Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type

Purely nonlocal Hamiltonian formalism for systems of hydr odynamic type John Gibbons ∗ , P aolo Lorenzoni ∗∗ , Andrea Raimondo ∗ * Departmen t of Mathematics, Imperial C olle ge 180 Queen’ s G ate, London SW7 2AZ, UK j.gibbo ns@imperia l.ac.uk, a.raimond o@imperial .ac.uk ** Dipartimento di Matematica e Applicazi oni Uni ver sit ` a di Milano-Bicocca V ia Roberto Cozzi 53, I-20125 Milano, Italy paolo.lo renzo ni@unimib .it Abstract W e study purely no nloca l Hamilton ian structures for syste ms of hydro dynamic type. In the case of a semi-Hamiltonian syste m, we sho w that such structure s are relate d to quadra tic e xp ansion s of t he diagona l metri cs naturall y as sociate d with the syst em. Introduction In the last t hree d ecades many papers hav e been de v oted t o Hami ltonian struct ures for sys- tems of hydrodynamic type: u i t = v i j ( u ) u j x , i = 1 , . . . , n. (0.1) The starting point of t his research was the paper [7] (see al so [8]) where Dubrovin and Novikov introduced an important class of local Hamiltonian structures, called Hamiltonian structur es of hydr odynami c type . Such operators ha ve the form P ij = g ij ( u ) d dx − g is Γ j sk ( u ) u k x , (0.2) where g ij are the contrav ariant components of a flat pseudo-Riemannian metric and Γ j sk are the Christo ff el sym bols of the a ssociated L e vi-Civita connection. Non local extensions of t he 1 bracket (0.2), related to met rics of constant curvature, were consid ered by Ferapontov and Mokhov in [ 11]; further generalizations, of the form P ij = g ij d dx − g is Γ j sk u k x + X α ε α ( w α ) i k u k x  d dx  − 1 ( w α ) j h u h x , ε α = ± 1 , (0.3) were considered by Ferapontov in [10]. Here we ha v e defined  d dx  − 1 = 1 2 Z x −∞ dx − 1 2 Z + ∞ x dx, and the index α can take values over a finite, infinite or e ven continuous set. In the case det g ij 6 = 0 , the operator (0.3) defines a Poiss on structure if and only if the tensor g ij defines a pseudo-Riemannian metric, t he coef ficients Γ j sk are the Christoffel symbol s of its Levi- Ci vita connection ∇ , and the af finors w α satisfy the conditions [ w α , w α ′ ] = 0 , (0.4a) g ik ( w α ) k j = g j k ( w α ) k i , (0.4b) ∇ k ( w α ) i j = ∇ j ( w α ) i k , (0.4c) R ij k h = X α ε α n ( w α ) i k ( w α ) j h − ( w α ) j k ( w α ) i h o , (0.4d) where R ij k h = g is R j sk h are th e component s of the Riemann curv ature tens or of the metri c g . As observ ed by Ferapontov , if the sum over α goes from 1 to m , then these equation s are the Gauss-Mainardi-Codazzi equat ions for an n -dim ensional subm anifold N with flat normal connection embedded in a ( n + m ) -dimensio nal pseudo-euclidean space. It is import ant to point out that the metric g , in general, does n ot uniquely fix the Hamil- tonian structure (0.3); this arbitrariness i s related to the fact that the affinors satisfying the Gauss-Peterson-Mainardi-Codazzi equations, and hence the corresponding embedding, m ay not be unique. More precisely , let the set of affinors w = { w α } sati sfy equations (0.4) for a gi ven metric g , wi th associated Le vi-Ci vita connection ∇ and curv ature t ensor R . Let another set of af finors W = { W β } satisfy the con ditions [ W β , w α ] = 0 , [ W β , W β ′ ] = 0 , g ik ( W β ) k j = g j k ( W β ) k i , ∇ k ( W β ) i j = ∇ j ( W β ) i k , X β ǫ β n ( W β ) i k ( W β ) j h − ( W β ) j k ( W β ) i h o = 0 , ǫ β = ± 1 . It th en follows trivially that the union w ∪ W als o satisfies equations (0.4) with the same 2 metric g . Thi s means that the expression P ij = g ij d dx − g is Γ j sk u k x + X α ε α ( w α ) i k u k x  d dx  − 1 ( w α ) j h u h x + X β ǫ β ( W β ) i k u k x  d dx  − 1 ( W β ) j h u h x is a Poisson biv ector; further , it is compatible with (0.3) as one can easily check by rescaling W → λW . In this way one obtains a pencil P 2 − λP 1 of Hamiltonian structures, where P 2 is the Hamil tonian structure of Ferapontov type (0.3 ) and P 1 is a pur ely nonlocal Hami ltonian structure of the form P ij 1 = X β ǫ β ( W β ) i k u k x  d dx  − 1 ( W β ) j h u h x . (0.5) Summarizing, the st udy of the arbitrariness of the nonl ocal tail in Ham iltonian operators of Ferapontov type (0.3) leads u s to consider nonl ocal operators of the form (0.5). Apart from purely nonlocal structures associated with flat metrics considered by Mokhov in [18] and a few isol ated examples [3, 22], such operators have not b een consi dered much in the literature. A more systematic stud y of som e such struct ures, a s ubclass of M okhov’ s, was giv en recently in [19], where it was sh own t hat s ome such purely nonlocal Poisson operators can appear as inv erses of local symplectic o perators. The aim of this paper i s to st udy Poisso n operators of the form (0.5) in greater generality , to find the conditi ons they m ust satisfy , and to construct s ome classes of examples. In the case of a semihamiltonian hierarchy , there is a remarkable relationship between the symmetries W α appearing in the operator , and t he metrics naturally a ssociated with the hierarchy - these are expanded as quadratic forms i n the W α . In particular we find such operators associat ed with reductio ns of the Benney equ ations, and with semisimp le Frobenius manifolds admitting a superpotential. 1 Pur ely nonlocal Ham iltonian f or malism of hydr odynamic type Let us con sider purely nonlo cal operators of th e form (0.5); the aim of this section is t o determine necessary and s uffi cient conditions for (0.5) t o be a Poisson op erator , namely to satisfy the ske w symmetry condition and the Jacobi id entity . For t his purpose it is mo re con venient to consider , instead of the differential operator (0.5), its associated brack et { F , G } = Z δ F δ u i P ij δ G δ u j dx = Z Z δ F δ u i ( x ) Π ij ( x, y ) δ G δ u j ( y ) dy dx, (1.1) 3 where we ha ve introduced Π ij ( x, y ) = X α ǫ α W α ( x ) i s u s x ν ( x − y ) W α ( y ) j l u l y , (1.2) and ν ( x − y ) = 1 2 sgn ( x − y ) . (1.3) The functi onals F and G app earing in the brack et are of local type, not depending on the x -deri vativ es of u . Th e ske w symmetry of th is bracket is trivially sati sfied, so we need t o find the conditions on the W α such that the Jacobi identity {{ G, H } , F } + {{ F , G } , H } + { { H , F } , G } = 0 holds for e very F, G, H . Pr oposition 1.1 Suppose tha t the affinors W α ar e n ot de generate an d th at they have a simple spectrum, then the bracke t (1.1) with (1.2) sa tisfies the Jacobi identity if an d only i f the following conditions ar e satisfied: ( W β ) m q ∂ m ( W α ) k l + ( W β ) m l ∂ m ( W α ) k q + ( W α ) k m ∂ l ( W β ) m q + ( W α ) k m ∂ q ( W β ) m l = = ( W α ) m q ∂ m ( W β ) k l + ( W α ) m l ∂ m ( W β ) k q + ( W β ) k m ∂ l ( W α ) m q + ( W β ) k m ∂ q ( W α ) m l (1.4) [ W α , W β ] = 0 , ∀ α , β , (1.5) X α ǫ α  ( W α ) i k ( W α ) j h − ( W α ) i h ( W α ) j k  = 0 . (1.6) Remark 1 The first two conditions say that the flows u i t α = ( W α ) i j u j x commute. Pr oof As not iced in [8], in order to p rove the Jacobi identity we can restrict our attenti on to linear functionals of the type F = Z f i ( x ) u i dx, G = Z g i ( x ) u i dx H = Z h i ( x ) u i dx. W e ha ve {{ G, H } , F } = Z Z δ { G, H } δ u m ( t ) Π mi ( t, x ) δ F δ u i ( x ) dx dt = Z Z δ { G, H } δ u m ( t ) Π mi ( t, x ) f i ( x ) dx dt, (1.7) 4 where δ { G, H } δ u m ( t ) = δ δ u m ( t ) Z Z δ G δ u j ( y ) Π j k ( y , z ) δ H δ u k ( z ) dy dz = Z Z g j ( y ) δ Π j k ( y , z ) δ u m ( t ) h k ( z ) dy dz . Hence, (1.7) can be reduced to {{ G, H } , F } = Z Z Z f i ( x ) g j ( y ) h k ( z ) S ij k ( x, y , z ) dx dy dz , where we ha ve introduced the quantity S j k i ( y , z , x ) = Z δ Π j k ( y , z ) δ u m ( t ) Π mi ( t, x ) dt. In this way , the Jacobi identity reads Z Z Z f i ( x ) g j ( y ) h k ( z )  S j k i ( y , z , x ) + S k ij ( z , x, x ) + S ij k ( x, y , z )  dx dy dz = 0 , This has to be satisfied for e very function f i , g j , h k , so that we must require S j k i ( y , z , x ) + S k ij ( z , x, x ) + S ij k ( x, y , z ) = 0 . (1.8) Let us consider the quantities S j k i ( y , z , x ) more explicitly . W e hav e δ Π j k ( y , z ) δ u m ( t ) = X α ǫ α  ∂ W α ( y ) j p ∂ u m ( t ) δ ( y − t ) u p y + W α ( y ) j m δ ′ ( y − t )  W α ( z ) k q u q z + W α ( y ) j p u p y " ∂ W α ( z ) k q ∂ u m ( t ) δ ( z − t ) u q z + W α ( z ) k m δ ′ ( z − t ) #) ν ( y − z ) , 5 and so S j k i ( y , z , x ) = X α ǫ α ∂ W α ( y ) j p ∂ u m ( y ) W α ( z ) k q ! Π mi ( y , x ) u p y u q z ν ( y − z ) + X α ǫ α W α ( y ) j m W α ( z ) k q ! ∂ Π mi ( y , x ) ∂ y u q z ν ( y − z ) + X α ǫ α W α ( y ) j p ∂ W α ( z ) k q ∂ u m ( z ) ! Π mi ( z , x ) u p y u q z ν ( y − z ) + X α ǫ α W α ( y ) j p W α ( z ) k m ! ∂ Π mi ( z , x ) ∂ z u p y ν ( y − z ) . Now we ev alu ate: ∂ Π mi ( y , x ) ∂ y = X α ǫ α ∂ W α ( y ) m p ∂ u l ( y ) W α ( x ) i s ! u s x u p y u l y ν ( y − x ) + X α ǫ α W α ( y ) m p W α ( x ) i s ! u s x u p y y ν ( y − x ) + X α ǫ α W α ( y ) m p W α ( x ) i s ! u s x u p y δ ( y − x ) , so that we can write S j k i ( y , z , x ) = A k j i q pls ( z , y , x ) u s x u p y u l y u q z ν ( y − x ) ν ( y − z ) + A j k i pq ls ( y , z , x ) u s x u p y u q z u l z ν ( z − x ) ν ( y − z ) + B k j i q ps ( z , y , x ) u s x u p y y u q z ν ( y − x ) ν ( y − z ) + B j k i pq s ( y , z , x ) u s x u p y u q z z ν ( z − x ) ν ( y − z ) + B k j i q ps ( z , y , x ) u s x u p y u q z δ ( y − x ) ν ( y − z ) + B j k i pq s ( y , z , x ) u s x u p y u q z δ ( z − x ) ν ( y − z ) , 6 Here we ha ve introduced the notatio n: A k j i q pls ( z , y , x ) = X α ǫ α W α ( z ) k q ∂ m W α ( y ) j p ! X α ǫ α W α ( y ) m l W α ( x ) i s ! + + X α ǫ α W α ( z ) k q W α ( y ) j m ! X α ǫ α ∂ l W α ( y ) m p W α ( x ) i s ! , (1.9) B k j i q ps ( z , y , x ) = X α ǫ α W α ( z ) k q W α ( y ) j m ! X α ǫ α W α ( y ) m p W α ( x ) i s ! . (1.10) Cyclically permuting wit h respect to i, j, k and x, y , z , and then rearranging t he terms, it is possible to re wri te condition (1.8) in the form h A ij k splq ( x, y , z ) − A k j i q pls ( z , y , x ) i u s x u p y u l y u q z ν ( x − y ) ν ( y − z ) +  B ij k spq ( x, y , z ) − B k j i q ps ( z , y , x )  u s x u p y y u q z ν ( x − y ) ν ( y − z ) +  B ij k spq ( x, y , z ) − B k j i q ps ( z , y , x )  u s x u p y u q z δ ( x − y ) ν ( y − z ) + ( cyclic permutations ) = 0 , and this reduces to the foll owing conditions (for if th ese are satisfied, then all the oth ers vanish identi cally): A ij k splq ( x, y , z ) + A ij k slpq ( x, y , z ) = A k j i q pls ( z , y , x ) + A k j i lpq s ( z , y , x ) , (1.11) B ij k spq ( x, y , z ) = B k j i q ps ( z , y , x ) (1.12) B ij k spq ( x, x, z ) + B ij k psq ( x, x, z ) = B k j i q ps ( z , x, x ) + B k j i q sp ( z , x, x ) (1.13) that foll ow i mmediately from (1.4), (1.5), (1.6). The con verse is also true if the affinors W α are not degenerate, a n d hav e a sim ple spectrum.  2 Semi-Ham iltonian systems Let us cons ider now t he important class of diagonalizable, s emi-Hamiltoni an systems o f hydrodynamic type. These sys tems were introduced by Tsare v in [23], and correspon d to the 7 class of systems which are integrable by the generalized hodograph method . More p recisely , a diagonal system of hydrodynamic type u i t = v i ( u ) u i x , (2.1) is called semi-Hamiltonia n [23] if t he coef ficients v i ( u ) satisfy the system of equations ∂ j  ∂ k v i v i − v k  = ∂ k  ∂ j v i v i − v j  ∀ i 6 = j 6 = k 6 = i, (2.2) where ∂ i = ∂ ∂ λ i . The v i are usu ally called characteristic velocities . Equations (2.2) are t he integrability c o nditions for three diffe rent systems: the first, giv en by ∂ j w i w i − w j = ∂ j v i v i − v j , (2.3) which provides the c h aracteristic velocities w i ( u ) of the symmetries of (0.1): u i τ = w i ( u ) u i x i = 1 , ..., n ; the second is a system whose solutions are the conserved densities H ( u ) of (0.1): ∂ i ∂ j H − Γ i ij ∂ i H − Γ j j i ∂ j H = 0 , Γ i ij = ∂ j v i v j − v i , (2.4) and the third is ∂ j ln √ g ii = ∂ j v i v j − v i , i 6 = j, (2.5) which relates the characteristic velocities of the system to a class of d iagonal metrics g ii ( u ) . In [10] Ferapontov not iced that these metrics represent all possible candidates for t he con- struction of Hamil tonian operators for t he system, whether of local typ e (0.2), or non local (0.3). Now let us consider a purely nonlocal Hamiltonian formalism. Let u i t = v i u i x , (2.6) be a semi-Hamilton ian sy stem and let u i t α = W i α u i x , be a set of symmetries s atisfying condition (1.6); if the affinors are di agonal, this takes the form: X α ǫ α W i α W j α = 0 i 6 = j. (2.7) Then, according to the results of Section 1, the operator P = X α ǫ α W i α u i x  d dx  − 1 W j α u j x (2.8) 8 defines a Hamiltonian structure. Moreover , the flo ws generated by t he Hamiltonian densit ies which solve the linear system (2.4) are symmetries of (2.6). More precisely , we consider u i τ = w i u i x , w i := P ij ∂ j H = X α ǫ α W i α K α , where the functions K α are the fluxes of conserv ation laws gi ven by ∂ t α H = ∂ x K α . (2.9) For i 6 = j we get: ∂ j w i = ∂ j X α ǫ α W i α K α ! = X α ǫ α ∂ j W i α K α + X α ǫ α W i α ∂ j K α , so by using equations (2.3) and (2.9) we obtain ∂ j w i = ∂ j v i v j − v i X α ǫ α  W j α − W i α  K α + X α ǫ α W i α W j α ∂ i H = ∂ j v i v j − v i  w j − w i  . Hence the Ham iltonian flows generated by conserved densities indeed belon g to t he semi- Hamiltonian hiera rchy containing (2.6). T he c on verse prob lem, namely whether an arbitrary flo w u i τ = X i = w i u i x (2.10) commuting with (2.6) is Hami ltonian with respect to the purely no nlocal Poisson structure (2.8), turns out t o be much more d iffi cul t to solve. Howe ver , we can say that t he the Hamil- tonian structure (2.8) is conserved along an y flow (2.10) of the hierarchy: Pr oposition 2.2 Let Π ij ( x, y ) = X α W i α ( x ) u i x ν ( x − y ) W j α ( y ) u j y (2.11) be a pur ely nonlocal P oisson bivector , and suppose that the commuting flows u i t α = W i α ( u 1 ( x ) , . . . , u n ( x )) u i x , belong to a semi-Hamiltonia n hierar chy . If u i t = X i ( x ) = w i ( x ) u i x , (2.12) is an arbitrary flow of this hierar chy , then Lie X Π = 0 . 9 Pr oof . For the bi vector (2.11) and the vector field (2.12) it is not dif ficult to sh ow [9] that the expression Lie X Π tak es th e form: [Lie X Π] ij = X k ( x ) ∂ Π ij ( x, y ) ∂ u k ( x ) + ∂ x X k ( x ) ∂ Π ij ( x, y ) ∂ u k x X k ( y ) ∂ Π ij ( x, y ) ∂ u k ( y ) + ∂ y X k ( y ) ∂ Π ij ( x, y ) ∂ u k y − ∂ X i ( x ) ∂ u k ( x ) Π k j ( x, y ) − ∂ X i ( x ) ∂ u k x ∂ x Π k j ( x, y ) − ∂ X j ( y ) ∂ u k ( y ) Π ik ( x, y ) − ∂ X j ( y ) ∂ u k y ∂ y Π ik ( x, y ) . Rearranging and simplifyin g, we ob tain [Lie X Π] ij = X α w k ( x ) u k x ∂ W i α ( x ) ∂ u k ( x ) u i x ν ( x − y ) W j α ( y ) u j y + X α  w k ( x ) u k xx + ∂ w k ( x ) ∂ u l ( x ) u k x u l x  δ i k W i α ( x ) ν ( x − y ) W j α ( y ) u j y + X α w k ( y ) u k y W i α ( x ) u i x ν ( x − y ) ∂ W j α ( y ) ∂ u k ( y ) u j y + X α  w k ( y ) u k y y + ∂ w k ( y ) ∂ u l ( y ) u k y u l y  δ j k W i α ( x ) u i x ν ( x − y ) W j α ( y ) − X α ∂ w i ( x ) ∂ u k ( x ) u i x W k α ( x ) u k x ν ( x − y ) W j α ( y ) u j y − X α δ i k w i ( x )  W k α ( x ) u k xx + ∂ W k α ( x ) ∂ u l ( x ) u k x u l x  ν ( x − y ) W j α ( y ) u j y − X α δ i k w i ( x ) W k α ( x ) u k x δ ( x − y ) W j α ( y ) u j y − X α ∂ w j ( y ) ∂ u k ( y ) u j y W i α ( x ) u i x ν ( x − y ) W k α ( y ) u k y − X α δ j k w j ( y ) W i α ( x ) u i x ν ( x − y )  W k α ( y ) u i y y + ∂ W k α ( y ) ∂ u l ( y ) u k y u l y  + X α δ j k w j ( y ) W i α ( x ) u i x δ ( x − y ) W k α ( y ) u k y . The terms con taining the second deriv atives vanish identicall y . Collecting t he remainin g 10 terms and using the properties of the delta function we obtain [Lie X Π] ij =  w j ( x ) − w i ( x )  X α W i α ( x ) W j α ( x ) ! u i x u j x δ ( x − y ) + X α  ∂ W i α ( x ) ∂ u k ( x )  w k ( x ) − w i ( x )  − ∂ w i ( x ) ∂ u k ( x )  W k α ( x ) − W i α ( x )   u k x u i x ν ( x − y ) W j α ( y ) u j y + X α  ∂ W j α ( y ) ∂ u k ( y )  w k ( y ) − w j ( y )  − ∂ w j ( y ) ∂ u k ( y )  W k α ( y ) − W j α ( y )   W i α ( x ) u i x ν ( x − y ) u j y u k y , which is identically zero, because: X α W i α ( x ) W j α ( x ) = 0 , i 6 = j, and ∂ k w i w k − w i = ∂ k W i α W k α − W i α , k 6 = i. Thus, indeed, Lie X Π = 0 .  3 Quadratic expansi on of the me tric Remarkably , in the ca se of sem i-Hamiltonian systems, the existence of pu rely nonlocal Pois - son structures is related to a quadratic expansion g ii δ ij = X α ǫ α W i α W j α , (3.1) of the contravariant components of a metric g , whose covariant components satisfy (2.5), namely: ∂ j ln p g ii = − ∂ j v i v j − v i , i 6 = j. For i 6 = j , the former i dentity follows from (2.7), whil e for the diagonal compo nents we hav e the following Pr oposition 3.3 Let a diagonal system (2.1) be semi-Hamiltonian, and suppose we h ave a set of symmetries W α satisfying condition (2.7) for certain ǫ α = ± 1 . Then, the set of functions Q i := X α ǫ α  W i α  2 , satisfies the system ∂ j ln p Q i = − ∂ j v i v j − v i , i 6 = j. 11 Pr oof . For i 6 = j, we ha ve ∂ j Q i = ∂ j X α ǫ α ( W i α ) 2 ! = 2 X α ǫ α W i α ∂ j W i α = 2 X α ǫ α W i α  W j α − W i α  ∂ j v i v j − v i = − 2 Q i ∂ j v i v j − v i , i 6 = j.  Remark 2 If g ii is a metric of Egor ov type, that is g ii = ∂ i H for a suitable function H , then it is known (see e .g. [20]) that the characteristic velociti es W i α can be written as W i α = − ∂ i K α ∂ i H = − ∂ i K α g ii , (3.2) wher e t he K α ar e densities o f conservation l aws. In this case, equatio n (3.1) can be writ ten as g ii δ ij = 1 g ii g j j X α ǫ α ∂ i K α ∂ j K α , that is g ii δ ij = X α ǫ α ∂ i K α ∂ j K α . This suggests that, in the case of Egor ov metrics, the existence of a quadratic e xpansio n for a solut ion of the linear system (2.5) is r elated to the existence of an embedding of our n dimensional mani fold N in a pseudo-euclid ean space with coor dinates K α , in whic h the metric g plays t he r ole of the first fundamental form. W e have proved that any purely nonlocal Ham iltonian s tructure constructed for a semi- Hamiltonian system i s necessarily related w ith one of the metrics which solves (2.5). This relation has been o btained with a di rect calculation using t he diagonal coordinates frame, in which both the symmetries and the metric are diagonal. In order to give a coordi nate-free formulation of this , it is con venient to interpret the characteristic velociti es of the s ymmetries entering in the quadratic expansion of the metri c as a vector fie l ds on our n -dimensio nal m anifold N . Such a change of point of view leads us naturally to i ntroduce an alg ebraic structure on the tangent b undle T N of our n -dimensional manifold N - each fibre T u N has the st ructure of an associative semisim ple multipl icativ e algebra, a n d the b undle a dmits a holonomic basis of idempotents. Thi s means that first, th ere exists a basis ( Z 1 , . . . , Z n ) of idempotents: Z i ( u ) · Z j ( u ) = δ ij Z j ( u ) . 12 Second, if this basis commutes, (i s holo nomic), then th ere exists a set of coordinates, call ed canonical coor dinates , ( u 1 , . . . , u n ) such that Z i = ∂ ∂ u i . An inv ariant form of the condition (3.1) can now be easily obtained b y not ing the follo w- ing. Lemma 3.1 Any diagonal izable system of hydr odynamic type u i t = v i j ( u ) u j x , ( 3 .3) can be written in the form v i j ( u ) = c i j k ( u ) X k ( u ) , (3.4) wher e the X k ar e n ow the components of a vector field and th e c i j k ar e t he ( u -depend ent) structur e “constants ” of a associati ve semisi mple a lgebra admitting a holonomi c bas is of idempotents. Pr oof . Ind eed system (3.4) becomes diagonal i n canonical coordinates - and in such coordi- nates, the structure constants are simply c i j k = δ i j δ i k . (3.5) These c i j k are evidently t he structu re constants of an associativ e algebra. Con versely , given a diagonal system of hydrodynamic type, we can define the structure constants by identi- fying t he canonical coordinates with the gi ven Riemann in variants b y means of (3.5). This identification will clearly depend on the choice of the Riemann in var iants.  Theor em 1 Let u i t α = ( W α ) i j ( u ) u j x = c i j k ( u ) X k α ( u ) u j x (3.6) be n commuting diagonal syst ems of hydr odynamic type defined by the structur e constants of an associ ative semis imple al gebra, admitting a holonomic basis of idempot ents and by n vector fields X α ( α = 1 , . . . , m ). Suppose that the metric g ij = ( n X α =1 ǫ α X α ⊗ X α ) ij . (3.7) is nonde generate and satisfies the following condition g k l c i lm = g il c k lm . (3.8) Then: 13 1. Denoting by ∇ the Le vi-Civita conn ection associated with g , we have g ik ( W α ) k j = g j k ( W α ) k i , ∇ k ( W α ) i j = ∇ j ( W α ) i k (3.9) 2. The affinors ( W α ) i j satisfy the conditions (1.4,1.5,1.6) and ther efor e the operator M X α =1 ǫ α ( W α ) i k u k x  d dx  − 1 ( W α ) j h u h x is a pur ely nonlocal Hamiltonia n oper a tor . Pr oof . 1. Condition (3.8 ) i mplies that the metric (3.7) is diagonal in canonical coordi nates. More- over it i mplies th e first of condi tions (3.9). In order to prove the second of condi tions (3.9) we observe that, in canonical coordinates, it reads ∂ j ln √ g ii = ∂ j W i α W j α − W i α , i 6 = j. T aking into account that, in canonical coordinates, the ve ct or fields W α ( α = 1 , . . . , m ) coincide with the characteristic v elocities of t he syst ems (3.6) and satisfy the conditi on (2.7 ), we obtain the result by computations of the Proposition 3.3. 2. Condi tions (1.4 ,1.5) fo llow from the com mutativity of t he flows (3.6). Condition (1.6 ) follows im mediately from (3.7,3.8). Indeed X α ǫ α  ( W α ) i k ( W α ) j h − ( W α ) i h ( W α ) j k  = X α ǫ α ( c i k l c j hm − c i hl c j k m )( X α ) l ( X α ) m = = ( c i k l c j hm − c i hl c j k m ) g lm . Using (3.8) we get ( c i k l c j hm − c i hl c j k m ) g lm = g j s ( c i k l c l hs − c i hl c l k s ) which v ani shes due to associativity .  Remark 3 W e should point out th at the second pa rt of the theor em only uses the a ssocia- tivity pr operty; the assu mption of semisimplicity is onl y needed for the first part, which uses canonical coor dinates. 14 4 Reductio ns of the Ben ney system W e recall the basic facts about the Benney chain and it s reductions (for details see for ex- ample [12 ] and references therein). T he Benney chain is the following infinit e syst em of quasilinear PDEs: A k t = A k +1 x + k A k − 1 A 0 x , k = 0 , 1 , 2 , . . . , (4.1) in the infinitely many v ariables A k ( x, t ) , which are usuall y called moments. Introducing the formal series λ = p + ∞ X k =0 A k p k +1 , we can encode the whole system in the single equation λ t = pλ x − A 0 x λ p , (4.2) which is the second flo w of t he dispersionless K P hierarchy; by considering th e in verse of λ with re spect to p , we obtain the series p = λ − ∞ X k =0 H k λ k +1 , (4.3) whose coeffi cient s are cons erved densities of the Benney chain, each of th em polynom ial in the moments. Remark 4 In many important e xampl es, and in par ticular for all the r edu ctions defined below , the series λ can be thought as the as ymptotic ex p ansion at infinity of a n analytic function λ ( p, x, t ) . In this case, the generating function (4.3) is obtained b y in vert ing the function λ wit h r espect to p , and then expanding asymptotically ar o und infinity . A r eduction of t he Be n ney chain is a suitable restriction of the system (4.1) to the case when all th e mom ents A k can be expressed in terms of finitely many variables; as proved in [13], all reductions of t he Benney chain are diagonalizable, that is they can be written in the form λ i t = v i ( λ ) λ i x , ( 4 .4) and they satis fy the semi-Hamiltoni an con dition (2.2). As can easily be understood, in the case of a reduction the corresponding function λ depends on the variables x and t only through λ 1 , . . . , λ n , that is λ ( p, x, t ) = λ ( p, λ 1 ( x, t ) , . . . , λ n ( x, t )) . In this case, and assuming the linear independence of the λ i x , (4.2) is equivalent to the s ystem ∂ λ ∂ λ j = ∂ A 0 ∂ λ j p − v j ∂ λ ∂ p , j = 1 , . . . , n, (4.5) 15 which is a system of n Loewner equa tions , which describe – for inst ance – families of con- formal maps from the upper complex half plane t o the upper complex half plane with n arbitrary slits [14]. The analyti c properties of the conformal map solut ions of (4.5) are in- timately related to the properties of t he corresponding reduction. For example, the critical points of λ are the characteristic velocities v i and its critical values ar e Riemann in variants: ∂ λ ∂ p ( v i ) = 0 , λ ( v i ) = λ i . Moreover , the coef ficients of the expansion at λ = ∞ of the functions W i ( λ, λ 1 , . . . , λ n ) = 1 p ( λ ) − v i = ∞ X n =1 w i ( n ) λ n (4.6) are characteristic velocities of symmetries. Finally , as proved by the present authors in [12], reductions o f t he Benney s ystem associated w ith t he function λ ( p, λ 1 , . . . , λ n ) are Ham ilto- nian with respect to the Hamiltonian structures P ij = ϕ i ( λ i ) λ ′′ ( v i ) δ ij d dx + Γ ij k λ k x − 1 2 π i n X l =1 Z C l w i ( λ ) λ i x  d dx  − 1 w j ( λ ) λ j x ϕ l ( λ ) dλ (4.7) where the contour C i is the image of a sufficiently s mall closed contour around the point p = v i in the p -plane with respect to the analy tic continuation of the conformal m ap λ ( p ) , the functions ϕ i are arbitrary functions of λ , and w i ( λ ) := − ∂ p ∂ λ ( p ( λ ) − v i ) 2 = ∂ W i ∂ λ . As all reductions of the Benney chain are semi-Hamilto nian sy stems, in addit ion to the Hamiltonian structures (4.7) we can obtain a family of pu rely nonlocal Hamilt onian struc- tures if we can expand the c ontrav ariant components of the metrics g ii ϕ = ϕ i ( λ i ) λ ′′ ( v i ) , (4.8) in terms of symmetries of the system. Theor em 2 The compo nents of a metric associa ted with a re duction of the Benne y chain admit the following quadratic e xpansion g ii ϕ δ ij = ϕ i ( λ i ) λ ′′ ( v i ) δ ij = 1 2 π i n X k =1 Z C k W i ( λ ) W j ( λ ) ϕ k ( λ ) dλ , (4.9) wher e the W i ( λ ) ar e t he gener a ting functions of the symmetr ies (4.6) and the contours C k ar e the same as in (4.7). 16 Pr oof The proof is a straightforward computation of the integral: 1 2 π i n X k =1 Z C k W i ( λ ) W j ( λ ) ϕ k ( λ ) dλ = n X k =1 Res λ = λ k  ϕ k ( λ ) dλ ( p ( λ ) − v i )( p ( λ ) − v j )  = n X k =1 Res p = v k " ∂ λ ∂ p ( p − v i )( p − v j ) ϕ k ( λ ( p )) dp # = ϕ i ( λ i ) λ ′′ ( v i ) δ ij , the last step being d ue to t he fact t hat p = v k are critical p oints of λ , so that the differential turns out to be regular at all these points for i 6 = j , and also for i = j and k 6 = i .  Remark 5 In the Benne y case, it is known [12] tha t the m etric associated with a r eductio n ar e of Egor ov type, and mor e pr ecisely of the for m ( g ϕ ) ii = 1 ϕ i ( λ i ) . Mor eover , the function p ( λ ) s atisfies a Loewner system of the form ∂ i p = − ∂ i A 0 p ( λ ) − v i = − W i ( λ ) ∂ i A 0 , obtained by (4.5) b y using the implicit function theore m. Using Remark 2 ab out t he Egor ov metrics, we conclude that the covariant metrics a ssociated with a r educt ion of the Benney chain can be w ri tten as ( g ϕ ) ii δ ij = 1 2 π i n X k =1 Z C k ∂ i p∂ j p ϕ k ( λ ) dλ. (4.10) 5 Semi-Ham iltonian s ystems related to sem isimple Fr o be- nius manif old s W e ha ve seen that these p urely nonlocal Hamil tonian structures are connected with a geo- metrical st ructure where the tangent space of a manifol d has the s tructure of an associativ e multipli cativ e algebra. The m ost important examples of these are F r ob enius manifolds . A Frobenius mani fold [4, 5] is a manifold M endowed with a commutat iv e, asso ciativ e m ulti- plicative structure · on the tangent sp aces together with a flat metric η , i n variant with respect to the product · . This means that the third order tensor c defined by c ( u, v , w ) = ( u · v , w ) (where u, v , w are arbi trary vector fields and ( , ) is the scalar product defined by η ) is symmetric. 17 It is easy to check that this condition , com bined with requiring t he sym metry of the fourth order tensor ∇ z c ( u, v , w ) implies that, in flat coordinates v 1 , . . . , v n , the structure constants of · can be writ ten (lo- cally) as third deriv atives of a funct ion F , called the F robenius potentia l : c αβ γ = η αδ c δ β γ = ∂ 3 F ∂ v α ∂ v β ∂ v γ . The definition of a Frobenius manifold also in volves two sp ecial vector fields: the first, usually denoted by e , is the unit of the p roduct · and can be i dentified with the v ector field ∂ ∂ v 1 ; the second, called the Euler vector field , encodes the quasi-ho megeneity properties of the Frobenius potential F : Lie E ( F ) = (3 − d ) F , (5.1) where d is a constant. In flat coordinates E is a linear vector field and the con dition (5.1) becomes X α ( d α t α + r α ) ∂ F ∂ v α = (3 − d ) F . where r α is a constant, non–vanishing only if d α = 0 . Any Frobeniu s manifold poss esses a s econd flat m etric defined, in flat coordinates for th e first metric, by the formula g αβ = E ǫ c αβ ǫ . Using the Du brovin-Noviko v results, starting from the two flat metrics η and g one can define the following p air of Hamiltonian structures of hydrodynamic type: P αβ 1 = η αβ ∂ x (5.2) P αβ 2 = g αβ ∂ x + Γ αβ γ u γ x = E ǫ c αβ ǫ ∂ x +  d − 1 2 + d β  c αβ γ . (5.3) It turns out [4] t hat P 1 and P 2 are com patible and t herefore define a bi-Hamilt onian hier- archy of hydrodynamic type. According to well-known result s (see for in stance [2]), the Hamiltonian densities o f such a hierarchy can be taken as the coefficients of the expansion at λ = ∞ c α ( x, λ ) = c α 1 ( x ) + c α 0 ( x ) λ + c α 1 ( x ) λ 2 + . . . λ → ∞ , (5.4) of the Casimirs of the pencil P 2 − λP 1 . Since the Casimirs of a Hamiltoni an structure of hy drodynamic type coincide with the flat coordinates of the correspondin g metric, it follows that the Casimirs (5.4) are gi ven by the flat coordinates of the pencil of metrics g − λη , (5.5) 18 and thus satisfy the Gauss-Manin system : ( ∇ ∗ − λ ∇ ) d c α = 0 . (5.6) Here ∇ ∗ is th e Le v i-Civita connection for the m etric g , and ∇ is th e Levi-Ci vita connection for the metric η . In thi s way , given a Frobenius m anifold, it is possibl e to define a bi -Hamiltonian hier- archy of hydrod ynamic type. In flat coordinates for the metric η th e equations of su ch a hierarchy read v β t α,k = η β γ ∂ x δ H α,k δ v γ , α, β = 1 , . . . , n, k = − 1 , 0 , 1 , . . . (5.7) where H α,k = Z c α,k dx. W e no w focus our attention on a special class of Frobenius manifolds. A Frobenius manifold M is called semisimpl e [4] if at a generic point v ∈ M t he Frobe- nius algebra T v M is semisim ple. The canonical coordinates ( u 1 , . . . , u n ) , whose existence is not an additional assumption but fol lows from the general properties of these manifolds, can be obtained as solution of the equation det( g − λη ) = 0 . It turns out that, in canoni cal coordinates, the metri cs η and g are diagonal and that the metric η is of Eg orov type. Moreover such canonical coordinat es are Riemann i n variants of the hierarchy (5.7). Giv en a semisimple Frobenius manifol d with d < 1 it is possib le to define a fun cion λ ( p, u 1 , . . . , u n ) called its superp otential , having the fol lowing p roperties (for details see [4, 5, 9]): - it is defined as the in verse of a special solution of the Gauss-Manin system (5.6) . - its critical values are the canonical coordinates. - the cov ariant components of the metric η in canonical coordinates can be written as η ij = − n X i =1 res p = p i ∂ i λ∂ j λ λ p dp = − 1 2 π i Z Γ ∂ i λ∂ j λ λ p dp (5.8) where Γ a re “small” closed contours around the critical points p 1 , . . . , p n of λ . Using these results it is easy to prove the following theorem Theor em 3 Let M be a semisimpl e F robenius mani fold with d < 1 a nd let λ ( p , u 1 , . . . , u n ) be its superpotential, then the covariant a nd contravariant component s o f th e metric η in 19 canonical coor dinates admit the following quadratic e xpansions η ij = 1 2 π i Z C ∂ i p ( λ, u 1 , . . . , u n ) ∂ j p ( λ, u 1 , . . . , u n ) dλ η ij = 1 2 π i Z C W i ( λ, u 1 , . . . , u n ) W j ( λ, u 1 , . . . , u n ) dλ wher e C ar e the i mages of the contour Γ in the λ plane and the functions W i ( λ, u 1 , . . . , u n ) , defined by: W i ( λ, u 1 , . . . , u n ) = ∂ i p ( λ, u 1 , . . . , u n ) η ii , ar e generating functions o f the s ymmetries of the semi-Hamiltonian hierar chy associated with M . Pr oof. The first quadratic expansion can be ob tained just by changin g the variable p → λ in the integral (5. 8). Raising t he indices we get the second quadratic expansion. In order t o prove th at the fun ctions W i are generating f u nctions of s ymmetries it is suffic i ent to observe that the metric η ii is of Egorov type and that the in verse of the superpotential is a generating function of Hamiltonian densities.  Remark 6 St arting fr om a F r o benius manifold one can define a hierar chy of inte grable PDEs also in the follo wing way . Let ∇ be the Levi Civita connection associat ed with the metric η and ( X ( α, 0) , α = 1 , . . . , n ) be a basis of co vari antly constant vector fields. One can define the primary flows of the hi erar chy as u i t ( α, 0) = c i j k X k ( α, 0) u j x , i = 1 , . . . , n. and the “higher flows” u i t ( α,n ) = c i j k X k ( α,n ) u j x , i = 1 , . . . , n, (5.9) r ecursively , by means of the r elati ons ∇ i X k ( α,n ) = c i k l X k ( α,n − 1) . The hierar chy defined in this way is usually call ed the principal hi erachy . It is equ ivalent to the hierachy defined above in terms of coefficients of the Casimirs of the pencil (5.5) since the flo ws (5.9) ar e re l ated to the flows (5.7) ju st by tr iangular linear transformat ions (see [4, 9] for detail s). This shows that in the case of h ierarchies of quasilinear PDEs associated wi th a Frobenius manifold the ”factorization” (3.4) has a natural interpretation: the structure constant s coin- cide with th e structure constants defining the Frobenius s tructure, and the vector fields X hav e a precise geometrical meaning. 20 6 The classical shallow water equations Let us consider the classical shallow water system, given by h t = ( hu ) x , (6.1) u t = uu x + h x . A related problem was solved by Riemann, [21], us ing the hodograph transformation. This system can be seen as th e elementary 2 − compon ent reduction of the Benney chain associated with the rational map [1, 24]: λ = p + h p − u , (6.2) Moreover , ( 6 .1) is a n element of a bi-Hamilton ian hierarchy associated with a 2 d imensional Frobenius manifold, with Frobenius potential F ( h, u ) = 1 2 hu 2 + h log h, in this sett ing, the functio n (6.2) is the su perpotential. Let us recall that under the change of coordinates r 1 = u − 2 √ h, r 2 = u + 2 √ h, the system takes the diagonal form r 1 t = 1 4  3 r 1 + r 2  r 1 x r 2 t = 1 4  r 1 + 3 r 2  r 2 x . The general solution of the linear system (2.5), in this case, is g ii = ϕ i ( r i ) ∂ i A 0 , i = 1 , 2 , (6.3) where ϕ i ( r i ) are arbitrary functions of a single variable and A 0 = ( r 1 − r 2 ) 2 16 . W e will show no w that th e quadratic expansion of the cont ra variant components of the m et- rics (6.3) can be reduced to a finite sum, s o that we can construct families of purely nonlocal Poisson bra ckets in volving only a finite number of symmetries. W e proceed as follo ws: first, we extend λ ( p ) from the up per half p lane to a rational fun ction on the wh ole Riemann sphere. Then, we note that although the e xt ended λ is un iv alent, its inv erse with respect to p is no t, 21 and we have to consider t wo fun ctions p + ( λ ) and p − ( λ ) each of them defined on o ne sheet o f a double c overing of the Riemann sphere, wit h branch points at r 1 and r 2 . The two functions are easily found to be p + ( λ ) = 1 2 λ + 1 4 ( r 1 + r 2 ) + 1 2 p ( r 2 − λ ) ( r 1 − λ ) , p − ( λ ) = 1 2 λ + 1 4 ( r 1 + r 2 ) − 1 2 p ( r 2 − λ ) ( r 1 − λ ) , their main dif ference bei ng in the beha vio ur at infinity , for: p + ( λ ) = λ + O  1 λ  , λ → ∞ + , while p − ( λ ) = u 2 + O  1 λ  , λ → ∞ − . Thus, we can construct two generating functions of the symmetries, the first is gi ven by w i ( λ ) = 1 p + ( λ ) − v i , whose expansion at infinity is w i ( λ ) = ∞ X n =1 w i n λ n , (6.4) and where the first fe w coefficients are gi ven by w 1 1 = 1 , w 2 1 = 1 , w 1 2 = 3 4 r 1 + 1 4 r 2 , w 2 2 = 1 4 r 1 + 3 4 r 2 , w 1 3 = 5 8 ( r 1 ) 2 + 1 4 r 1 r 2 + 1 8 ( r 2 ) 2 , w 2 3 = 1 8 ( r 1 ) 2 + 1 4 r 1 r 2 + 5 8 ( r 2 ) 2 . For the sec o nd generating function, an easy calculation shows th at the analogous generating function constructed from p − ( λ ) is related with the first by 1 p − ( λ ) − v i = w i 0 − 1 p + ( λ ) − v i , i = 1 , 2 , where w 1 0 = − 4 r 1 − r 2 , w 2 0 = 4 r 1 − r 2 , is an extra symmetry , not appearing in the e x pansion (6.4). 22 Reducing the integral (4.10) t o the sum of residues around the t wo points at infinity , ∞ + , ∞ − , we obtain a finite quadratic e xpansi on of the compon ents of the metric tensor in terms of symmetries: g ii ( k ) δ ij = ( r i ) k δ ij ∂ i A 0 = − Res λ = ∞ + λ k dλ ( v i − p + ( λ ))( v j − p + ( λ )) − Res λ = ∞ − λ k dλ ( v i − p − ( λ ))( v j − p − ( λ )) = Res λ = ∞ + w i 0 λ k dλ p + ( λ ) − v j + Res λ = ∞ + w j 0 λ k dλ p + ( λ ) − v i − 2 Res λ = ∞ + λ k dλ ( v i − p + ( λ ))( v j − p + ( λ )) = w i 0 w j k +1 + w i k +1 w j 0 − 2 k X s =1 w i s w j k − s +1 . Therefore, for k = 0 , 1 , . . . , the corresponding purely nonlocal Poisson operators ha ve the form P ij ( k ) = w i 0 r i x  d dx  − 1 w j k +1 r j x + w i k +1 r i x  d dx  − 1 w j 0 r j x − 2 k X s =1 w i s r i x  d dx  − 1 w j k − s +1 r j x ! . W e consid er now the flows generated by the sim plest of t hese st ructures, nam ely P (0) . As Hamiltonian density we cons ider the generating function p + ( λ ) ; the quantities we want to e valuate are thus the flo ws µ i ( λ ) := 2 X j =1 P ij (0) ∂ p + ( λ ) ∂ λ j . Explicitly , and ass uming v ani shing boundary conditions lim | x |→ ∞ r i ( x, t ) = 0 , these are found to be µ 1 ( λ ) = − 4 p + ( λ ) − λ r 1 − r 2 − 2 ln  p λ − r 1 + p λ − r 2  + ln 4 λ, µ 2 ( λ ) = 4 p + ( λ ) − λ r 1 − r 2 − 2 ln  p λ − r 1 + p λ − r 2  + ln 4 λ. By comparing th e coefficients of the asymptotic expansions at i nfinity of µ i ( λ ) and p + ( λ ) we o btain, for instance, that t he characteristic velocities w 2 of the systems are generated by the Hamiltonian d ensity H 0 , while the sym metry w 3 is ob tained from H 1 . For th e shallo w water h ierarchy , there exist anoth er Poisson structure which sends the Hamiltoni an densi ty H 0 to the sys tem wi th characteristi c velociti es w 2 . This i s the thi rd local Poisson structure of the sys tem, g enerated by the flat metric g ii = 2 ( λ i ) 2 ∂ i A 0 . W e denote this structu re as P loc , and we call z i ( λ ) the corresponding flo ws, so that z i ( λ ) = 2 X j =1 P ij loc ∂ p + ( λ ) ∂ λ j . 23 W ith little difficulty , it can be shown that the two Hamiltonian hierarchies z i ( λ ) and µ i ( λ ) are related by z i ( λ ) = λ 2 2 d 2 µ i ( λ ) d λ 2 , so that the coef ficients of th e expansion a t infinity are related by z i k = k ( k + 1) 2 µ i k . Moreover , the original generating fun ction of the s ymmetries w i ( λ ) can written in terms of µ i ( λ ) as w i ( λ ) = 1 λ − d µ i ( λ ) d λ , so that the coef ficients sati sfy the relation w i k +1 = − ( k + 1) µ i k . Acknowledgm ents W e would like to thank th e E SF grant MISGAM 141 4, for it s sup port of P aolo Lorenzoni’ s visit to Imperial. W e are also grateful to the European Commissions FP6 programme for support of this work throu gh the ENIGMA network, and particularly for their supp ort of Andrea Raimondo, who also recei ved an EPSRC DT A. References [1] D. J. Benney , Some pr operties of l ong nonli near waves , Stud.Appl. Math., 52 (1973) 45–50. [2] P . Casati, F . M agri, M. Pedroni, Bihamilto nian M anifolds and τ -function , M athematical Aspects of Classi cal Field Theory (M.J. Go tay et al., eds.), Contemporary M athematics, 132 , AMS, Providence, ( 1 992) 213–234. [3] J en-Hsu Chang, On the waterbag model of the dispersionless KP hierar chy . II , J. Phys. 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