϕ α,βi . The definingequations for ϕ and ϕ i are ℓ + E ( ϕ ) = 0 ,D + y ( ϕ i ) = D + y ( ϕ ) q i − ,x + u y D + x ( ϕ i − ) (10) D + z ( ϕ i ) = D + z ( ϕ ) q i − ,x + u z D + x ( ϕ i − ) − D + t ( ϕ i − ) . As above, we introduce generating vector-functions Φ = ( ϕ, ϕ , . . . ) and using the notation S = S Φ define the bracket between these functions. Nonlocal shadows in the positive coveringare identified with solutions of ℓ + E ( ϕ ) = 0, while invisible symmetries Φ = (0 , ϕ , . . . ), where ϕ i satisfy the system D + y ( ϕ i ) = u y D + x ( ϕ i − ) D + z ( ϕ i ) = u z D + x ( ϕ i − ) − D + t ( ϕ i − ) . (11)Symmetries in the Whitney product are vector fields S = E ± ϕ + ∞ X i =1 ∞ X α,β =0 D αy D βz ( ϕ − i ) ∂∂r α,βi + D αx D βt ( ϕ + i ) ∂∂q α,βi ! , where the functions ϕ ± , ϕ − i , ϕ + i ∈ C ∞ ( E − × E E + ) enjoy the relations ℓ ± E ( ϕ ± ) = 0 ,D ± t ( ϕ i ) = u − y ( u y D ± z ( ϕ ) − u z D ± y ( ϕ )) r i − ,y + u z u − y D ± y ( ϕ i − ) − D ± z ( ϕ i − ) ,D ± x ( ϕ i ) = − u − y D ± y ( ϕ ) r i − ,y + u − y D ± y ( ϕ i − ) ,D ± y ( ϕ i ) = D ± y ( ϕ ) q i − ,x + u y D ± x ( ϕ i − ) D ± z ( ϕ i ) = D ± z ( ϕ ) q i − ,x + u z D ± x ( ϕ i − ) − D ± t ( ϕ i − ) . (12) D MASH EQUATION 7
Remark . A useful instrument in analysis of Lie algebra structures is the weights (gradings)that may be assigned to all the variables in the coverings under consideration and all polynomialfunctions in these variables. Namely, if we set the weights on independent variables to be x
7→ | x | , y
7→ | y | , z
7→ | z | , t
7→ | t | , then from Equations (1), (6) and (7) it follows that | u | = | x | + | z | − | t | , | r i | = | y | + i ( | t | − | z | ) , | q i | = | x | + ( i + 1)( | z | − | t | ) . To a vector field
A∂/∂a we assign the weight | A | − | a | . Then for any two fields one has | [ A, B ] | = | A | + | B | . Thus, Lie algebras spanned by homogeneous fields become graded.So, we have four independent way to introduce weights reflects existence of four independentscaling symmetries in sym E (see Section 3). Weights of differential polynomials are computedin an obvious way. In what follows, it will be convenient to use the following choice: | x | = − , | t | = | y | = | u | = 0 , | z | = 1 , and thus | r i | = − i | q i | = i. To conclude the discussion of structures inherent to the equation under study, we mentionthe recursion operator found in [11]. The tangent equation corresponding to (1) is of the form u ty = u z u xy − u y u xz ,p yt = u xy p z − u zt p y + u z p xy − u y p xz . The B¨acklund transformation that relates two copies of TE is D y ( ϕ ) = u y D x ( ϕ ′ ) − u xy ϕ ′ ,D z ( ϕ ) = − D t ( ϕ ′ ) + u z D x ( ϕ ′ ) − u xz ϕ ′ . (13)If ϕ is a solution of Equation (2) then ϕ ′ also solves it and vice versa. The correspondence ϕ ϕ ′ defined by relations (13) will be denoted by −→ R and the opposite one by ←− R . The operator −→ R changes the weight by +1, while ←− R changes it by − Algebras of nonlocal symmetries
We accomplish the construction of the desired algebra in several steps that are: • explicit computation of basic shadows and their lifts to τ − , τ + , and τ ± (Proposition 1); • construction of hierarchies by means of commutators of the basic symmetries; • construction of new hierarchies by somewhat artificial trick (Theorems 1 and 2); • computation of the Lie algebra structure (Theorem 3). Notation.
In what follows, A = A ( y, z ) and B = B ( x, t ) are arbitrary smooth functions.Notation S ji for a symmetry indicates its weight i (and the position in a hierarchy), while thesuperscript j (if any) enumerates the hierarchies. If a symmetry contains a function A , wecompute its weight assuming A = y ; if it contains B , the assumption is B = x .The coefficient of S ji at ∂/∂u (the shadow) will be denoted by s ji, , while its coefficientsat ∂/∂r α and ∂/∂q α will be s j, − i,α and s j, + i,α , respectively. Thus, any symmetry is presented by itsgenerating vector-function S ji ∼ h s ji, , s j, − i, , s j, + i, , . . . , s j, − i,α , s j, + i,α , . . . i , where s ji, , s j, − i,α , s j, + i,α are smooth functions on E − ⊗ E E + . The basic shadows.
The following shadows are found by direct computations: ψ − , = − u z , ψ , = u t , ψ , = q ,t − u t u x , For the convenience of the subsequent exposition, we present it a slightly different from [11] form, which is ofcourse equivalent to the original one.
I.S. KRASIL ′ SHCHIK AND P. VOJˇC ´AK ω − , = u y (2 r + zr ,z − r ,y ( r + zr ,z )) , ω − , = u y ( r + zr ,z ) , ω , = u − zu z ,ω , = 2 q − uu x + zu t , ω , = 3 q − u x q − uq ,x + zq ,t + uu x − zu t u x ,ξ − , ( A ) = u y ( Ar ,y − A y r + A z t ) , ξ , ( A ) = − Au y , ξ , ( A ) = 0 ,υ − , ( B ) = B, υ , ( B ) = − Bu x + B x , u − B t z,υ , ( B ) = B ( u x − q ,x ) + B x ( q − uu x ) + B t zu x + 12 B xx u + 12 B tt z − B tx zu. Proposition 1.
All the above listed shadows admit lifts to τ ± .Proof. The lifts of the shadows ψ ji, and ω ji, are described explicitly. Namely, we set ψ , −− ,α = − r α,z , ψ , + − ,α = − q α,z ,ψ , − ,α = r α,t , ψ , +0 ,α = q α,t ,ψ , − ,α = r α − ,t − u t r α,x , ψ , − ,α = q α +1 ,t − u t q α,x , and ω , −− ,α = − ( α + 2) r α +2 − zr α +2 ,z + ( r + zr ,z ) r α +1 ,y + (2 r + zr ,z − ( r + zr ,z ) r ,y ) r α,y ,ω , + − ,α = zq α − ,t − zu z q α − ,x + ( α − q α − + u y ( r + zr ,z ) q α − ,x + u y (2 r + zr ,z − ( r + zr ,z ) r ,y ) q α − ,x ,ω , −− ,α = − ( α + 1) r α +1 − zr α +1 ,z + ( r + zr ,z ) r α,y ,ω , + − ,α = zq α − ,t − zu z q α − ,x + αq α − + u y ( r + zr ,z ) q α − ,x ,ω , − ,α = − αr α − zr α,z ,ω , +0 ,α = ( α + 1) q α − zq α,z ,ω , − ,α = − ( α − r α − − zr α − ,z + ( zr ,t − ur ,x ) r α − ,y ,ω , +1 ,α = ( α + 2) q α +1 − uq α,x + zq α,t ,ω , − ,α = − ( α − r α − − zr α − ,z − ( u − zu z ) r α − ,x − (2 q − uu x + zu t ) r α,x ,ω , +2 ,α = ( α + 3) q α +2 − uq α +1 ,x + zq α +1 ,t − (2 q − uu x + zu t ) q α,x . To lift the shadows ξ i, ( A ) and υ i, ( B ), let us introduce the operators Y = − t ∂∂z + ∞ X i =0 ( i + 1) r i +1 ∂∂r i , X = − z ∂∂t + u ∂∂x + ∞ X i =0 ( i + 2) q i +1 ∂∂q i (recall that r = y and q = u ) and the quantities P α ( A ), Q α ( B ), j = 0 , , , . . . , defined byinduction as follows: P ( A ) = A, P α ( A ) = 1 α Y ( P α − ( A )) , Q ( B ) = B, Q α ( B ) = 1 α X ( Q α − ( B )) , α ≥ . We also tacitly assume that P α ( A ) and Q α ( B ) vanish if α is negative. Then ξ −− ,α ( A ) = A ( r ,y r α,y − r α +1 ,y ) − A y r r α,y + A z tr α,y + P α +1 ( A ) ,ξ + − ,α ( A ) = u y ( A ( r ,y q α − ,x − q α − ,x ) − A y r q α − ,x + A z tq α − ,x ) ,ξ − ,α ( A ) = − Ar α,y + P α ( A ) ,ξ +0 ,α ( A ) = − Aq α,y ,ξ − ,α ( A ) = P α − ( A ) ,ξ +1 ,α ( A ) = 0 , D MASH EQUATION 9 and υ −− ,α ( B ) = 0 ,υ + − ,α ( B ) = Q α ( B ) ,υ − ,α ( B ) = − Br α,x ,υ +0 ,α ( B ) = − Bq α,x + Q α +1 ( B ) ,υ − ,α ( B ) = B ( u x r ,x r α − ,y − r α − ,x ) − B x ur ,x r α − ,y + B t zr ,x r α − ,y ,υ +1 ,α ( B ) = B ( u x q α,x − q α +1 ,x ) − B x uq α,x + B t zq α,x + Q α +2 ( B ) . It is atrightforward to check that these are indeed the needed lifts. (cid:3)
Thus, we obtained fourteen symmetriesΨ − , Ψ , Ψ , Ω − , Ω − , Ω , Ω , Ω , Ξ − ( A ) , Ξ ( A ) , Ξ ( A ) , Υ − ( B ) , Υ ( B ) , Υ ( B )in τ ± which will serve as seeds for construction the entire algebra of nonlocal symmetries. Remark . It is worth to note that the operators X and Y used in the proof of Proposition 1 have atransparent geometrical interpretation. Namely, consider the system consisting of Equations (1)and (4) and let us treat the parameter λ as an additional independent variable with the condition u λ = 0. Then the total derivative D λ transforms to X when passing from the covering (4) to τ + and to Y when passing to τ − . Construction of hierarchies 1.
Now we use the symmetries Ω ± as hereditary ones andconstruct two infinite hierarchiesΨ i = i + 1 { Ω − , Ψ i +1 } , if i ≤ − , i − { Ω , Ψ i − } , if i ≥ , and Ω i = i + 2 { Ω − , Ω i +1 } , if i ≤ − , i − { Ω , Ω i − } if i ≥ . These hierarchies will be used below to construct new ones.
Construction of hierarchies 2.
Define the functions ψ ji, = j X m =0 ( − m (cid:18) jm (cid:19) t j − m z m ψ i − m, , j ≥ . (14) Theorem 1.
Formula (14) defines shadows of symmetries. These shadows can be lifted to τ ± and thus define infinite number of hierarchies { Ψ ji } of nonlocal symmetries.Proof. The proof is accomplished in two steps: first we establish that ψ ji, are shadows and thenshow that they can be lifted. Step
1. Induction on j . To this end, let us rewrite (14) recursively. Namely, we write ψ ji, = tψ j − i, − zψ j − i − , , i ≥ . (15)Let j = 1. Consider the linearization operator lifted to τ ± ℓ ± E = D ± y D ± t − u xy D ± z + u xz D ± y − u z D ± x D ± y + u y D ± x D ± z . (16) ′ SHCHIK AND P. VOJˇC ´AK
Then due to (16) for j = 1 one obviously has ℓ ± E ( ψ i, ) = ℓ ± E ( tψ i, − zψ i − , ) = tℓ ± E ( ψ i, ) − zℓ ± E ( ψ i − , ) + D ± y ( ψ i, ) + u xy ψ i − , − u y D ± x ( ψ i − , ) = D ± y ( ψ i, ) + u xy ψ i − , − u y D ± x ( ψ i − , ) , since ψ i, and ψ i − , are shadows. But the last term in the equalities above is exactly the firstequation in the formula (13) for the recursion operator. It can be checked that this operator,modulo the image of zero (see discussion in the end of the paper) connects the shadows ψ i, and ψ i − , . In particular, D ± y ( ψ i, ) + u xy ψ i − , − u y D ± x ( ψ i − , ) = 0 , (17)because all ψ α, are shadows. Moreover, since (17) does not contain the total derivatives in z and t , we deduce, using (15),that D ± y ( ψ i, ) + u xy ψ i − , − u y D ± x ( ψ i − , ) = 0 . Let now j > l < j and i ∈ Z the functions ψ li, are shadows thatenjoy the relations D ± y ( ψ li, ) + u xy ψ li − , − u y D ± x ( ψ li − , ) = 0. Then the proof of the inductionstep is exactly the same as the one for the case j = 1. Step
2. We shall now prove that the functions ψ j, ± i,α = tψ j − , ± i,α − zψ j − , ± i − ,α (18)satisfy System (12) for all i ∈ Z , j ≥ α ≥
1. We also use induction on j here.Consider the case j = 1. Substituting the expression ψ , ± i,α = tψ , ± i,α − zψ , ± i − ,α to the definingequations (12), we obtain for τ − D − t ( tψ , − i, − zψ , − i − , ) = 1 u y (cid:16) u y D − z ( tψ , − i, − zψ , − i − , ) − u z D − y ( tψ , − i, − zψ , − i − , ) (cid:17) in the case α = 1 and D − t ( tψ , − i,α − zψ , − i − ,α ) = 1 u y (cid:16) u y D − z ( tψ , − i, − zψ , − i − , ) − u z D − y ( tψ , − i, − zψ , − i − , ) (cid:17) + 1 u y (cid:16) ( u y D − z ( tψ , − i, − zψ , − i − , ) − u z D − y ( tψ , − i,α − − zψ , − i − ,α − ) (cid:17) , when α >
1. But the functions ψ , − i,α are the components of the nonlocal symmetries Ψ i andhence we obtain the conditions ψ , − i, = − u y ψ , − i − , , ψ , − i,α = − u y ψ , − i − , + ψ , − i − ,α − , α > . (19)from the above equations.Similar computations show that the conditions ψ , + i − , = u z ψ , + i − , , ψ , + i − ,α = q α − ,x ψ , + i − , + ψ , + i,α − , α > . (20)must hold in τ + . Lemma 1.
Conditions (19) and (20) do hold for all α > and i ∈ Z .Proof of Lemma . The proof comprises two inductions on i (for i ≥ i ≤
0) and consists ofvoluminous computations based on explicit descriptions from Proposition 1 and on the definitionof the symmetries Ψ i . We omit the details. (cid:3) Note now that the functions ψ , ± i,α = tψ , ± i,α − zψ , ± i − ,α satisfy the conditions similar to (19)and (20) by linearity. This finishes the proof of the induction base. The proof of the inductionstep does not differ from the latter. (cid:3) D MASH EQUATION 11
In a similar way, we define the functions ω ji, = j X m =0 ( − m (cid:18) jm (cid:19) t j − m z m ω i − m, , j ≥ , (21)and prove the following Theorem 2.
Formula (21) defines shadows of symmetries. These shadows can be lifted to τ ± and thus define infinite number of hierarchies { Ω ji } of nonlocal symmetries.The proof almost exactly copies the one of Theorem . (cid:3) Remark . As it follows from Theorems 1 and 2, the hierarchies { Ψ ji } , { Ω ji } , i ∈ Z , j ≥
0, existin the Whitney product τ ± , but this result may be clarified. More detailed information on theΨ-hierarchies is presented in Table 1. Note that the symmetries Ψ − , Ψ , and Ψ are local.Additional properties of the Ω-hierarchies are given in Table 2. Of all these symmetries, only Ω is a local one. Ψ ji j < i + 2 j ≥ i + 2 i ≤ τ − , τ + , τ ± in τ − , τ ± i > τ + , τ ± in τ ± only Table 1.
Distribution of Ψ ji over τ − , τ + , and τ ± Ω ji j < i + 1 j ≥ i + 1 i ≤ τ − , τ + , τ ± in τ − , τ ± i > τ + , τ ± in τ ± only Table 2.
Distribution of Ω ji over τ − , τ + , and τ ± Construction of hierarchies 3.
The last step is the construction of the ( x, t )- and ( y, x )-dependent hierarchies. To this end, we setΞ i ( A ) = i + 1 { Ω − + Ψ − , Ξ i +1 ( A ) } , if i ≤ − , i − { Ω + Ψ , Ξ i − ( A ) } , if i ≥ , and Υ i ( B ) = i + 1 { Ω − , Υ i +1 ( B ) } , if i ≤ − , i − { Ω , Υ i − ( B ) } if i ≥ A = A ( y, z ) and B = B ( x, t ) are arbitrary smooth functions). Remark . As above, the structure of these hierarchies may be clarified in some respects. Namely,we have the following facts:Ξ i ( A ) is a symmetry in τ − , τ ± , if i ≤ − ,τ − , τ + , τ ± , if i = 0 ,τ ± , if i ≥ . Moreover, the symmetry Ξ ( A ) is local, while Ξ i ( A ) are invisible symmetries for i ≥ ′ SHCHIK AND P. VOJˇC ´AK
In a similar way, Υ i ( B ) is a symmetry in τ ± , if i ≤ − ,τ − , τ + , τ ± , if i = − , ,τ + , τ ± , if i ≥ . The symmetries Υ i ( B ) are invisible for all i ≤ − − ( B ), Υ ( B ) are localones. Lie algebra structure.
Let us now describe the structure of the Lie algebra formed bythe above constructed symmetries. To this end, relabel some of them to make the results lookneater. Namely, we change notation as follows:Ψ ji
7→ − Ψ j +1 i , Ξ i ( A ) Ξ i ( A · z − i ) . Then we have the following result:
Theorem 3.
The Lie algebra g = sym τ ± ( E ) of the τ ± -nonlocal symmetries for the 4D MAShequation as an R -vector space is generated by the elements { Ψ ji } j ≥ i ∈ Z , { Ω ji } j ≥ i ∈ Z , { Υ i ( B ) } i ∈ Z , { Ξ i ( A ) } i ∈ Z , where B = B ( x, t ) and A = A ( y, z ) are arbitrary smooth functions. They enjoy the commutatorrelations presented in Table . Ψ lk Ω lk Υ k ( ¯ B ) Ξ k ( ¯ A )Ψ ji ( l − j )Ψ j + li + k l Ω j + li + k − i Ψ j + li + k Υ i + k ( t j +1 ¯ B t ) ( − j Ξ k + i − j ( z i +1 ¯ A z − kz i ¯ A )Ω ji ( k − i )Ω j + li + k Υ i + k ( kt j ¯ B ) ( − j Ξ k + i − j ( z i +1 ¯ A z )Υ i ( B )) Υ i + k ([ B, ¯ B ]) Ξ i ( A ) Ξ i + k ([ A, ¯ A ]) Table 3.
The Lie algebra structure
Here the notation [ A, ¯ A ] = A ¯ A y − ¯ AA y , [ B, ¯ B ] = B ¯ B x − ¯ BB x was used.Proof. The proof is omitted due to its extreme length. It consists of a number of inductionswith explicit computations in the bases of these inductions. (cid:3)
Remark . Denote by h ⊂ g the subalgebra spanned by the elements Ψ ji , Ω ji , and Υ i ( B ), andlet i ( A ) ⊂ g denote the ideal { Ξ i ( A ) } . Then g is the semi-direct product h ⋉ i ( A ). In its turn, h = h ⋉ i ( B ), where h = { Ψ ji , Ω ji } , i ( B ) = { Υ i ( B ) } . The structure of h is quite clear. Consider the correspondenceΨ ji t j +1 z i ∂∂t , j ≥ ji t j z i +1 ∂∂z , j ≥ i ( B ) z i B ∂∂y , i ∈ Z . Then we obtain an isomorphism between h and the Lie algebra of the corresponding vectorfields. The action of h on i ( A ) is less conventional (see the last column of Table 3). Action of the recursion operator.
Let us now describe the action of the recursion oper-ator (13) on the shadows our symmetries. First of all note that −→ R (0) = ξ , ( A ) , ←− R (0) = υ , ( B ) , D MASH EQUATION 13 and thus the action is defined modulo the images of zero. Keeping this in mind we have . . . ←− R , , ψ j , − −→ R j j ←− R , , ψ j , − −→ R l l ←− R + + ψ j , −→ R l l ←− R + + ψ j , −→ R k k ←− R + + ψ j , −→ R k k ←− R ) ) . . . −→ R k k . . . ←− R , , ω j , − −→ R j j ←− R , , ω j , − −→ R l l ←− R + + ω j , −→ R l l ←− R + + ω j , −→ R k k ←− R + + ω j , −→ R k k ←− R ) ) . . . −→ R k k . . . −→ R - - υ , ( B ) −→ R - - ←− R j j υ , ( B ) −→ R * * ←− R m m −→ R - - ←− R m m ξ , ( A ) −→ R - - ←− R j j ξ , ( A ) −→ R ) ) ←− R m m . . . ←− R m m Remark . To conclude, recall that in [12] an infinite series of pair-wise commuting nonlocalsymmetries was presented. The algebra sym τ ± ( E ) described above contains infinite number ofsuch hierarchies. Namely, for any i ∈ Z and j ∈ N each of the families Ψ j = { Ψ ji } i ∈ Z and Ω i = { Ω ji } j ≥ consists of pair-wise commuting symmetries. In addition, if we fix the functions A ( y, z ) and B ( x, t )then the families Ξ ( A ) = { Ξ i ( A ) } i ∈ Z and Υ ( B ) = { Υ i ( B ) } i ∈ Z will possess the same property. Acknowledgments
Computations were supported by the
Jets software, [2]. The work of I.K. was partiallysupported by Russian Foundation for Basic Research Grant 18-29-10013 and Simons-IUM Fel-lowship Grant 2020.
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