On integrability of the Camassa-Holm equation and its invariants. A geometrical approach

On integrability of the Camassa–Holm equation and its in variants A geometr ical appro ach V . Golo vko · P . Kersten · I. Krasil ′ shchik · A. V erbov etsky Abstract Using geometrical approach exposed i n R efs. [12, 13], we ex plore the C amassa– Holm equ ation (both in its initial scalar fo rm, and in th e form of 2 × 2-system). W e describe Hamiltonian and symplectic structur es, recursion operator s and infinite series of s ymmetries and conserv ation laws (local and nonlocal). K eywords C amassa–Holm equ ation · Inte grability · Hamiltonian structures · Symplectic structures · Recursion operator s · Symmetries · C onserv ation la ws · Geometrical approach P A CS 02.30.Ik · 11.30.-j Mathematics Subject Classification (2000) 37K05 · 35Q53 1 Introduction The Camassa–Holm equation was intro duced in [4] in the form u t + µ u x − u xxt + 3 uu x = 2 u x u xx + uu xxx , µ ∈ R , (1) This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consorti um E.I.N.S.T .E .I.N. g rant 06-01-92060. V alen tina Golovko Lomonosov MSU, Facult y of Ph ysics, Depa rtment of Mathematic s, V orob ′ e vy Hills, Moscow 119902, Russia E-mail: golov ko@mccme.ru Paul K ersten Uni versity of T wente, Postbu s 217, 7500 AE Enschede, the Netherlan ds E-mail: kerste nphm@ewi.ut wente.nl Iosif Krasil ′ shchik Independ ent Univ ersity of Moscow , B. Vlasev sky 11, 119002 Moscow , Russia E-mail: josephk@di ffiety .ac.ru Alex ander V erbove tsky Independ ent Univ ersity of Moscow , B. Vlasev sky 11, 119002 Moscow , Russia E-mail: verbo vet@mccme . ru 2 and was intensi vely explored afterwar ds (see, for example, Refs. [5, 6 , 7, 1 7]). Its superiza- tions were also constructed , see [1, 19]. Since (1) is n ot an ev olution equation, its integrabil- ity properties (existence and ev en definition of Hamiltonian structures, conserv ation laws, etc.) are not standard to establish. One of the ways widely used t o ov ercome this dif ficulty is to introduce a ne w un- kno wn m = u − u xx and transform Eq. (1) to the system ( m t = − um x − ( 2 m + µ ) u x , m = u − u xx , (2) which has almost e volutio nary form. W e stress this “almost”, because the second equatio n in (2) (that ca n be considered as a constrain to th e fi rst on e) disrup ts the picture and, at best, necessitates to inv ert the operator 1 − D 2 x . At wor s t, dealin g with Eq. ( 2) as with an ev olution equation may lead to fallacious results. In our approach based on the geometrical framew ork exposed in Ref. [3], we treat the equation at hand as a s ubmanifo ld in the manifold of infinite jets and consider two natur al extension s of this eq uation, cf. with R ef [16]. The first one is called the ℓ -cov ering and serves the role of the tangent bun dle. The second e xtension, ℓ ∗ -cov ering, is the counterp art to the cotangent b undle. The ke y property o f these extensions is that t he spaces of their nonlocal (in the sense of [15]) symmetries and cosymmetries contain all essential integrability in v ariants of the initial equation. The efficienc y of the method was tested for a number of problems (see Refs. [12, 13, 14]) and we apply it to the Camassa–Holm equation here. In S ection 2 we briefly e xpose the necessary definition and facts. Section 3 contains computation s for the Camassa–Holm equation in its matrix version (computations and re- sults are more com pact in this representation), while in Section 4 we reformulate them for the original for m (1) an d compare later the results obtained for the tw o alternativ e presenta- tions. Finally , Section 5 contains discussion of the results obtain ed. Throu ghout our exp osi- tion we use a very stimulating con ceptual parallel between categories of smooth manifolds and diffe rential equations proposed initially by A.M. V inogrado v and in it s modern form presented in T able 1. T able 1 Conceptual parallel between two catego ries Manif olds Equation s Smooth manifold Infinitel y prolonged equation Point Formal solu tion Smooth functi on Conserv ation law V ecto r field Higher symmetry Dif ferential 1-form Cosymmetry de Rham comple x ( n − 1 ) st line of V inogrado v’ s C -spectral sequence T angent bundle ℓ -cov ering Cotange nt bundl e ℓ ∗ -cov ering This table is not just a toy dictionar y but a quite helpful tool to formulate important definitions and results. For exam ple, a biv ector on a smoo th manifold M may b e understood as a deri v ation of the ring C ∞ ( M ) with v alues in C ∞ ( T ∗ M ) . Tr anslating this statement to the language of differential equations we co me to the definition of v ariational biv ectors and their description as s hado ws of symmetries in the ℓ ∗ -cov ering (see Theorem 2 below). Another examp le: any v ector field (differential 1-form ) on M may be treated as a function on T ∗ M 3 (on T M ). Hence, to any symmetry (cosymmetry ) there corresp onds a conserv ation law on the s pace of the ℓ ∗ -cov ering ( ℓ -cov ering). This leads to the notions of nonlocal vectors and forms that, in turn, provide a basis to construct weakly nonlocal structures (see Subsec- tions 3.4 and 3.5 ). Of course, these p arallels are not completely straightforward (in technical aspects, especially), b ut e xtremely enlightening and fruitful. The idea of this paper arose in the discussions one of the authors had with V olodya Roubtsov in 2007. W e agreed to write two parallel t exts on integrab ilit y of the Camass a– Holm equ ation that reflect our vie wpoints. The reader can no w com pare our results with the ones presented in [20]. 2 Underlying theory W e present here a concise exposition of the theoretical backgro und used in the subsequent sections, see Refs. [3, 13, 15]. 2.1 Equations, symmetries, etc. Let π : E → M be a fiber bu ndle and π ∞ : J ∞ ( π ) → M be the bun dle of i ts infinite jets. T o simplify our exposition we shall assume that π is a vector bu ndle. In all applications be- lo w π is the t ri vial bundle R m × R n → R n . W e conside r infinite pr olongations of dif ferential equations as submanifold s E ⊂ J ∞ ( π ) and retain the notation π ∞ for the restriction π ∞ | E . Any such a manifold is endowed with the Cartan distribu tion which spans at ev ery point tangent spaces to the graphs of jets. A symmetr y of E is a vector fi eld that preserv es this distrib ution. The set of symmetr ies is a Lie algebr a o ver R denoted by sym E . For any equation E its linearization operator ℓ E : κ → P is defined, where κ is the module 1 of sections of the pullback π ∞ ( π ) and P is the module of sections of some vecto r b undle o ver E . Then sym E can be iden tified with s olutions of the equation ℓ E ( ϕ ) = 0 , ϕ ∈ κ . (3) For tw o symmetries ϕ 1 , ϕ 2 ∈ sym E their commutator is denoted by { ϕ 1 , ϕ 2 } . Denote by Λ i h the module of horizontal i -forms on E and intr oduce the notatio n ˆ Q = H om F ( Q , Λ n h ) , n = dim M , for any mo dule Q . The adjo int to ℓ E operator ℓ ∗ E : ˆ P → ˆ κ (4) arises and solutions of the equation ℓ ∗ E ( ψ ) = 0 , ψ ∈ ˆ P , (5) are called cosymmetries of E ; the s pace of cosymmetr ies is denoted by cosym E . Let d h : Λ i h → Λ i + 1 h be t he horizontal de Rham differ ential . A conservation law of the equation E is a closed form ω ∈ Λ n − 1 h . T o an y conserv ation law there corr es ponds its gener - ating function δ ω ∈ cosym E , where δ : E 0 , n − 1 1 → E 1 , n − 1 1 is the differen tial in th e E 1 term of 1 All the modules bel ow are modules over the ring F of smooth functions on E . 4 V inogradov ’ s C -spectral sequence, see [21]. In the ev olutionary case δ coincides with the Euler –Lagrange operator . A conserv ation law is trivial if its generating function vanishes. In particular , d h -exact con servation laws are tri vial. A vector field on E is called a C -field if it lies in the Cartan distribution . A dif ferential operator ∆ : P → Q , P and Q being F -modules, is called a C -differ ential operator if it is locally exp ressed in terms of C -fields. For e xample, ℓ E is a C -dif ferential op erator . A C -dif ferential operator H : ˆ P → κ is said to be a variational bivector on E if ℓ E ◦ H = H ∗ ◦ ℓ ∗ E . (6) This condition means that H takes cosymmetries of E to symmetries. If E is an ev olu- tion Eq. (6) implies also t hat H ∗ = − H . A biv ector H i s a Hamiltonian str uctur e 2 on E if [ [ H , H ] ] s = 0, where [ [ · , · ] ] s is the variational Schouten brac ket (see also [11]). T wo Hamil- tonian structures are compatible if [ [ H 1 , H 2 ] ] s = 0. A C -dif ferential operator S : κ → ˆ P is called a variational 2 -form on E if ℓ ∗ E ◦ S = S ∗ ◦ ℓ E . (7) Such operator s tak e symmetries to cosymmetries and in ev olutionary case are ske w-adjoint. They are elements of the term E 2 , n − 1 1 of V inogrado v’ s C -spectral sequence. A v ariational form is a s ymplectic structure on the equation E if it is variationally closed , i.e., δ S = 0, where δ : E 2 , n − 1 1 → E 3 , n − 1 1 is the correspon ding dif ferential. W e shall also consider recursio n C -differ ential operators R : κ → κ and ˆ R : ˆ P → ˆ P satisfying the conditions ℓ E ◦ R = R ′ ◦ ℓ E , ℓ ∗ E ◦ ˆ R = ˆ R ′ ◦ ℓ ∗ E (8) for some C -dif ferential operators R ′ : P → P and ˆ R ′ : ˆ κ → ˆ κ . An operator R satis fies the Nijenhuis cond ition if [ [ R , R ] ] n = 0, where [ [ · , · ] ] n is the variational N ijenhuis br acket . A re- cursion operator R is compatible with a Hamiltonian structure H is the composition R ◦ H i s a Hamiltonian structure as well. 2.2 Nonlocal theory Let E and ˜ E be equations and ξ : ˜ E → E be a fiber b undle. Deno te by C and ˜ C the Cartan distrib utions on E and ˜ E , resp. W e say that ξ is a covering if for an y ˜ θ ∈ ˜ E the dif feren- tial d ˜ θ ξ isomorphically maps ˜ C ˜ θ onto C ξ ( ˜ θ ) . A particular case of cov erings (the so-called Abelian cov erings) is naturally as sociated with closed horizon tal 1-for ms 3 . By definition, any C -field X on E can be uniquely lifted to a C -field ˜ X on ˜ E s uch that d ξ ( ˜ X ) = X . Consequently , any C -dif ferential operator ∆ : P → Q is e xtended to a C - dif ferential operator ˜ ∆ : ˜ F ⊗ F P → ˜ F ⊗ F Q , ˜ F being the algebra of smooth functions on ˜ E . 2 It is m ore appropri ate to call these object s P oisson structur es , but we foll ow the traditi on accept ed in the theory of inte grable systems. 3 When dim M = 2, Abelia n cov erings are associated with conserva tion laws of the equat ion E . 5 A nonloc al ξ -(co)symm etry of E is a (co)symmetry of the cov ering equation ˜ E . They are s olutions of the equations ℓ ˜ E ϕ = 0 and ℓ ∗ ˜ E ψ = 0, resp. Along with these two equations one can consider the equations ( 1 ) ˜ ℓ E ϕ = 0 , ( 2 ) ˜ ℓ ∗ E ψ = 0 . (9) Their solutions are called ξ -shadows of symmetries and cosymm etries, resp. A shado w of symmetry is a deriv ation F → ˜ F that preserv es the Cartan distrib utions. For an y two shad- o ws of symmetries ϕ 1 and ϕ 2 their comm utator { ϕ 1 , ϕ 2 } can be defined. This commutator is a shado w in a ne w cov ering that is canonically deter mined by ϕ 1 and ϕ 2 . 2.3 The ℓ - and ℓ ∗ -cov erings Let E ⊂ J ∞ ( π ) be an equation. Its ℓ -covering τ : L ( E ) → E is obtained by adding to E the equatio n ℓ E ( q ) = 0, where q is a ne w v ariable. Dually , the ℓ ∗ -covering τ ∗ : L ∗ ( E ) → E is constru cted by adding the equ ation ℓ ∗ E ( p ) = 0 with a new variable p . They are the exact counterpar ts of the tangent and cotangen t b undles in the category of dif ferential equations. By the reasons that w ill become clear later , we regar d both q and p as odd variables. The main poin t of our method is the fundamental relation between inte grability in variants of E and shado ws in τ and τ ∗ . T o formulate this relation, let us gi ve an auxiliary definition: for an arbitrar y oper ator equation A ◦ ∆ = ∇ ◦ B we say that ∆ is a trivial s olutions if ∆ is of the form ∆ = ∆ ′ ◦ B . Classes o f solutions m odulo triv ial ones will be called non trivi al solution s . Then the follo wing results hold. Theorem 1 (shadows in the ℓ -cov ering) There is a one-to-one corr espondence between nontrivial solutions of the equation ℓ E ◦ R = R ′ ◦ ℓ E , R : κ → κ , and τ -shadows of symmetries linear w . r .t. t he variables q. In a similar way , there is a on e- to-one corr espondence between nontrivial solutions of the equation ℓ ∗ E ◦ S = S ′ ◦ ℓ E , S : κ → ˆ P , and τ -shadows of cosymmetries linear w .r .t. the variables q. Theorem 2 (shadows in the ℓ ∗ -cov ering) Ther e is a one-to-one corr espondence between nontrivial solutions of the equation ℓ E ◦ H = H ′ ◦ ℓ ∗ E , H : ˆ P → κ , and τ ∗ -shadows of s ymmetries linear w .r .t. the variables p. In a s imilar way , there is a one-to-o ne corr espondence between nontrivial solutions of the equation ℓ ∗ E ◦ ˆ R = ˆ R ′ ◦ ℓ ∗ E , ˆ R : ˆ P → ˆ P , and τ ∗ -shadows of cosymmetries linear w .r .t. the variables p. Theorem 3 Let R 1 and R 2 be r ecursion operators for symmetries on E and Φ R denote the τ -shadow corr esponding to R . Then [ [ R 1 , R 2 ] ] n = 0 iff { Φ R 1 , Φ R 1 } = 0 . Similarly , if H 1 and H 2 ar e bivectors then [ [ H 1 , H 2 ] ] s = 0 iff { Φ ∗ H 1 , Φ ∗ H 1 } = 0, wher e Φ ∗ H denotes the τ ∗ - shadow corr esponding to H . 6 In both cas es the curly brackets deno te the super brack et of s hado ws that arises due to oddness of the v ariables q and p . Additional discussion of Theo rem 3 the reader will fin d in Remark 2. Theorem 4 T o any cosymmetry of E there canonically corr esponds a conservation law of L ( E ) . Dua lly , to any symmetry of E there canonically corr esponds a conservation law of L ∗ ( E ) . 2.4 Computational scheme Let locally the equation E be given by the system      F 1 ( x 1 , . . . , x n , . . . , ∂ | σ | u j ∂ x σ , . . . ) = 0 , . . . F r ( x 1 , . . . , x n , . . . , ∂ | σ | u j ∂ x σ , . . . ) = 0 , (10) where j = 1 , . . . , m and | σ | ≤ k . Step 1 consists of writing out defining equations for s ymmetries and cosymmetries of E . Let { u j σ } j ∈ J σ ∈ S be internal coo rdina tes on E , S and J being some sets of (multi)in dices and u j σ correspond ing to ∂ | σ | u j / ∂ x σ . Then any C -field on E is a linear combination of the total derivatives D x i = ∂ ∂ x i + ∑ σ ∈ S , j ∈ J u j σ i ∂ ∂ u j σ , i = 1 , . . . , n . (11) The linearization of E is the matrix o perator with the entries ( ℓ E ) l j = ∑ σ ∈ S ∂ F l ∂ u j σ D σ , j ∈ J , l = 1 , . . . , r . (12) A symmetry ϕ = ( ϕ 1 , . . . , ϕ m ) enjoys the equation ∑ σ ∈ S , j ∈ J ∂ F l ∂ u j σ D σ ( ϕ j ) = 0 , l = 1 , . . . , r , (13) and the correspond ing field is the evolution ary vector field З ϕ = ∑ σ ∈ S , j ∈ J D σ ( ϕ j ) ∂ ∂ u j σ , (14) while the brack et of symmetries ϕ 1 , ϕ 2 is gi ven by { ϕ 1 , ϕ 2 } j = З ϕ 1 ( ϕ j 2 ) − З ϕ 2 ( ϕ j 1 ) , j = 1 , . . . , m . (15) The operator adjoint to (12) is ( ℓ ∗ E ) j l = ∑ σ ∈ S ( − 1 ) | σ | D σ ◦ ∂ F l ∂ u j σ , j ∈ J , l = 1 , . . . , r , (16) and a cosymmetry ψ = ( ψ 1 , . . . , ψ r ) satisfies the equation ∑ σ , l ( − 1 ) | σ | D σ ( ∂ F l ∂ u j σ ψ l ) = 0 , j ∈ J . (17) 7 Step 2. Here we look for closed 1-forms and construct Abelian cov erings associated to them. A horizontal form ω = ∑ i A i x . i is closed if D x α ( A β ) = D x β ( A α ) , 1 ≤ α < β ≤ n . (18) Such a form gi ves rise to a nonlocal variab le w = w ω that satisfies the equations ∂ w ∂ x i = A i , i = 1 , . . . , n . (19) These equation s are compatible on (10) due to (18). Recall that for n = 2 closed 1-forms coincide with conserv ation la ws. The total deriv ativ es lifted to the co vering equation ˜ E are ˜ D x i = D x i + A i ∂ ∂ w , i = 1 , . . . , n . (20) Step 3. At this step we compute a number o f particular sym metries and cosymmetries (using equations (13) and (17) , resp.). Th ey are used to construct cano nical nonlocal variab les on the ℓ ∗ -cov ering ( non local vectors ) and on t he ℓ -covering ( nonlocal forms ), resp., at Step 4. W e also use them as seed elements in series generated by recursion operators. Step 4 consists of construction of the ℓ - and ℓ ∗ -cov erings and introductio n of canonical nonlocal v ariables ov er them (see Step 3). The ℓ -cov ering is obtained by adding to (9) the system of equations ∑ σ ∈ S , j ∈ J ∂ F l ∂ u j σ ∂ q j ∂ x σ = 0 , l = 1 , . . . , r , (21) cf. with Eq. (12), while the ℓ ∗ -cov ering is given by ∑ σ , l ( − 1 ) | σ | ∂ ( ∂ F l ∂ u j σ p l ) ∂ x σ = 0 , j ∈ J , (22) that comes from (16). If ϕ is a sym metry of E then one can intro duce a co vering o ver L ∗ ( E ) described b y the system ∂ ¯ p ∂ x i = ∑ σ , l ∆ l σ , i ( ϕ ) p l σ , i = 1 , . . . , n , (23) where ∆ l σ , i are C -dif ferential operators (see Theorem 4). I n a similar way , to any co symme- try ψ there correspon ds a cov ering ∂ ¯ q ∂ x i = ∑ σ , j ∇ j σ , i ( ψ ) q j σ i = 1 , . . . , n , (24) ∇ j σ , i being C -differ ential operators as well. W e omit here a general description of these operators and refer the read er to the particular case of ou r interest exp osed in Sections 4.4 and 4.5. 8 Step 5. W e now use Th eorems 1 and 2 to con s truct recursion operators an d Hamiltonian and symplectic structures. Let ψ 1 , . . . , ψ s cosymmetries of E . Let us consider the covering ^ L ( E ) ov er L ( E ) with the nonlocal variab les ¯ q 1 , . . . , ¯ q s defined by (24) and lift the operators ℓ E and ℓ ∗ E to this cov ering. Then the following result specifies Theor em 1: Theorem 5 Let Φ = ( Φ 1 , . . . , Φ m ) be a solution o f the equation ˜ ℓ E ( Φ ) = 0 on ^ L ( E ) linear w .r .t. q α σ and ¯ q β : Φ j = ∑ α , σ a α , j σ q α σ + ∑ β b j β ¯ q β . Then the operator R = ∑ σ a α , j σ D σ + ∑ β b j β D − 1 x i ◦ ∑ σ ∇ α σ , i ( ψ β ) D σ , takes shadow of symmetries to shad ows of symmetries. In a similar way , to an y solution Ψ = ( Ψ 1 , . . . , Ψ r ) , Ψ j = ∑ α , σ c α , j σ q α σ + ∑ β d j β ¯ q β ther e correspo nds the op erator S = ∑ σ c α , j σ D σ + ∑ β d j β D − 1 x i ◦ ∑ σ ∇ α σ , i ( ψ β ) D σ that takes shad ows of symmetries to shadows of cosymmetries. In a dual way , co nsider sym metries ϕ 1 , . . . , ϕ s of the equation E and the cov ering ^ L ∗ ( E ) ov er L ∗ ( E ) w ith the nonlocal var iables ¯ p 1 , . . . , ¯ p s defined by (23). Then, lifting ℓ E and ℓ ∗ E , we obtain a similar specification of Theorem 2: Theorem 6 Let Φ = ( Φ 1 , . . . , Φ m ) be a solution of the equation ˜ ℓ E ( Φ ) = 0 on ^ L ∗ ( E ) linear w .r .t. p α σ and ¯ p β : Φ j = ∑ α , σ a α , j σ p α σ + ∑ β b j β ¯ p β . Then the operator H = ∑ σ a α , j σ D σ + ∑ β b j β D − 1 x i ◦ ∑ σ ∆ α σ , i ( ϕ β ) D σ , takes shadow of cosymmetries to shadows of symmetries. In a similar way , to any solu- tion Ψ = ( Ψ 1 , . . . , Ψ r ) , Ψ j = ∑ α , σ c α , j σ p α σ + ∑ β d j β ¯ p β ther e correspo nds the op erator ˆ R = ∑ σ c α , j σ D σ + ∑ β d j β D − 1 x i ◦ ∑ σ ∆ α σ , i ( ϕ β ) D σ that takes shad ows of cosymmetries to shadows of cosymmetries. After finding the operators R , S , H and ˆ R we check conditions (6 ) and (7) and compute necessary Schouten and Nijenhuis brackets. 9 Step 6 The las t step consists of establishing algebraic relations between the inv ariants con- structed abo ve. 3 The matrix version W e consider Eq. (1) in the form u t − u t xx + 3 uu x = 2 u x u xx + uu xxx , (25) i.e., set µ = 0, and, similar to (2), introduce a new v ariable w = α u − u xx , where α is a ne w real constant. Consequently , the initial equation transforms to the system ( w t = − 2 u x w − uw x , w = α u − u xx . (26) W e choose the follo wing v ariables for intern al local coordinates on th e infinite prolon gation of Eq. (26): x , t , w l = ∂ k w ∂ x k , u 0 , k = ∂ k u ∂ t k , u 1 , k = ∂ k + 1 u ∂ x ∂ t k , k = 0 , 1 , . . . Then the total deri vati ves in these coordinates will be of the form D x = ∂ ∂ x + ∑ k ≥ 0 w k + 1 ∂ ∂ w k + ∑ k ≥ 0 u 1 , k ∂ ∂ u 0 , k + ∑ k ≥ 0 D k t ( α u − w ) ∂ ∂ u 1 , k , D t = ∂ ∂ t − ∑ k ≥ 0 D k x ( 2 u 1 , 0 w + uw 1 ) ∂ ∂ w k + ∑ k ≥ 0 u 0 , k + 1 ∂ ∂ u 0 , k + ∑ k ≥ 0 u 1 , k + 1 ∂ ∂ u 1 , k . W e introd uce the follo wing grad ings: | x | = − 1 , | t | = − 2 , | u | = 1 , | w | = 3 , | α | = 2 and extend them in a natural way to all polyn omial functions of the internal coordinates. Then all computatio ns can be restricted to homogene ous components. 3.1 Nonlocal variables In subsequent computations we shall need the follo wing nonlocal v ariables arising from conserv ation la ws and defined by the equations ( s 2 ) x = w , ( s 2 ) t = ( − u 2 α − 2 uw + u 2 1 ) / 2; ( s 3 ) x = uw , ( s 3 ) t = − 2 u 2 w + uu 1 , 1 − u 1 u 0 , 1 ; ( s 6 ) x = w ( uw − u 1 , 1 ) , ( s 6 ) t = ( − u 4 α 2 − 4 u 3 w α + 2 u 2 u 2 1 α + 4 u 2 u 1 w 1 − 4 u 2 w 2 − 4 uu 0 , 2 α + 12 uu 2 1 w + 4 uu 1 , 1 w − u 4 1 + 4 u 1 u 1 , 2 − 4 u 1 u 0 , 1 w ) / 4; 10 ( s 7 ) x = − 2 u 3 w 2 + 60 u 2 w 2 − 36 uu 1 , 1 w + 30 u 0 , 2 w + 27 u 2 1 u 1 , 1 , ( s 7 ) t = − 104 u 4 w α − 28 u 4 w 2 − 132 u 3 u 1 w 1 + 36 u 3 u 1 , 1 α − 40 u 3 w 2 − 18 u 2 u 0 , 2 α − 48 u 2 u 2 1 w − 36 u 2 u 1 u 0 , 1 α + 144 u 2 u 1 , 1 w + 66 u 2 u 0 , 1 w 1 − 78 uu 0 , 2 w − 36 uu 2 1 u 1 , 1 + 18 uu 1 u 1 , 2 − 36 uu 1 u 0 , 1 w + 30 uu 1 , 3 + 18 uu 2 1 , 1 − 36 uu 2 0 , 1 α + 9 u 0 , 2 u 2 1 + 18 u 0 , 2 u 1 , 1 − 30 u 0 , 3 u 1 + 36 u 3 1 u 0 , 1 + 36 u 1 u 1 , 1 u 0 , 1 − 18 u 1 , 2 u 0 , 1 + 18 u 2 0 , 1 w . The variable s i is of grading i an d com putational e xperiment shows that for ev ery grading i = 4 n − 2 + ε , ε = 0 , 1, there exist an s i such that | s i | = i . In addition, we found conserv ation laws of fractional gradings: ( s 1 / 2 ) x = w 1 / 2 , ( s 1 / 2 ) t = − w 1 / 2 u ; ( s − 1 / 2 ) x = w − 3 / 2 ( 6 w α + w 2 ) , ( s − 1 / 2 ) t = w − 3 / 2 ( 4 uw α − uw 2 + 14 w 2 ) ; ( s − 3 / 2 ) x = w − 7 / 2 ( 12 w 2 α 2 + 12 ww 2 α − 2 ww 4 + 7 w 2 2 ) , ( s − 3 / 2 ) t = w − 7 / 2 ( 16 uw 2 α 2 + 16 uww 2 α + 2 uww 4 − 7 uw 2 2 − 124 w 3 α − 32 w 2 w 2 ) ; ( s − 5 / 2 ) x = w − 11 / 2 ( 216 w 3 α 3 + 540 w 2 w 2 α 2 − 180 w 2 w 4 α + 20 w 2 w 6 + 882 ww 2 2 α − 378 ww 2 w 4 − 16 ww 2 3 + 837 w 3 2 ) , ( s − 5 / 2 ) t = w − 11 / 2 ( 320 uw 3 α 3 + 1008 uw 2 w 2 α 2 − 360 uw 2 w 4 α − 20 uw 2 w 6 + 1908 uww 2 2 α + 378 uww 2 w 4 + 16 uww 2 3 − 837 uw 3 2 − 64 u 1 w 2 w 3 α − 1400 w 4 α 2 − 1868 w 3 w 2 α + 580 w 3 w 4 + 64 w 2 w 1 w 3 − 2322 w 2 w 2 2 ) , etc. 3.2 Symmetries A symmetry ϕ = ( ϕ w , ϕ u ) of Eq. (26) mu s t satisfy the linearized equation ( D t ( ϕ w ) + uD x ( ϕ w ) + 2 u 1 ϕ w + 2 wD x ( ϕ u ) + w 1 ϕ u = 0 , ϕ w + D 2 x ( ϕ u ) − α ϕ u = 0 . Direct computation s lead to the follo wing results. (x,t)-indepen dent symmetries. One can observ e two types of symmetries that are indepen - dent of x and t . The first one consists of s ymmetries of inte ger gr adings: ϕ w 1 = w 1 , ϕ u 1 = u 1 ; ϕ w 2 = uw 1 + 2 u 1 w , ϕ u 2 = − u 0 , 1 ; ϕ w 5 = u 2 w 1 α + 2 uww 1 − u 2 1 w 1 − 2 u 1 , 1 w 1 − 4 u 0 , 1 w α , ϕ u 5 = u 2 u 1 α + 2 u 2 w 1 + 6 uu 1 w − u 3 1 + 2 u 1 , 2 − 2 u 0 , 1 w ; 11 ϕ w 6 = u 3 w 1 α + 2 u 2 u 1 w α + 8 u 2 ww 1 − uu 2 1 w 1 + 12 uu 1 w 2 − 2 uu 1 , 1 w 1 + 2 u 0 , 2 w 1 − 2 u 3 1 w + 2 u 1 u 0 , 1 w 1 + 4 u 1 , 2 w − 4 u 0 , 1 w 2 , ϕ u 6 = 2 u 3 w 1 + 6 u 2 u 1 w − 3 u 2 u 0 , 1 α − 6 uu 0 , 1 w − 2 u 0 , 3 + 3 u 2 1 u 0 , 1 , etc. Symmetries of the second type ha ve semi-integer grading s: ϕ w − 3 / 2 = ( − 4 w 2 w 1 α + 4 w 2 w 3 − 18 ww 1 w 2 + 15 w 3 1 ) / ( 2 w 7 / 2 ) , ϕ u − 3 / 2 = − 2 w 1 / ( w 3 / 2 ) ; ϕ w − 5 / 2 = ( − 48 w 4 w 1 α 2 + 80 w 4 w 3 α − 32 w 4 w 5 − 520 w 3 w 1 w 2 α + 320 w 3 w 1 w 4 + 560 w 3 w 2 w 3 + 560 w 2 w 3 1 α − 1820 w 2 w 2 1 w 3 − 2520 w 2 w 1 w 2 2 + 6930 ww 3 1 w 2 − 3465 w 5 1 ) / ( 12 w 13 / 2 ) , ϕ u − 5 / 2 = ( − 12 w 2 w 1 α + 8 w 2 w 3 − 40 ww 1 w 2 + 35 w 3 1 ) / ( 3 w 9 / 2 ) , etc. All symmetries are local and   ϕ γ   = γ . If one adds to the non local setting the v ariables s γ (see above) then an additional series of nonlo cal symmetries arises: ¯ ϕ w − 1 = ( s 1 / 2 w − 1 / 2 ( 4 w 2 w 1 α − 4 w 2 w 3 + 18 ww 1 w 2 − 15 w 3 1 ) + 16 t w 3 w 1 + 2 w ( − 4 w 2 α − 4 ww 2 + 5 w 2 1 )) / ( 16 w 3 ) , ¯ ϕ u − 1 = ( s 1 / 2 w − 1 / 2 w 1 + 4 t u 1 w − 2 w ) / ( 4 w ) ; ¯ ϕ w − 2 = ( 3 s 1 / 2 w − 1 / 2 ( − 48 w 4 w 1 α 2 + 80 w 4 w 3 α − 32 w 4 w 5 − 520 w 3 w 1 w 2 α + 320 w 3 w 1 w 4 + 560 w 3 w 2 w 3 + 560 w 2 w 3 1 α − 1820 w 2 w 2 1 w 3 − 2520 w 2 w 1 w 2 2 + 6930 ww 3 1 w 2 − 3465 w 5 1 ) + 2 s − 1 / 2 w − 1 / 2 w 3 ( − 4 w 2 w 1 α + 4 w 2 w 3 − 18 ww 1 w 2 + 15 w 3 1 ) + 2 w ( 96 w 4 α 2 + 384 w 3 w 2 α − 192 w 3 w 4 − 740 w 2 w 2 1 α + 1364 w 2 w 1 w 3 + 960 w 2 w 2 2 − 5514 ww 2 1 w 2 + 3465 w 4 1 )) / ( 4 w 6 ) , ¯ ϕ u − 2 = ( 3 s 1 / 2 w − 1 / 2 ( − 12 w 2 w 1 α + 8 w 2 w 3 − 40 ww 1 w 2 + 35 w 3 1 ) − 2 s − 1 / 2 w − 1 / 2 w 3 w 1 + 2 w ( 24 w 2 α + 24 ww 2 − 35 w 2 1 )) / w 4 , etc. (x,t)-depend ent symmetries. The first three symmetries that depend on x and t are ϕ w 0 = t ( − uw 1 − 2 u 1 w ) + w , ϕ u 0 = t u 0 , 1 + u ; ϕ w 3 = t ( − u 2 w 1 α − 2 uww 1 + u 2 1 w 1 + 2 u 1 , 1 w 1 + 4 u 0 , 1 w α ) + 2 ( s 2 w 1 + 2 uw α + u 1 w 1 ) , ϕ u 3 = t ( − u 2 u 1 α − 2 u 2 w 1 − 6 uu 1 w + u 3 1 − 2 u 1 , 2 + 2 u 0 , 1 w ) + 2 ( s 2 u 1 + 2 uw − 2 u 1 , 1 ) ; ϕ w 4 = t ( − u 3 w 1 α − 2 u 2 u 1 w α − 8 u 2 ww 1 + uu 2 1 w 1 − 12 uu 1 w 2 + 2 uu 1 , 1 w 1 − 2 u 0 , 2 w 1 + 2 u 3 1 w − 2 u 1 u 0 , 1 w 1 − 4 u 1 , 2 w + 4 u 0 , 1 w 2 ) + 2 ( s 2 ( uw 1 + 2 u 1 w ) + s 3 w 1 + 4 uw 2 − 4 u 1 , 1 w − 2 u 0 , 1 w 1 ) , ϕ u 4 = t ( − 2 u 3 w 1 − 6 u 2 u 1 w + 3 u 2 u 0 , 1 α + 6 uu 0 , 1 w + 2 u 0 , 3 − 3 u 2 1 u 0 , 1 ) + 2 ( − s 2 u 0 , 1 + s 3 u 1 12 + u 3 α + 3 u 2 w − uu 2 1 + 3 u 0 , 2 ) . All these symmetries, except for the first one, are nonlocal (description of the nonlocal v ariable is g iven in Subsection 3.1) and , as abo ve, the subscript denotes the grading. 3.3 Cosymmetries The defining equation for cosymmetries ψ = ( ψ w , ψ u ) is the adjoint to the linearization of (26): ( D t ( ψ w ) + uD x ( ψ w ) − u 1 ψ w − ψ u = 0 , 2 wD x ( ψ w ) + w 1 ψ w − D 2 x ( ψ u ) + α ψ u = 0 . Similar to symmetries, we consider two types of cosymmetries. (x,t)-indepen dent cosymmetries. They are local and may be of inte ger and semi-inte ger gradings: ψ w 3 = 1 , ψ u 3 = − u 1 ; ψ w 4 = u , ψ u 4 = u 0 , 1 ; ψ w 7 = u 2 α + 2 uw − u 2 1 − 2 u 1 , 1 , ψ u 7 = − u 2 u 1 α − 2 u 2 w 1 − 6 uu 1 w + u 3 1 − 2 u 1 , 2 + 2 u 0 , 1 w ; ψ w 8 = u 3 α + 4 u 2 w − uu 2 1 − 2 uu 1 , 1 + 2 u 0 , 2 + 2 u 1 u 0 , 1 , ψ u 8 = − 2 u 3 w 1 − 6 u 2 u 1 w + 3 u 2 u 0 , 1 α + 6 uu 0 , 1 w + 2 u 0 , 3 − 3 u 2 1 u 0 , 1 etc. and ψ w 3 / 2 = w − 1 / 2 , ψ u 3 / 2 = 0; ψ w 1 / 2 = ( 4 w 2 α + 4 ww 2 − 5 w 2 1 ) / ( 4 w 3 w 1 / 2 ) , ψ u 1 / 2 = 2 w 1 / ( ww 1 / 2 ) ; ψ w − 1 / 2 = ( 48 w 4 α 2 + 160 w 3 w 2 α − 64 w 3 w 4 − 280 w 2 w 2 1 α + 448 w 2 w 1 w 3 + 336 w 2 w 2 2 − 1848 ww 2 1 w 2 + 1155 w 4 1 ) / ( 48 w 6 w 1 / 2 ) , ψ u − 1 / 2 = ( 12 w 2 w 1 α − 8 w 2 w 3 + 40 ww 1 w 2 − 35 w 3 1 ) / ( 3 w 4 w 1 / 2 ) , etc. Similar to the case of symmetries, when one adds nonlocal variables s γ an additional series of nonlocal cosymmetries arises: ¯ ψ w 1 = ( 3 s 1 / 2 w − 1 / 2 ( − 4 w 2 α − 4 ww 2 + 5 w 2 1 ) − 2 s − 1 / 2 w − 1 / 2 w 3 + 96 t w 3 − 10 ww 1 ) / ( 96 w 3 ) , ¯ ψ u 1 = ( − s 1 / 2 w − 1 / 2 w 1 − 4 t u 1 w + 2 w ) / ( 4 w ) ; 13 ¯ ψ w 0 = ( 3 s 1 / 2 w − 1 / 2 ( 48 w 4 α 2 + 160 w 3 w 2 α − 64 w 3 w 4 − 280 w 2 w 2 1 α + 448 w 2 w 1 w 3 + 336 w 2 w 2 2 − 1848 ww 2 1 w 2 + 1155 w 4 1 ) + 4 s − 1 / 2 w − 1 / 2 w 3 ( 4 w 2 α + 4 ww 2 − 5 w 2 1 ) + 12 s − 3 / 2 w − 1 / 2 w 6 + 2 w ( 400 w 2 w 1 α − 276 w 2 w 3 + 1366 ww 1 w 2 − 1155 w 3 1 )) / ( 16 w 6 ) , ¯ ψ u 0 = ( 3 s 1 / 2 w − 1 / 2 ( 12 w 2 w 1 α − 8 w 2 w 3 + 40 ww 1 w 2 − 35 w 3 1 ) + 2 s − 1 / 2 w − 1 / 2 w 3 w 1 + 2 w ( − 24 w 2 α − 24 ww 2 + 35 w 2 1 )) / w 4 ; ¯ ψ w − 1 = ( 27 s 1 / 2 w − 1 / 2 ( 320 w 6 α 3 + 2240 w 5 w 2 α 2 − 1792 w 5 w 4 α + 512 w 5 w 6 − 5040 w 4 w 2 1 α 2 + 16128 w 4 w 1 w 3 α − 6912 w 4 w 1 w 5 + 12096 w 4 w 2 2 α − 14592 w 4 w 2 w 4 − 8832 w 4 w 2 3 − 81312 w 3 w 2 1 w 2 α + 52800 w 3 w 2 1 w 4 + 181632 w 3 w 1 w 2 w 3 + 42944 w 3 w 3 2 + 60060 w 2 w 4 1 α − 274560 w 2 w 3 1 w 3 − 569712 w 2 w 2 1 w 2 2 + 1021020 w w 4 1 w 2 − 425425 w 6 1 ) + 18 s − 1 / 2 w − 1 / 2 w 3 ( 48 w 4 α 2 + 160 w 3 w 2 α − 64 w 3 w 4 − 280 w 2 w 2 1 α + 448 w 2 w 1 w 3 + 108 s − 3 / 2 w − 1 / 2 w 6 ( 4 w 2 α + 4 ww 2 − 5 w 2 1 ) + 40 s − 5 / 2 w − 1 / 2 w 9 + 2 w ( 69120 w 4 w 1 α 2 + 336 w 2 w 2 2 − 1848 ww 2 1 w 2 + 1155 w 4 1 ) + 3416 0 w 4 w 5 + 624816 w 3 w 1 w 2 α − 349880 w 3 w 1 w 4 − 605776 w 3 w 2 w 3 − 665280 w 2 w 3 1 α + 1991880 w 2 w 2 1 w 3 + 2772036 w 2 w 1 w 2 2 − 7636860 w w 3 1 w 2 + 3828825 w 5 1 )) / ( 72 w 9 ) , ¯ ψ u − 1 = ( 3 s 1 / 2 w − 1 / 2 ( 240 w 4 w 1 α 2 − 320 w 4 w 3 α + 128 w 4 w 5 + 2240 w 3 w 1 w 2 α − 1344 w 3 w 1 w 4 − 2240 w 3 w 2 w 3 − 2520 w 2 w 3 1 α + 7728 w 2 w 2 1 w 3 + 10416 w 2 w 1 w 2 2 − 29568 ww 3 1 w 2 + 15015 w 5 1 ) + 4 s − 1 / 2 w − 1 / 2 w 3 ( 12 w 2 w 1 α − 8 w 2 w 3 + 40 ww 1 w 2 − 35 w 3 1 ) + 12 s − 3 / 2 w − 1 / 2 w 6 w 1 + 2 w ( − 384 w 4 α 2 − 1536 w 3 w 2 α + 768 w 3 w 4 + 3360 w 2 w 2 1 α − 5732 w 2 w 1 w 3 − 3840 w 2 w 2 2 + 23422 ww 2 1 w 2 − 15015 w 4 1 )) / w 7 , etc. (x,t)-depend ent cosymmetries. All them are nonlocal: ψ w 5 = t ( − u 2 α − 2 uw + u 2 1 + 2 u 1 , 1 ) + 2 ( s 2 + u 1 ) , ψ u 5 = t ( u 2 u 1 α + 2 u 2 w 1 + 6 uu 1 w − u 3 1 + 2 u 1 , 2 − 2 u 0 , 1 w ) + 2 ( − s 2 u 1 − 2 uw + 2 u 1 , 1 ) ; ψ w 6 = t ( − u 3 α − 4 u 2 w + uu 2 1 + 2 uu 1 , 1 − 2 u 0 , 2 − 2 u 1 u 0 , 1 ) + 2 ( s 2 u + s 3 − 2 u 0 , 1 ) , ψ u 6 = t ( 2 u 3 w 1 + 6 u 2 u 1 w − 3 u 2 u 0 , 1 α − 6 uu 0 , 1 w − 2 u 0 , 3 + 3 u 2 1 u 0 , 1 ) + 2 ( s 2 u 0 , 1 − s 3 u 1 − u 3 α − 3 u 2 w + uu 2 1 − 3 u 0 , 2 ) , etc. 3.4 Nonlocal forms Recall that nonlocal forms are nonlocal v ariables of a special type on the ℓ -coverin g. The ℓ -cov ering itself is obtained from Eq. (26) by adding two additional equations ( q w t = − uq w x − 2 u x q w − 2 wq u x − w x q u , q u xx = α q u − q w , 14 where q w and q u are ne w odd variables. The total deri vati ves on the ℓ -cover ing are ˜ D x = D x + ∑ k ≥ 0 q w k + 1 ∂ ∂ q w k + ∑ k ≥ 0 q u 1 , k ∂ ∂ q u 0 , k + ∑ k ≥ 0 ˜ D k t ( α q u − q w ) ∂ ∂ q u 1 , k , ˜ D t = D t − ∑ k ≥ 0 ˜ D k x ( uq w 1 + 2 u 1 q w + 2 wq u 1 + w 1 q u ) ∂ ∂ q w k + ∑ k ≥ 0 q u 0 , k + 1 ∂ ∂ q u 0 , k + ∑ k ≥ 0 q u 1 , k + 1 ∂ ∂ q u 1 , k . The nonlocal form Q i associated to a cosymmetry ψ i = ( ψ w i , ψ u i ) (see Subsection 3.3) is defined by the equations ˜ D x ( Q i ) = ψ w i q w , ˜ D t ( Q i ) = − u ψ w i q w + ( D x ( ψ u i ) − 2 w ψ w i ) q u − ψ u i q u 1 . 3.5 Nonlocal vectors Dually to nonlo cal forms, nonloca l vector s arise as s pecial nonlo cal variab les on the ℓ ∗ - cov ering ass ociated to symmetries of the initial equation . The ℓ ∗ -cov ering is the exten si on of Eq. (26) by two ne w equations ( p w t = − u p w x + u x p w + p u , p u xx = 2 w p w x + w x p w + α p u , where p = ( p w , p u ) is a new od d v ariable. The total deriav ativ es are given by ˜ D x = D x + ∑ k ≥ 0 p w k + 1 ∂ ∂ p w k + ∑ k ≥ 0 p u 1 , k ∂ ∂ p u 0 , k + ∑ k ≥ 0 ˜ D k t ( 2 w p w 1 + w 1 p w + α p u ) ∂ ∂ p u 1 , k , ˜ D t = D t + ∑ k ≥ 0 ˜ D k x ( − u p w 1 + u 1 p w + p u ) ∂ ∂ p w k + ∑ k ≥ 0 p u 0 , k + 1 ∂ ∂ p u 0 , k + ∑ k ≥ 0 p u 1 , k + 1 ∂ ∂ p u 1 , k . The nonlocal vector P i associated to a symmetry ϕ = ( ϕ w , ϕ u ) (see Subsection 3.2 ) is defined by the equations ˜ D x ( P i ) = ϕ w i p w , ˜ D t ( P i ) = − ( u ϕ w i + 2 w ϕ u i ) p w − D x ( ϕ u i ) p u + ϕ u i p u 1 . 3.6 Recursion operators for symmetries The defining equations for these operators are ( ˜ D t ( R w ) + u ˜ D x ( R w ) + 2 u 1 R w + 2 w ˜ D x ( R u ) + w 1 R u = 0 , R w + ˜ D 2 x ( R u ) − α R u = 0 (see Theorem 5), where the total deriv ativ es are those described in Subsection 3.4. The follo wing two solutions are essential: R w − 1 = ( Q 3 / 2 w − 1 / 2 ( − 4 w 2 w 1 α + 4 w 2 w 3 − 18 ww 1 w 2 + 15 w 3 1 ) − 4 q w 2 w 2 + 10 q w 1 ww 1 + q w ( 4 w 2 α + 8 ww 2 − 15 w 2 1 )) / w 3 , 15 R u − 1 = 4 ( − Q 3 / 2 w − 1 / 2 w 1 + q w ) / w and R w 3 = Q 3 w 1 + q u 1 w 1 + 2 q u w α , R u 3 = Q 3 u 1 + q w u − q u 1 , 1 + q u w . The correspon ding operators are of the form R − 1 = 1 8 w 3  h 12 w − 1 / 2 D − 1 x ◦ w − 1 / 2 / 2 − 4 w 2 D 2 x + 10 ww 1 D x + h 8 0 − 4 w 2 w 1 w − 1 / 2 D − 1 x ◦ w − 1 / 2 / 2 + 4 w 2 0  and R 3 =  w 1 D − 1 x w 1 D x + 2 w α u 1 D − 1 x + u − D xt + w  , where h 12 = − 4 w 2 w 1 α + 4 w 2 w 3 − 18 ww 1 w 2 + 15 w 3 1 , h 8 = 4 w 2 α + 8 ww 2 − 15 w 2 1 . All other solutions obtained in our computatio ns corresponded to operators that are generated by the two abo ve. 3.7 Symplectic structures Symplectic structures, as it follo ws from Theorem 5, are defined by the equations ( − ˜ D t ( S w ) − u ˜ D x ( S w ) + u 1 S w + S u = 0 , − 2 w ˜ D x ( S w ) − w 1 S w + ˜ D 2 x ( S u ) − α S u = 0 , where the total deri vati ves were defined in Subsection 3.4. Here are the simplest no ntrivial solutions: S w 2 = Q 3 / 2 w − 1 / 2 , S u 2 = − q u ; S w 5 = Q 3 + q u 1 , S u 5 = − Q 3 u 1 − q w u + q u 1 , 1 − q u w ; S w 6 = Q 4 + Q 3 u − q u 0 , 1 + q u 1 u − q u u 1 , S u 6 = − Q 4 u 1 + Q 3 u 0 , 1 − q w u 2 − q u 0 , 2 + q u ( − u 2 α − 2 uw + u 2 1 ) with the correspond ing symplectic operators S 2 =  w − 1 / 2 D − 1 x ◦ w − 1 / 2 / 2 0 0 − 1  , S 5 =  D − 1 x D x − u 1 D − 1 x − u D xt − w  , S 6 =  D − 1 x ◦ u + uD − 1 x − D t + uD x − u 1 − u 1 D − 1 x ◦ u + u 0 , 1 D − 1 x − u 2 − D 2 t − u 2 α − 2 uw + u 2 1  . 16 3.8 Hamiltonian structures The equations that should be satisfied by a Hamiltonian operator (see Theorem 6) are ( ˜ D t ( H w ) + u ˜ D x ( H w ) + 2 u 1 H w + 2 w ˜ D x ( H u ) + w 1 H u = 0 , H w + ˜ D 2 x ( H u ) − α H u = 0 , where the total deriv ativ es are from Subsection 3.5. In particular , we found the follo wing solutions: H w − 3 = − p w 3 + p w 1 α , H u − 3 = p w 1 ; H w − 2 = 2 p w 1 w + p w w 1 , H u − 2 = − p u ; H w 1 = P 1 w 1 − 2 p w ww 1 + p u 1 w 1 + 2 p u w α , H u 1 = P 1 u 1 − 2 p w 1 uw − p w ( uw 1 + 2 u 1 w ) − p u 1 , 1 + p u w . The correspon ding Hamiltonian operators are H − 3 =  − D 3 x + α D x 0 D x 0  , H − 2 =  2 wD x + w 1 0 0 − 1  , H 1 =  w 1 D − 1 x ◦ w 1 − 2 ww 1 w 1 D x + 2 w α u 1 D − 1 x ◦ w 1 − 2 uwD x − ( uw 1 + 2 u 1 w ) − D xt + w  . 3.9 Recursion operators for cosymmetries By Theorem 6, the equation to find recursion operators for cosymmetries are ( − ˜ D t ( ˆ R w ) − u ˜ D x ( ˆ R w ) + u 1 ˆ R w + ˆ R u = 0 , − 2 w ˜ D x ( ˆ R w ) − w 1 ˆ R w + ˜ D 2 x ( ˆ R u ) − α ˆ R u = 0 with the total deri vati ves giv en in Sub sect ion 3.5. One of solutions is presented belo w: ˆ R w 3 = P 1 − 2 p w w + p u 1 , ˆ R u 3 = − P 1 u 1 + 2 p w 1 uw + p w ( uw 1 + 2 u 1 w ) + p u 1 , 1 − p u w The correspon ding recursion operator is ˆ R 3 =  D − 1 x ◦ w 1 − 2 w D x − u 1 D − 1 x ◦ w 1 + 2 uwD x + uw 1 + 2 u 1 w D xt − w  . 17 3.10 Interrelation W e expose here basic facts on structural relations between the above described in variants. The main one is the follo wing Theorem 7 Recursion operator R 3 and Hamiltonian operator H − 3 constitute a P oisson– Nijenhuis structur e on the Camassa–Holm equation. Consequently , all the operato rs R n 3 ◦ H − 3 ar e Hamiltonian and pair- w ise compatible. In particular , R 3 ◦ H − 3 = α H − 2 . Pr oof The proof consists of direct computations using the results and techniques of R ef. [8] 4 . A visual presentation of how symmetries are distrib uted ov er gradings is giv en in T a- ble 2. Ho w to pro ve locality of the first two series of s ymmetries will be discussed in Sec- T able 2 Distributi on of symm etries o ver gradings Gradings − 5 2 − 2 − 3 2 − 1 − 1 2 0 1 2 3 4 5 6 7 8 Local positi ve • • • • Local nega tiv e • • Nonlocal positi ve • • • • • Nonlocal nega tiv e • • tion 5. Similar presentation for cosymmetries see in T able 3. T able 3 Distributi on of cosymmetries over gradi ngs Gradings − 3 2 − 1 − 1 2 0 1 2 1 3 2 2 3 4 5 6 7 8 Local positi ve • • • • Local nega tiv e • • • • Nonlocal positi ve • • • Nonlocal nega tiv e • • • The action of Hamiltonian and recursion operators for symmetries (up to a constant multiplier) is gi ven in Diagram (27): . . . R 3 / / ϕ − 5 / 2 R 3 / / R − 1 u u ϕ − 3 / 2 R 3 / / R − 1 s s 0 ϕ 1 R 3 / / R − 1 w w ϕ 2 R 3 / / R − 1 v v . . . R − 1 v v . . . ψ 1 / 2 H − 2 = = { { { { { { { { { H − 3 O O ψ 3 / 2 H − 2 ? ?           H − 3 O O ψ 3 H − 2 A A          H − 3 O O ψ 4 H − 2 A A          H − 3 O O ψ 7 H − 2 A A          H − 3 O O . . . (27) The action of recursion operator s for cosymmetries and simplectic structures is similar . W e shall no w prove commutati vity of the local hierarchies. 4 W e are pret ty sure that the pair ( H − 3 , R − 1 ) is also a Poisson–Nije nhuis s tructure and generates an infinite nega tiv e hierarchy of Hamiltonian operators, but could not prove this fact because of computer capacit y limitat ions. 18 Lemma 1 The symmet ry ¯ ϕ 3 is a positive hered itary symmetry , i.e. , its action on local sym - metries , ϕ 7→ { ¯ ϕ 3 , ϕ } , coincides , up to a multiplier , w ith the one of the r ecursion opera- tor R 3 . The only symmetries that vanish under this action are ϕ − 3 / 2 and ϕ 1 . In a similar way , the symmetry ϕ − 1 is a ne gative her editary symmetry and the only symmetry that is taken to zer o under its action is ϕ 1 . A direct coro llary of this result is Theorem 8 Local positive and ne gative symmetries form co mmutative hier arc hies. 4 The scalar version Let us consider no w the Camassa–Holm equation i n its initial form (1) with µ = 0 and, similar to the matrix case introduce a ne w real param eter α : α u t − u t xx + 3 α uu x = 2 u x u xx + uu xxx . (28) For the intern al coord inates we choose the function s u l = ∂ l u ∂ x l , u l , k = ∂ k + l u ∂ x l ∂ t k , l = 0 , 1 , 2 , k ≥ 1 . The total deri vati ves in these coordinates are of the form D x = ∂ ∂ x + 2 ∑ l = 0 u l + 1 ∂ ∂ u l + ∑ k ≥ 1  u 1 , k ∂ ∂ u 0 , k + u 2 , k ∂ ∂ u 1 , k + D k t ( u 3 ) ∂ ∂ u 2 , k  , D t = ∂ ∂ t + 2 ∑ l = 0 u l , 1 ∂ ∂ u l + 2 ∑ l = 0 ∑ k ≥ 1 u l , k + 1 ∂ ∂ u l , k , where u 3 = ( α u 0 , 1 − u 2 , 1 + 3 α uu 1 − 2 u 1 u 2 ) / u . The equation becomes homogeneou s if assign the follo wing grad ings: | x | = − 1 , | t | = − 2 , | u | = 1 , | α | = 2 . 4.1 Nonlocal variables W e introduce nonlocal v ariable associated to conserv at ion laws of the equation at hand: ( s 2 ) x = u α − u 2 , ( s 2 ) t = ( − 3 u 2 α + 2 uu 2 + u 2 1 ) / 2; ( s 3 ) x = u ( u α − u 2 ) , ( s 3 ) t = − 2 u 3 α + 2 u 2 u 2 + uu 1 , 1 − u 0 , 1 u 1 ; ( s 6 ) x = u 3 α 2 − 2 u 2 u 2 α − uu 1 , 1 α + uu 2 2 + u 1 , 1 u 2 , ( s 6 ) t = ( − 9 u 4 α 2 + 12 u 3 u 2 α + 6 u 2 u 2 1 α + 4 u 2 u 1 , 1 α − 4 u 2 u 2 2 − 8 uu 0 , 1 u 1 α − 4 uu 0 , 2 α − 4 uu 2 1 u 2 + 4 uu 1 u 2 , 1 − 4 uu 1 , 1 u 2 + 4 u 0 , 1 u 1 u 2 − u 4 1 + 4 u 1 u 1 , 2 ) / 4; ( s 7 ) x = 60 u 4 α 2 − 116 u 3 u 2 α − 12 u 2 u 2 1 α − 40 u 2 u 1 , 1 α + 56 u 2 u 2 2 − 6 uu 0 , 1 u 1 α 19 + 28 uu 0 , 2 α + 12 uu 2 1 u 2 + 10 uu 1 u 2 , 1 + 40 uu 1 , 1 u 2 + 2 uu 2 , 2 + 2 u 2 0 , 1 α − 4 u 0 , 1 u 1 u 2 − 2 u 0 , 1 u 2 , 1 − 30 u 0 , 2 u 2 + 27 u 2 1 u 1 , 1 , ( s 7 ) t = − 144 u 5 α 2 + 240 u 4 u 2 α + 48 u 3 u 2 1 α + 124 u 3 u 1 , 1 α − 96 u 3 u 2 2 − 156 u 2 u 0 , 1 u 1 α − 124 u 2 u 0 , 2 α − 48 u 2 u 2 1 u 2 + 8 u 2 u 1 u 2 , 1 − 88 u 2 u 1 , 1 u 2 + 28 u 2 u 2 , 2 − 56 uu 2 0 , 1 α + 112 uu 0 , 1 u 1 u 2 + 38 uu 0 , 1 u 2 , 1 + 78 uu 0 , 2 u 2 − 36 uu 2 1 u 1 , 1 + 18 uu 1 u 1 , 2 + 18 uu 2 1 , 1 + 30 uu 1 , 3 − 18 u 2 0 , 1 u 2 + 36 u 0 , 1 u 3 1 + 36 u 0 , 1 u 1 u 1 , 1 − 18 u 0 , 1 u 1 , 2 + 9 u 0 , 2 u 2 1 + 18 u 0 , 2 u 1 , 1 − 30 u 0 , 3 u 1 . 4.2 Symmetries A symmetry ϕ must satisfy the linearized equation α D t ( ϕ ) − D 2 x D t ( ϕ ) − uD 3 x ( ϕ ) − 2 u 1 D 2 x ( ϕ ) + ( 3 α u − 2 u 2 ) D x ( ϕ ) + ( 3 α u 1 − u 3 ) ϕ = 0 . W e computed tw o types of symmetries. Everywh ere below the subscrip t was chosen in a way to correspo nd the enum eration taken for the matrix case. ( x , t ) -independent symmetries. W e present the first four of them: ϕ 1 = u 1 , ϕ 2 = − u 0 , 1 , ϕ 5 = 3 u 2 u 1 α − 4 uu 0 , 1 α − 2 uu 1 u 2 + 2 uu 2 , 1 + 2 u 0 , 1 u 2 − u 3 1 + 2 u 1 , 2 , ϕ 6 = 2 u 3 u 1 α − 11 u 2 u 0 , 1 α − 2 u 2 u 1 u 2 + 2 u 2 u 2 , 1 + 6 uu 0 , 1 u 2 + 3 u 0 , 1 u 2 1 − 2 u 0 , 3 . All these symmetries are local. ( x , t ) -dependent symmetries. The first three ( x , t ) -depen dent s ymmetries are ϕ 0 = t u 0 , 1 + u , ϕ 3 = 2 s 2 u 1 + t ( − 3 u 2 u 1 α + 4 uu 0 , 1 α + 2 uu 1 u 2 − 2 uu 2 , 1 − 2 u 0 , 1 u 2 + u 3 1 − 2 u 1 , 2 ) + 4 ( u 2 α − uu 2 − u 1 , 1 ) , ϕ 4 = − 2 s 2 u 0 , 1 + 2 s 3 u 1 + t ( − 2 u 3 u 1 α + 11 u 2 u 0 , 1 α + 2 u 2 u 1 u 2 − 2 u 2 u 2 , 1 − 6 uu 0 , 1 u 2 − 3 u 0 , 1 u 2 1 + 2 u 0 , 3 ) + 2 ( 4 u 3 α − 3 u 2 u 2 − uu 2 1 + 3 u 0 , 2 ) The only local symmetry in this series is ϕ 1 . 4.3 Cosymmetries The defining equation for cosymmetries is − α D t ( ψ ) + D 2 x D t ( ψ ) + uD 3 x ( ψ ) + u 1 D 2 x ( ψ ) + ( u 2 − 3 α u ) D x ( ψ ) = 0 . 20 ( x , t ) -independent cosymmetries. These cosymmetries are local. The first four of them are ψ 3 = 1 , ψ 4 = u , ψ 7 = 3 u 2 α − 2 uu 2 − u 2 1 − 2 u 1 , 1 , ψ 8 = 5 u 3 α − 4 u 2 u 2 − uu 2 1 − 2 uu 1 , 1 + 2 u 0 , 1 u 1 + 2 u 0 , 2 . ( x , t ) -dependent cosymmetries. These cosymmetries are nonlocal, except for the first one: ¯ ψ 2 = − t u , ¯ ψ 5 = 2 s 2 + t ( − 3 u 2 α + 2 uu 2 + u 2 1 + 2 u 1 , 1 ) + 2 u 1 , ¯ ψ 6 = 2 s 2 u + 2 s 3 + t ( − 5 u 3 α + 4 u 2 u 2 + uu 2 1 + 2 uu 1 , 1 − 2 u 0 , 1 u 1 − 2 u 0 , 2 ) − 4 u 0 , 1 , etc. 4.4 Nonlocal forms Nonlocal forms arise in the ℓ -cov ering which is given b y the equ ation α q t − q xxt − uq xxx − 2 u x q xx + ( 3 α u − 2 u xx ) q x + ( 3 α u x − u xxx ) q = 0 with the total deri vati ves ˜ D x = D x + 2 ∑ l = 0 q l + 1 ∂ ∂ q l + ∑ k ≥ 1  q 1 , k ∂ ∂ q 0 , k + q 2 , k ∂ ∂ q 1 , k + D k t ( q 3 ) ∂ ∂ q 2 , k  , ˜ D t = D t + 2 ∑ l = 0 q l , 1 ∂ ∂ q l + 2 ∑ l = 0 ∑ k ≥ 1 q l , k + 1 ∂ ∂ q l , k , where q 3 = ( α q 0 , 1 − q 2 , 1 + ( 3 u 1 α − u 3 ) q + ( 3 u α − 2 u 2 ) q 1 − 2 u 1 q 2 ) / u . T o any cosym metry ψ i we associate a nonlocal form Q i defined by the equations ˜ D x ( Q i ) = ( α ψ i − D 2 x ( ψ i )) q , ˜ D t ( Q i ) = (( u 2 − 3 u α ) ψ i + uD 2 x ( ψ i )) q u + ( u 1 ψ i − uD x ( ψ i )) q 1 + u ψ i q 2 − D x ( ψ i ) q 0 , 1 + ψ i q 1 , 1 . 4.5 Nonlocal vectors Nonlocal vecto rs arise in the ℓ ∗ -cov ering. The latter is the extension of the initial equ ation by the equation − α p t + p xxt + u p xxx + u x p xx + ( u xx − 3 α u ) p x = 0 with the total deri vati ves ˜ D x = D x + 2 ∑ l = 0 p l + 1 ∂ ∂ p l + ∑ k ≥ 1  p 1 , k ∂ ∂ p 0 , k + p 2 , k ∂ ∂ p 1 , k + D k t ( p 3 ) ∂ ∂ p 2 , k  , 21 ˜ D t = D t + 2 ∑ l = 0 p l , 1 ∂ ∂ p l + 2 ∑ l = 0 ∑ k ≥ 1 p l , k + 1 ∂ ∂ p l , k , where p 3 = (( 3 u α − u 2 ) p 1 − u 1 p 2 + α p 0 , 1 − p 2 , 1 ) / u . T o any symm etry ϕ i there correspon ds a nonlocal vector P i defined by the relations ( P i ) x = ( ϕ i α − D 2 x ( ϕ i )) p , ( P i ) t = (( u 2 − 3 u α ) ϕ i + u 1 D x ( ϕ i ) + uD 2 x ( ϕ i )) p − uD x ( ϕ i ) p 1 + u ϕ i p 2 − D x ( ϕ i ) p 0 , 1 + ϕ i p 1 , 1 . 4.6 Recursion operators for symmetries The defining equation for these operator s is α ˜ D t ( R ) − ˜ D 2 x ˜ D t ( R ) − u ˜ D 3 x ( R ) − 2 u 1 ˜ D 2 x ( R ) + ( 3 α u − 2 u 2 ) ˜ D x ( R ) + ( 3 α u 1 − u 3 ) R = 0 where the total deriv ativ es are those presented in Subsection 4.4. W e consider here two nontri vial solutions: R − 1 = ( 4 Q 3 / 2 ( u α − u 2 ) − 5 / 2 u ( 2 u 2 u 1 α 2 + uu 0 , 1 α 2 − 4 uu 1 u 2 α − uu 2 , 1 α − u 0 , 1 u 2 α + 2 u 1 u 2 2 + u 2 u 2 , 1 ) − 4 q 2 u 2 ( u α − u 2 ) − 1 + 4 q 1 ( u α − u 2 ) − 3 u ( − 2 u 2 u 1 α 2 − uu 0 , 1 α 2 + 4 uu 1 u 2 α + uu 2 , 1 α + u 0 , 1 u 2 α − 2 u 1 u 2 2 − u 2 u 2 , 1 ) + 2 q (( u α − u 2 ) − 3 ( 4 u 2 u 2 1 α 2 + 4 uu 0 , 1 u 1 α 2 − 8 uu 2 1 u 2 α − 4 uu 1 u 2 , 1 α + u 2 0 , 1 α 2 − 4 u 0 , 1 u 1 u 2 α − 2 u 0 , 1 u 2 , 1 α + 4 u 2 1 u 2 2 + 4 u 1 u 2 u 2 , 1 + u 2 2 , 1 ) + 2 u 2 α ( u α − u 2 ) − 1 )) / u 2 and R 3 = Q 3 u 1 − q 1 , 1 − q 2 u − q 1 u 1 + q ( 2 u α − u 2 ) . The operator correspond ing to the second one is R 3 = α u 1 D − 1 x − D xt − uD 2 x − u 1 D x + 2 u α − u 2 . The first operator is too complicated to present it here. 4.7 Symplectic structures The equation defining symplectic structures is − α ˜ D t ( S ) + ˜ D 2 x D t ( S ) + u ˜ D 3 x ( S ) + u 1 ˜ D 2 x ( S ) + ( u 2 − 3 α u ) ˜ D x ( S ) = 0 , where the total deri vati ves are from Subsection 4.4. The simplest solutions are S 5 = Q 3 , S 6 = Q 4 + Q 3 u − q 0 , 1 − q 1 u while the correspond ing operators ha ve the form S 5 = α D − 1 x , S 6 = D − 1 x ◦ ( α u − u 2 ) + α uD − 1 x − D t − uD x . 22 4.8 Hamiltonian structures Hamiltonian structures are defined by the equation α ˜ D t ( H ) − ˜ D 2 x ˜ D t ( H ) − u ˜ D 3 x ( H ) − 2 u 1 ˜ D 2 x ( H ) + ( 3 α u − 2 u 2 ) ˜ D x ( H ) + ( 3 α u 1 − u 3 ) H = 0 , where the total deri va tives are giv en in Sub s ection 4.5. W e present here three solutions. T wo of them are local and the third one is nonlocal: H − 3 = p 1 , H − 2 = − p 0 , 1 − p 1 u + pu 1 , H 1 = P 1 u 1 − p 1 , 2 − p 2 , 1 u + p 0 , 1 u α − p 2 u 0 , 1 + p 1 u ( − u α + u 2 ) + p ( − uu 1 α + u 0 , 1 α + u 1 u 2 ) The correspon ding operators are H − 3 = D x , H − 2 = − D t − uD x + u 1 , H 1 = u 1 D − 1 x ◦ (( − 2 uu 1 α − u 0 , 1 α + 2 u 1 u 2 + u 2 , 1 ) / u ) − uD 2 x D t − D x D 2 t + u α D t − u 0 , 1 D 2 x + u ( − u α + u 2 ) D x − 4 uu 1 α + uu 3 + 3 u 1 u 2 + u 2 , 1 . 4.9 Recursion operators for cosymmetries These recursion operators are defined by the equation − α ˜ D t ( ˆ R ) + ˜ D 2 x D t ( ˆ R ) + u ˜ D 3 x ( ˆ R ) + u 1 ˜ D 2 x ( ˆ R ) + ( u 2 − 3 α u ) ˜ D x ( ˆ R ) = 0 , where the total deri vati ves are giv en in Subsection 4.5 . One of the solutions is ˆ R 3 = P 1 + p 1 , 1 + p 2 u − 2 u α + u 2 to which the operator ˆ R 3 = D − 1 x ◦ (( − 2 uu 1 α − u 0 , 1 α + 2 u 1 u 2 + u 2 , 1 ) / u ) + D xt + uD 2 x − 2 u α + u 2 correspond s. 5 Conclusion W e finish this paper with a number of remarks. Remark 1 First of all, let us stress again what has been done abov e. W e treated the C amassa– Holm equatio n directly , without artificial assumptions abou t its e vo lution or pseudo-e v olu- tion nature. The pass from the original form (1) to system (26) (which is not in ev olution form) was done by technical reasons only . On this w ay , we fo und an infinite family of pair- wise compatible Hamiltonian structures, recursion operators for symm etries and cosym me- tries, and s ymplectic operators. These structures lead to ex is tence of two commutativ e series of local symmetries and conserv ation laws and thus the equation is integrable. Remark 2 Sev eral comments are wor th to be made in relation to Theorem 3. T wo cases must be distinguished: local and nonlocal. In bo th cases the problem reduces to r econstruction of nonlocal shado ws up to symmetries and subsequent computation of the bracket. 23 Local case. This case is simple and the result is as follo ws: any shadow in the ℓ -cover ing correspond ing to a recursion operator can be canonically lifted up to a symmetry of the cov ering equation and the commutator of these symmetries correspond s to the Nijenhuis brack et of the operators. A similar result is valid for the shadows in the ℓ ∗ -cov ering cor - responding t o l ocal bi vectors (in this case the commutator of symmetries is related to the Schouten brack et). Nonlocal case. The case of nonlo cal operator s is much more complicated and rests on the problem of how to co mmute nonlocal shad ows. Seemingly there exists no natur al definition of such a commutator , but in R ef. [9] we proposed a procedure both to reconstruct and commute shado ws . A weak poin t of this procedure is that it is based on rather cumbersome and intuiti vely not obviou s notion of shado w equiv alence. Dealing with equiv alence classes of shado ws necessitates elimination of certain kind of nonlocal var iables th at we call pseudo- constants an d that are intrinsically related to the covering at hand. This n ot always simple to do and we plan to simplify and clarify the procedu re. Remark 3 Of course, all in varian ts of the Camassa–Holm equation in its initial form can be obtain ed from those computed for the s ystem by simple transformatio ns. E.g., to obtain symmetries of Eq. (28) from t hose of (26) one should substitute in the u -com ponent the v ariables w l by D l x ( u α − u 2 ) . On the other way , any cosymmetry in the scalar case can be constructed directly from the corresponding cosymmetry of Eq. (26) by changing the v ariables w l in the w -compon ent by D l x ( u α − u 2 ) . Similar transformations are applicable to other structures. Remark 4 Computation of symmetries and cosymmetries in the matrix representation can be simplified using the follo wing ob s erv ation: Propositio n 1 The following correspon dence between the components of symmetri es and cosymmetries of matrix equation (26) is valid : ψ w k  D t + uD x − u 1 / / ψ u k  · ( − 1 ) / / ϕ u k − 2  α − D 2 x / / ϕ w k − 2 . Remark 5 Now we shall sho w how to prove locality of symmetry hierarchies. Let us intro- duce a ne w v ariable v = w 2 . Then Eq. (26) transforms to ( v t = − u x v − uw x , v 2 = α u − u xx , (29) i.e., the first equation acquires potential form 5 . This means that for any symmetry ϕ = ( ϕ v , ϕ u ) the form ω ϕ = ϕ v d x − ( u ϕ v + v ϕ u ) d t i s a conserv ation law of Eq. (29). Compar - ing gradings, it is easily checked that all conserv ation laws of the form ω ϕ are tri vial and consequently ϕ v lies in the image of D x for an y symmetry ϕ . From this f act it follo ws that the action of recursion operators is local. 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