Bi-Hamiltonian nature of the equation u tx = u xy u y − u yy u x
aa r X i v : . [ m a t h - ph ] F e b Bi-Hamiltonian nature of the equation u tx = u xy u y − u yy u x V. Ovsienko
Abstract
We study non-linear integrable partial differential equations naturally arising as bi-Hamiltonian Eulerequations related to the looped cotangent Virasoro algebra. This infinite-dimensional Lie algebra (con-structed in [16]) is a generalization of the classical Virasoro algebra to the case of two space variables.Two main examples of integrable equations we obtain are quite well known. We show that the rela-tion between these two equations is similar to that between the Korteweg-de Vries and Camassa-Holmequations.
Mathematics Subject Classification (2000) :
Key Words :
Generalized Virasoro algebra,integrability, bi-Hamiltonian systems.
The differential equation u tx = u xy u y − u yy u x , (1.1)where u = u ( t, x, y ) and where u x , u y , etc. are the partial derivatives, is a nice example of anon-linear integrable model. This equation is quite well known and appears in the Mart´ınezAlonzo-Shabat “universal hierarchy” (see [13], formula (8)).The main purpose of this note is to show that this equation (coupled together with anotherdifferential equation, see formula (3.6) below) naturally appears as a bi-Hamiltonian vector field(in particular, the variable t plays the rˆole of time while x, y are space variables). More precisely,we will show that this equation is an Euler equation on the space dual to the “looped cotangentVirasoro algebra” introduced in [16]. This, in particular, implies its integrability in the (weakalgebraic) sense of existence of a hierarchy of first integrals in involution.The bi-Hamiltonian approach to the same Lie algebra has already been considered in [16]and led to another non-linear differential equation: f t = f x ∂ − x f y − f y u + c ∂ − x f yy , which can be rewritten without non-local terms: u tx = u xx u y − u xy u x + c u yy , (1.2)after the substitution f = u x . Here c ∈ R is an arbitrary constant (the “central charge”). Notethat this equation is also a quite well known integrable system (see [5, 6] and also [4]) thatappears both in differential geometry and hydrodynamic.quations (1.2) and (1.1) look alike but they are not equivalent to each other. We will showthat the relation between these equations is similar to that between the classical Korteweg-deVries equation (KdV) and the Camassa-Holm equation (CH). Recall that both KdV and CHare bi-Hamiltonian systems on the dual of the Virasoro algebra, see [12, 3, 11, 14, 10]. Aninteresting “tri-Hamiltonian” viewpoint was suggested in [15], in order to establish a certainduality between KdV and CH. Equations (1.2) and (1.1) are dual in the same sense.This paper fits into the general framework due to V.I. Arnold, see [1]. Non-linear partialequations are viewed as Euler equations on the dual of a Lie algebra (for instance, the Liealgebra of vector fields). This approach explains the geometric meaning of the equations: everyEuler equation describes geodesics of some left-invariant metric on the corresponding group (ofdiffeomorphisms).The bi-Hamiltonian Euler equations are of special interest. Most of the known bi-Hamiltoniannon-linear partial differential equations (KdV,CH, etc.) are of dimension 1 + 1 (i.e., contain onlyone space variable). Equations (1.1) and (1.2) provide with examples of such equations in the(2 + 1)-dimensional case. In this section we recall the general construction of pairs of compatible Poisson structures onthe space dual to a Lie algebra. We also give the standard construction of bi-Hamiltonian vectorfields on this space, due to F. Magri [12].Let a be a (finite-dimensional) Lie algebra, the canonical Lie-Poisson(-Berezin-Kirillov-Kostant) bracket on a ∗ is given by { F, G } ( m ) = h [ d m F, d m G ] , m i , (2.1)where m ∈ a ∗ and where d m F and d m G are the differentials of F and G at m understood aselements of a , namely dF m ∈ ( a ∗ ) ∗ ∼ = a . This Poisson structure is linear, i.e., the space of linearfunctions equipped with the bracket (2.1) is a Lie subalgebra of C ∞ ( a ∗ ) (isomorphic to a ).Given a skew-symmetric bilinear form ω : a ∧ a → R , one defines another Poisson structureon a : { F, G } ω ( m ) = ω ( d m F, d m G ) . (2.2)This structure is with constant coefficients, i.e., the bracket of two linear functions is a constantfunction on a ∗ .Two Poisson structures are called compatible (or a Poisson pair) if their linear combinationis again a Poisson structure. The following simple fact is well known (see, e.g., [2], Section 5.2). Proposition 2.1.
The Poisson structures (2.1) and (2.2) are compatible if an only if ω is a2-cocycle on a . The simplest example of a constant Poisson structure (2.2) corresponds to the case wherethe 2-cocycle ω is trivial (i.e., a coboundary). Every such structure is of the following form. Fixa point m ∈ a ∗ and set ω ( x, y ) = h m , [ x, y ] i . (2.3)It worth noticing that one can understand this particular case of constant Poisson structure on a ∗ as the most general one. Indeed, it suffices to replace a by its central extension.Every function H on a ∗ defines two vector fields that we denote X H and X ωH on a ∗ : the firstone is Hamiltonian with respect to the linear structure (2.1) and is given by X H ( m ) = ad ∗ d m H m, (2.4)2hile the vector field X ωH is Hamiltonian with respect to the constant bracket (2.2). In theparticular case (2.3), one has explicitly X ωH ( m ) = ad ∗ d m H m . (2.5)Given two compatible Poisson structures, a vector field which is Hamiltonian with respect tothe both structures is called bi-Hamiltonian. The usual way to construct bi-Hamiltonian vectorfields on a ∗ is as follows. Consider the following 1-parameter family of Poisson structures { , } λ = { , } ω − λ { , } , (parameterized by λ ∈ R ). Assume that H is a Casimir function of this bracket, i.e., one has { H, F } λ = 0 , for all F ∈ C ∞ ( a ∗ ) . Assume also that H is written in a form of a series H = H + λ H + λ H + · · · (2.6)One immediately obtains the following facts:1. the function H is a Casimir function of { , } ω ;2. the Hamiltonian vector field corresponding to H k are bi-Hamiltonian, namely X H k = X ωH k +1 , for all k ;3. all the functions H k are in involution with respect to the both Poisson structures, indeed,for k ≤ ℓ one has { H k , H ℓ } = { H k +1 , H ℓ } ω = { H k +1 , H ℓ − } = · · · = 0 , and therefore are first integrals of every vector field X H k .Let us summarize the method. To construct an integrable hierarchy, one chooses a function H which is a Casimir function of the constant Poisson structure { , } ω ; one then considersits Hamiltonian vector field, X H , with respect to the Lie-Poisson structure. This vector fieldis again Hamiltonian with respect to the constant Poisson structure, with some Hamiltonianfunction H , so that one has: X H = X ωH . One then iterates the procedure to find H , H , etc. In this section we recall the definition [16] of the looped cotangent Virasoro algebra. We alsodescribe its coadjoint representation.Let us start with the definition of the classical Virasoro algebra. Consider the Lie algebra,Vect( S ), of vector fields on the circle: f ( x ) ∂∂x where f ∈ C ∞ ( S ) and x is a coordinate on S ,we assume x ∼ x + 2 π . To simplify the formulæ, we will identify Vect( S ) with C ∞ ( S ); theLie bracket in Vect( S ) is then given by[ f, g ] = f g x − f x g. S ). It is defined on the space Vect( S ) ⊕ R , the commutator being given by[( f, α ) , ( g, β )] = (cid:18) f g x − f x g, Z S f g xxx dx (cid:19) . (3.1)Note that the constants α and β do not enter the right hand side of the above formula sincethey belong to the center of Vir.The Virasoro algebra was found by Gelfand and Fuchs [8], the constant term in the righthand side of (3.1) is called the Gelfand-Fuchs cocycle. This Lie algebra plays an importantrˆole in mathematical physics, essentially because of the applications of its representations toconformal field theory, but also because of its applications to integrable systems.The dual space, Vect( S ) ∗ , is the space of distributions. One often considers only a subspace,Vect( S ) ∗ reg , called the “regular dual” (cf. [9]). As a vector space, this regular dual is, again,isomorphic to C ∞ ( S ), the pairing h ., . i : Vect( S ) ⊗ C ∞ ( S ) → R being give by (cid:28) f ( x ) ∂∂x , a ( x ) (cid:29) := Z S f ( x ) a ( x ) dx. The regular dual to the Virasoro algebra is Vir ∗ reg = C ∞ ( S ) ⊕ R ; the coadjoint action of Vir onits regular dual is: ad ∗ ( f,α ) ( a, c ) = ( f a x + 2 f x a + c f xxx , . This formula easily follows from (3.1) and the definition of ad ∗ , see [9]. Note that the constant c is preserved by the action, it is therefore a parameter called the central charge. Remark 3.1.
The Virasoro algebra is, indeed, exceptional. The reason is that the Lie algebrasof vector fields on a manifold of dimension ≥
2, has no central extensions, cf. [7]. The problemof generalization of the Virasoro algebra is an interesting subject studied by many authors.The looped cotangent Virasoro algebra [16] is a generalization of Vir in the case of twovariables. We consider the 2-torus T and define a Lie algebra structure on the space g = C ∞ ( T ) ⊕ C ∞ ( T ) ⊕ R . given by the commutator fa ( α, α ′ ) , gb ( β, β ′ ) = f g x − f x gf b x + 2 f x b − g a x − g x a (cid:16) Z S × S f g xxx dxdy , Z S × S ( f b y − g a y ) dxdy (cid:17) (3.2)where ( x, y ) are the usual coordinates on T and where f, g, a, b are smooth functions in x, y ;the constants α, α ′ , β, β ′ ∈ R are elements of the center. Note that, unlike the Virasoro algebra,the center of g is two-dimensional. Remark 3.2.
One notices that the quotient-algebra g / R (by the center) is the loop algebrawith coefficients in the semidirect sum Vect( S ) ⋉ Vect( S ) ∗ reg . The dependence in y -variablein this quotient-algebra is somehow trivial. The second 2-cocycle in (3.2), however, makes thisdependence in y non-trivial. Note also that this cocycle is rather similar to the Kac-Moodycocycle. 4e will need the coadjoint representation of g and the notion of regular dual space. Considerthe pairing h ., . i : g ⊗ g → R * fa ( α , α ) , gb ( α , α ) + = Z S × S ( f b + g a ) dxdy + α β + α β , that identifies g with a part of its dual space: g ֒ → g ∗ , we call this subspace the regular dualspace of g and denote it by g ∗ reg . The coadjoint action of g on g ∗ reg can be easily calculated: c ad ∗ fa ( α , α ) gb ( c , c ) = f g x − f x g + c f y f b x + 2 f x b − a x g − a g x + c f xxx + c a y (0 , . (3.3)Note that the center R ⊂ g acts trivially.The Lie algebra g is infinite-dimensional. In order to define the brackets (2.1) and (2.2) inthis case, we consider only the space of so-called pseudodifferential polynomials on g ∗ reg : H ( f, a ) = Z S × S h (cid:0) f, a, f x , a x , f y , a y , ∂ − x f, ∂ − x a, ∂ − y f, ∂ − y a, f xy , a xy , . . . (cid:1) dxdy, where h is a polynomial and f, a, f x , a x , f y , a y , ∂ − x f, . . . are understood as independent variables.The differential d m H is replaced by the standard variational derivative: d ( f, a ) H := ( δ a H, δ f H )understood as element of g / R . The Lie-Poisson structure (2.1) then makes sense on g ∗ reg andthe Hamiltonian vector fields are again given by (2.4). Example 3.3.
Recall that the Euler-Lagrange equation provides an explicit formula for varia-tional derivatives. For instance, one has δ a H = h a − ∂ x ( h a x ) − ∂ y (cid:0) h a y (cid:1) − ∂ − x (cid:16) h ∂ − x a (cid:17) − ∂ − y (cid:16) h ∂ − y a (cid:17) +( ∂ x ) ( h a xx ) + ∂ x ∂ y (cid:0) h a xy (cid:1) + ( ∂ y ) (cid:0) h a yy (cid:1) ± · · · where, as usual, h u means the partial derivative ∂h∂a , similarly h a x = ∂h∂a x , etc..One of course should be careful with the definition of the non-local operators ∂ − x and ∂ − y .We use the expression ( ∂ − x f )( x, y ) = Z x f ( ξ, y ) dξ − Z π f ( x, y ) dx, and similarly for ∂ − y .We refer to [3] for further details on Hamiltonian formalism on infinite-dimensional (func-tional) Lie algebras. 5 .1 Calculating the bi-Hamiltonian equations Let us fix the following point of g ∗ reg : m = ( f ( x ) , a ( x ) , c , c ) = (1 , , , c ) , (3.4)with arbitrary c ∈ R , and consider the constant Poisson structure (2.2) corresponding to thecoboundary (2.3). The Hamiltonian vector field X ωH with the Hamiltonian H is then given by f t = − ( δ a H ) x + c ( δ a H ) y a t = 2 ( δ a H ) x − ( δ f H ) x + c ( δ f H ) y . The limit case c → ∞ corresponds to the following structure f t = ( δ a H ) y a t = ( δ f H ) y . (3.5)We are ready to formulate our main result. Theorem 3.4.
The following system on g ∗ reg u tx = u xy u y − u yy u x v tx = 2 ( u yy v x − u xy v y ) + u y v xy − u x v yy − u yy u x + 2 u xy u y ) (3.6) is bi-Hamiltonian with respect to the standard Lie-Poisson structure on g reg , together with (3.5),where f = u y and a = v y .Proof. The simplest class of Casimir functions of this constant Poisson structure are linearcombinations of the functionals R f dxdy and R u dxdy . We will choose the Casimir function H ( f, a ) = Z S × S ( a − f ) dxdy. The Hamiltonian vector field, X H , with respect to the Lie-Poisson structure defines the followingvector field f t = f x a t = 2 f x + a x . (3.7)Indeed, one obviously has ( δ a H , δ f H ) = ( − ,
1) (understood as an element of g / R ) and onethen applies the definition (2.4).One thus looks for a function H ( f, a ) on g ∗ reg such that its Hamiltonian vector field withrespect to the constant Poisson structure satisfies X ωH = X H , which leads to the following system of equation on the variational derivatives δ f H and δ u H : − ( δ a H ) x + c ( δ a H ) y = f x δ a H ) x − ( δ f H ) x + c ( δ f H ) y = 2 f x + a x . − ∂ x + c ∂ y , one shows by a simple straightforward calculation that following function: H ( f, a ) = Z S × S (cid:0) Λ − ( f x ) a + Λ − ( f x ) f − Λ − ( f xx ) f (cid:1) dxdy (3.8)is a solution of the above system.The Hamiltonian vector field X H is then as follows f t = Λ − ( f x ) f x − Λ − ( f xx ) f + c Λ − ( f xy ) a t = Λ − ( f x ) a x + 2 Λ − ( f xx ) a − Λ − ( a xx ) f − − ( a x ) f x − (cid:0) Λ − ( f xx ) − Λ − ( f xxx ) (cid:1) f − (cid:0) Λ − ( f x ) − Λ − ( f xx ) (cid:1) f x c Λ − ( f xxxx ) + c (cid:0) Λ − ( a xy ) + 2 Λ − ( f xy ) − − ( f xxy ) (cid:1) In the same way as in [14], we substitute to this equation f = Λ( u ) and a = Λ( v ) and rewrite itin the following form: − u tx + c u ty = c ( u xy u x − u xx u y ) + c u xy − v tx + c v ty = c (2 u xx v y − u xy v x + u x v xy − u y v xx )2 (cid:0) u xx − Λ − ( u xxx ) (cid:1) ( u x − c u y ) + 4 (cid:0) u x − Λ − ( u xx ) (cid:1) ( u xx − c u xy ) c u xxxx + c (cid:0) v xy + 2 u xy − − ( u xxy ) (cid:1) . (3.9)It is very easy to check that, in the limit case c → ∞ , this system coincides with (3.6) withexchanged notation for the variables ( x, y ) ↔ ( y, x ).Theorem 3.4 implies the existence of an infinite series of first integrals in involution for theequation (1.1), as well as of an infinite hierarchy of commuting flows, see [16], Section 5.5. Remark 3.5.
1) The special case c = 0 in (3.4) was considered in the details in [16]. This caseis related to the equation (1.2).2) One can also choose a non-zero value of the first central charge c in (3.4). This will,however, only change the second equation in (3.9).3) Consider the first equation in (3.9). The term c u xy can be removed by the transformation u u − c c x . Furthermore, the coordinate transformation ( x, y ) → ( x, y + c x ) leads to thefollowing family: u tx = c ( u xy u y − u yy u x ) + u xx u y − u xy u x depending on c as parameter. This family gives one an interpolation between the equations (1.1)and (1.2), but with zero central charge. Acknowledgments . I am grateful to G. Misiolek, E. Ferapontov, A. Reiman and C. Rogerfor enlightening discussions. A part of this work was done during the meeting “M´ecaniqueg´eom´etrique”, November 2007 at CIRM; I am pleased to thank the organizers A. Constantinand B. Kolev. 7 eferences [1] V.I. Arnold, Mathematical methods of classical mechanics. Third edition, Nauka, Moscow,1989.[2] V. Arnold, A. Givental,
Symplectic geometry, in: Encycl. of Math. Sci., Dynamical Systems,4, Springer-Verlag, 1990, 1–136.[3] L.A Dickey, Soliton equations and Hamiltonian systems. Second edition, Adv. Ser. in Math.Phys., 26. World Scientific, 2003.[4] M. Dunajski,
A class of Einstein-Weyl spaces associated to an integrable system of hydro-dynamic type , J. Geom. Phys. :1 (2004) 126–137.[5] E.V Ferapontov, K.R. Khusnutdinova, On the integrability of (2+1) -dimensional quasilinearsystems , Comm. Math. Phys. :1 (2004) 187–206.[6] E.V Ferapontov, K.R. Khusnutdinova,
Hydrodynamic reductions of multi-dimensional dis-persionless PDEs: the test for integrability , J. Math. Phys. :6 (2004) 2365-2377.[7] D.B Fuks, Cohomology of infinite-dimensional Lie algebras. Consultants Bureau, New York,1986.[8] I.M. Gelfand, D.B Fuks, Cohomologies of the Lie algebra of vector fields on the circle , Func.Anal. Appl. :4 (1968) 92–93.[9] A.A. Kirillov, Infinite-dimensional Lie groups: their orbits, invariants and representations.The geometry of moments , Lecture Notes in Math., , 101–123, Springer, Berlin, 1982.[10] B. Khesin, G. Misiolek,
Euler equations on homogeneous spaces and Virasoro orbits , Adv.Math. :1 (2003) 116–144.[11] B. Khesin, V. Ovsienko,
The super Korteweg-de Vries equation as an Euler equation , Func.Anal. Appl. :4 (1987) 81–82.[12] F. Magri, A simple model of the integrable Hamiltonian equation , J. Math. Phys. :5 (1978)1156–1162.[13] L. Mart´ınez Alonso, A.B. Shabat, Hydrodynamic reductions and solutions of the universalhierarchy , Theoret. and Math. Phys. :2 (2004), 1073–1085.[14] G. Misiolek,
A shallow water equation as a geodesic flow on the Bott-Virasoro group , J.Geom. Phys. :3 (1998) 203–208.[15] P. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutionshaving compact support , Phys. Rev. E (3) :2 (1996), 1900–1906.[16] V. Ovsienko, C. Roger, Looped cotangent Virasoro algebra and non-linear integrable systemsin dimension273