Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions
aa r X i v : . [ n li n . S I] A ug Dispersive deformations of Hamiltonian systems ofhydrodynamic type in dimensions
E.V. Ferapontov, V.S. Novikov and N.M. Stoilov
Department of Mathematical SciencesLoughborough UniversityLoughborough, Leicestershire LE11 3TUUnited Kingdome-mails:
[email protected]@[email protected]
Dedicated to Professor Boris Dubrovin on the occasion of his 60th birthday
Abstract
We develop a theory of integrable dispersive deformations of dimensional Hamil-tonian systems of hydrodynamic type following the scheme proposed by Dubrovin and hiscollaborators in dimensions. Our results show that the multi-dimensional situation isfar more rigid, and generic Hamiltonians are not deformable. As an illustration we discussa particular class of two-component Hamiltonian systems, establishing the triviality of firstorder deformations and classifying Hamiltonians possessing nontrivial deformations of thesecond order.MSC: 35L40, 37K05, 37K10, 37K55.Keywords: Hamiltonian systems of Hydrodynamic Type, Dispersive Deformations, Hy-drodynamic Reductions. Introduction
Deformation theory of dimensional Hamiltonian systems has been thoroughly investigated byDubrovin and his collaborators in [8, 9, 11, 12, 13]: given a Hamiltonian system of hydrodynamictype, u it = { u i , H } = P ij δH /δu j , (1) i, j = 1 , . . . , n , where P ij = ǫ i δ ij d/dx is the Hamiltonian operator and H = R h ( u ) dx is aHamiltonian with the density h ( u ) , one looks for deformations of the form H = H + ǫH + ǫ H + . . . (2)where the density of H i is assumed to be a homogeneous polynomial of degree i in the x -derivatives of u . Here the Hamiltonian operator P ij can be assumed undeformed due to thegeneral results of [19, 5]. Deformation (2) is called integrable (to the order ǫ m ) if any hydro-dynamic Hamiltonian F = R f ( u ) dx commuting with H can be deformed in such a way that { H, F } = 0 (mod ǫ m +1 ) . It is assumed that H generates an integrable system of hydrodynamictype [23, 24, 7]: any system of this kind possesses an infinity of commuting Hamiltonians F parametrised by n arbitrary functions of one variable. The classification of integrable deforma-tions is performed modulo canonical transformations of the form H → H + ǫ { K, H } + ǫ { K, { K, H }} + . . . (3)where K is any functional of the form (2). The richness of this deformation scheme is due to thefollowing basic facts: • The variety of integrable ‘seed’ Hamiltonians H is parametrised by n ( n − / arbitraryfunctions of two variables; • For a fixed integrable Hamiltonian H , the deformation procedure introduces extra arbi-trary functions of one variable known, in bi-Hamiltonian context, as ‘central invariants’.One should point out that it is still an open problem to extend a deformation, for arbitraryvalues of these functions, to all orders in the deformation parameter ǫ .The main goal of this paper is to discuss the analogous deformation scheme in dimensions.One again starts with the Hamiltonian system (1) where P ij is a -dimensional Hamiltonianoperator of hydrodynamic type, see [6, 21, 22] for the general theory and classification results.In the two-component case there exist only three types of such operators: the first two of themcan be reduced to constant-coefficient forms, P = (cid:18) d/dx d/dy (cid:19) , P = (cid:18) d/dxd/dx d/dy (cid:19) , while the third one is essentially non-constant, P = (cid:18) v ww (cid:19) ddx + (cid:18) vv w (cid:19) ddy + (cid:18) v x v y w x w y (cid:19) , here v, w are the dependent variables. We will refer to them as Hamiltonian operators of typeI, II and III, respectively. To be specific, we will concentrate on case II. The correspondingHamiltonian systems take the form (cid:18) vw (cid:19) t = (cid:18) d/dxd/dx d/dy (cid:19) (cid:18) δH /δvδH /δw (cid:19) , (4)2 = R h ( v, w ) dxdy , or, explicitly, v t = ( h w ) x , w t = ( h v ) x + ( h w ) y . We will be looking at deformations of the form (2) where the density of H i is a homogeneouspolynomial of degree i in the x - and y -derivatives of v and w . The Hamiltonian operator willbe assumed undeformed (although we are not aware of any results establishing the triviality ofPoisson cohomology in higher dimensions). Since a system of the form (4) does not possess anynontrivial conservation laws of hydrodynamic type other than the Casimirs and the Hamiltonian,the definition of integrability needs to be modified. Thus, the ‘seed’ system (4) will be calledintegrable if it possesses infinitely many hydrodynamic reductions [20, 14]. This requirementimposes strong constraints on the Hamiltonian density h ( v, w ) , providing an efficient classificationcriterion (see Sect. 2 for more details). Following [17, 18], a deformation of H will be calledintegrable (to the order ǫ m ) if it inherits all hydrodynamic reductions of the seed system (4) tothe same order (the deformation procedure is outlined in Sect. 3). The main features of the dimensional deformation scheme can be summarised as follows: • The variety of integrable ‘seed’ Hamiltonians H is finite dimensional. • Generic integrable Hamiltonians H possess no nontrivial deformations.Nevertheless, there exist deformable (non-generic) Hamiltonians. Example 1.
Let H = R w + f ( v ) dxdy . In this case the integrability conditions reduce toa single fourth order ODE, f ′′′′ f ′′ = f ′′′ , so that without any loss of generality one can set f ( v ) = e v . Modulo canonical transformations, this Hamiltonian possesses a unique integrabledispersive deformation of the form H = Z w f ( v ) − ǫ f ′′′ v x + O ( ǫ ) dxdy. For f ( v ) = e v it can be rewritten in the equivalent form H = Z w e v + ǫ e v v xx + O ( ǫ ) dxdy. It is quite remarkable that this deformation can be extended to all orders in the deformationparameter ǫ , providing a Hamiltonian formulation of the D Toda system, H = Z w exp (cid:18) v + ǫ v xx + ǫ v xxxx + . . . (cid:19) dxdy, see Sect. 6 for further details. Example 2.
Let H = R α v + βvw + f ( w ) dxdy . Here the integrability conditions reduce to asingle fourth order ODE, f ′′′′ f ′′ ( αf ′′ − β ) = f ′′′ (3 αf ′′ − β ) . Modulo canonical transformations,this Hamiltonian possesses a unique integrable dispersive deformation of the form H = Z α v βvw + f ( w ) + ǫ f ′′′ (cid:18) − α f ′′ v x + β f ′′ v y + 2 αw x + 2 βv y w y + f ′′ w y (cid:19) + O ( ǫ ) dxdy. β = 0 the Hamiltonian H gives rise to the dispersionless KP (dKP) equation, thedeformation presented here is not equivalent to the full KP equation: see Sect. 7 for furtherdiscussion.It will be demonstrated (Theorem 2 of Sect. 5) that, modulo certain equivalence transfor-mations, these two examples exhaust the list of Hamiltonians of type II which possess nontrivialintegrable deformations to the order ǫ . In Sect. 4 we prove the triviality of ǫ -deformations. Thestructure of ǫ -deformations is analysed in Sect. 5. Deformations of Hamiltonians of type II/IIIare discussed in Sect. 8/9. In this section we review the classification of integrable Hamiltonian systems of the form (4).Following [14, 16] we require the existence of N -phase solutions of the form v = v ( R , R , . . . , R N ) , w = w ( R , R , . . . , R N ) , (5)where the phases R i ( x, y, t ) satisfy the commuting equations R it = λ i ( R ) R ix , R iy = µ i ( R ) R ix ; (6)recall that the assumption of commutativity imposes the following restrictions on the chrarac-teristic speeds λ i and µ i : ∂ j λ i λ j − λ i = ∂ j µ i µ j − µ i , (7) ∂ j = ∂/∂ R j , i = j , see [24]. Equations (6) are said to define an N -component hydrodynamicreduction of the original system (4). It was observed in [14] that the requirement of the exis-tence of such reductions imposes strong constraints on the original system (4), and provides anefficient classification criterion. Recall that the key property is the existence of three-componentreductions: in this case one also has N -component reductions for arbitrary N . This property isreminiscent of the three-soliton condition in the theory of integrable systems. On the contrary,the existence of one- or two-component reductions is a common phenomenon which is not gen-erally related to the integrability (at least for two-component systems as in the present paper).As shown in [16], the requirement of existence of three-component reductions leads to a systemof fourth order PDEs for the Hamiltonian density h ( v, w ) , which constitute the integrabilityconditions: h ww ( h vv h ww − h vw ) h vvvv = − h vw h vvw + 3 h ww h vv h vvw +4 h vw h vww h vvv − h ww h vw h vvw h vvv + h ww h vvv ,h ww ( h vv h ww − h vw ) h vvvw = − h vw h vww h vvw + 3 h ww h vww h vvw h vv − h ww h vw h vvw + h vw h vvv h + h ww h vw h vww h vvv + h ww h vvw h vvv ,h ww ( h vv h ww − h vw ) h vvww = − h vw h vww + 2 h vv h vww h ww − h ww h vw h vww h vvw + h ww h vv h h wvv + h ww h vw h h vvv + h ww h vww h vvv ,h ww ( h vv h ww − h vw ) h = − h vw h h vww − h ww h vww h vw +3 h ww h vv h h vww + h ww h vw h h vvw + h ww h h vvv ,h ww ( h vv h ww − h vw ) h = − h vw h − h ww h h vw h vww − h ww h vww + 3 h ww h vv h + 4 h ww h h vvw . (8)4his system is in involution, and is invariant under the -parameter group of Lie-point symme-tries, v → av + b,w → pv + cw + d,h → αh + βv + γw + δ. These transformations form the equivalence group of the problem. They preserve the Hamil-tonian structure, and will be used to simplify the classification results. Under the Legendretransformation, V = h v , W = h w , H = vh v + wh w − h, H V = v, H W = w, the integrability conditions (8) simplify to H V V V V = 2 H V V V H V V ,H V V V W = 2 H V V W H V V V H V V ,H V V W W = 2 H V V W H V V ,H V W W W = 3 H V W W H V V W − H W W W H V V V H V V ,H W W W W = 6 H V W W − H W W W H V V W H V V . These equations were explicitly solved in [16], leading to the following classification result:
Theorem 1
Modulo the natural equivalence group, the generic integrable potential H ( V, W ) oftype II is given by the formula H = V ln Vσ ( W ) , where σ is the Weierstrass sigma-function: σ ′ /σ = ζ, ζ ′ = − ℘, ℘ ′ = 4 ℘ − g . Its degenerationscorrespond to H = V ln VW , H = V ln V, H = V W + αW , as well as the following polynomial potentials: H = V V W W , H = V W . Taking the inverse Legendre transform, one can obtain a complete list of integrable Hamiltoniandensities h ( v, w ) . Just to mention a few of them, one gets h ( v, w ) = w e v , h ( v, w ) = v w / , h ( v, w ) = 12 ( w + v / , h ( v, w ) = 12 ( w + e v ) , etc. However, we would prefer to avoid case-by-case considerations, and work with the full setof integrability conditions (8). 5 Dispersive deformations in dimensions
Given a Hamiltonian system of the form (4), its deformation H = H + ǫH + · · · + ǫ m H m + O ( ǫ m +1 ) will be called integrable (to the order ǫ m ) if both equations (5) and (6) defining N -phase solutionscan be deformed to the same order in ǫ , in other words, the deformed dispersive system is requiredto ‘inherit’ all hydrodynamic reductions of its dispersionless limit [17, 18]. More precisely, werequire the existence of expansions v = v ( R , R , . . . , R N ) + ǫv + · · · + ǫ m v m + O ( ǫ m +1 ) ,w = w ( R , R , . . . , R N ) + ǫw + · · · + ǫ m w m + O ( ǫ m +1 ) , (9)where v i and w i are assumed to be homogeneous polynomials of degree i in the x -derivatives of R ’s (thus, both R ixx and R ix R kx have degree two, etc). Similarly, hydrodynamic reductions (6)are deformed as R it = λ i ( R ) R ix + ǫa + · · · + ǫ m a m + O ( ǫ m +1 ) ,R iy = µ i ( R ) R ix + ǫb + · · · + ǫ m b m + O ( ǫ m +1 ) , (10)where a i and b i are assumed to be homogeneous polynomials of degree i + 1 in the x -derivativesof R ’s. We require that the substitution of (9), (10) into the deformed system (4) satisfies theequations up to the order O ( ǫ m +1 ) . This requirement proves to be very restrictive indeed, andimposes strong constraints on the structure of the deformed Hamiltonian H . Remark.
Expansions (9)-(10) are invariant under Miura-type transformations of the form R i → R i + ǫr + ǫ r + . . . , where r i denote terms which are polynomial of degree i in the x -derivatives of R ’s. These trans-formations can be used to simplify calculations. For instance, working with one-phase solutionsone can assume that v remains undeformed. Similarly, working with two-phase solutions onecan assume that both v and w remain undeformed. For three-phase solutions this normalisationstill leaves some extra Miura-freedom which can be used to simplify expressions for a i and b i (to the best of our knowledge there exist no general theory of normal forms under Miura-typetransformations). ǫ -deformations In this section we prove that all ǫ -deformations are trivial and can be eliminated by an appropriatecanonical transformation. Thus, we consider deformations of the form (cid:18) vw (cid:19) t = (cid:18) d/dxd/dx d/dy (cid:19) (cid:18) δH/δvδH/δw (cid:19) (11)where H = Z h ( v, w ) + ǫ ( av x + bv y + pw x + qw y ) + O ( ǫ ) dxdy. Here a, b, p, q are functions of v and w . We require that all N -phase solutions (5) can be extendedto the order ǫ , v = v ( R , R , . . . , R N ) + ǫv + O ( ǫ ) , w = w ( R , R , . . . , R N ) + ǫw + O ( ǫ ) , (12)6here v and w are polynomials of order one in the x -derivatives of R ’s. Similarly, hydrodynamicreductions (6) are deformed as R it = λ i ( R ) R ix + ǫa + O ( ǫ ) , R iy = µ i ( R ) R ix + ǫb + O ( ǫ ) , (13)where a and b are polynomials of order two in the x -derivatives of R ’s. We thus require thatrelations (12), (13) satisfy the original system (11) up to the order O ( ǫ ) .It was verified by a direct calculation that all one- and two-component reductions can bedeformed in this way, for any a, b, p, q and any density h ( v, w ) , not necessarily integrable. Onthe contrary, the requirement of the inheritance of three-component reductions (recall that theexistence of three-component reductions forces h ( v, w ) to satisfy the integrability conditions (8)),is nontrivial, and leads to the following single relation: (cid:18) h vv N w − h vw ( M w − N v ) h vv h ww − h vw (cid:19) w = (cid:18) h vw N w − h ww ( M w − N v ) h vv h ww − h vw (cid:19) v , (14)here M = ( a w − p v ) /h ww , N = ( b w − q v ) /h ww . It remains to show that the relation (14) isnecessary and sufficient for the existence of a canonical transformation of the form H → H + ǫ { K, H } + O ( ǫ ) , with K = R k ( v, w ) dxdy , which eliminates all ǫ -terms. Since the density of the functional H + ǫ { K, H } is given by the formula h ( v, w ) + ǫ ( av x + bv y + pw x + qw y ) + ǫ ( k v , k w ) (cid:18) d/dxd/dx d/dy (cid:19) (cid:18) h v h w (cid:19) + O ( ǫ ) = h ( v, w ) + ǫ ( Av x + Bv y + P w x + Qw y ) + O ( ǫ ) , where A = a + k v h vw + k w h vv , B = b + k w h vw , P = p + k v h ww + k w h vw , Q = q + k w h ww , the conditions that ǫ -terms are trivial (form a total derivative), take the form A w = P v , B w = Q v .This leads to the following linear system for k ( v, w ) : k vv h ww − k ww h vv = a w − p v , k vw h ww − k ww h vw = b w − q v . The compatibility conditions of these equations for k can be obtained by introducing the auxiliaryvariable p via the relation k ww = ph ww , and solving for the remaining second order derivativesof k , k vv = M + ph vv , k vw = N + ph vw , k ww = ph ww . Cross-differentiating and solving for p v and p w we obtain p v = h vv N w − h vw ( M w − N v ) h vv h ww − h vw , p w = h vw N w − h ww ( M w − N v ) h vv h ww − h vw . Ultimately, the compatibility condition p vw = p wv gives the required relation (14), thus finishingthe proof. 7 Reconstruction of ǫ -deformations In this section we analyse the structure of ǫ -deformations. The result of the previous sectionallows us to set all ǫ -terms equal to zero. Thus, we consider deformations of the form (cid:18) vw (cid:19) t = (cid:18) d/dxd/dx d/dy (cid:19) (cid:18) δH/δvδH/δw (cid:19) (15)where H = Z h ( v, w ) + ǫ h ( v, w, v x , w x , v y , w y ) + O ( ǫ ) dxdy. Here h is assumed to be of second order in the x - and y -derivatives of v and w , h = f v x + f v y + f w x + f w y + f v x w x + f ( v x w y + v y w x ) + f v y w y + f v x v y + f w x w y , where f , . . . , f are functions of v and w . Note that all terms which are linear in the second orderderivatives of v and w can be removed via integration by parts. Furthermore, any expression ofthe form f ( v, w )( v x w y − v y w x ) can be omitted, since its variational derivative is identically zero.We require that all N -phase solutions (5) can be extended to the order ǫ , v = v ( R , R , . . . , R N ) + ǫ v + O ( ǫ ) , w = w ( R , R , . . . , R N ) + ǫ w + O ( ǫ ) , (16)where v and w are polynomials of order two in the x -derivatives of R ’s. Similarly, hydrodynamicreductions (6) are deformed as R it = λ i ( R ) R ix + ǫ a + O ( ǫ ) , R iy = µ i ( R ) R ix + ǫ b + O ( ǫ ) , (17)where a and b are polynomials of order three in the x -derivatives of R ’s. We thus require thatrelations (16), (17) satisfy the deformed system (15) up to the order O ( ǫ ) . The classification isperformed modulo canonical transformations of the form H → H + ǫ { K, H } + O ( ǫ ) , here K = ǫ R ( av x + bv y + pw x + qw y ) dxdy . Note that the density of the functional ǫ { K, H } isgiven by the following formula (set m = a w − p v , n = b w − q v ): ǫ ( δK/δv , δK/δw ) (cid:18) d/dxd/dx d/dy (cid:19) (cid:18) h v h w (cid:19) + O ( ǫ ) = ǫ m ( h vv v x + h vw v x v y + h ww v x w y − h ww w x )+ ǫ n ( h vv v x v y + h vw v y + h ww v y w y − h ww w x w y )+ O ( ǫ ) . Our calculations demonstrate that generic integrable Hamiltonians H do not possess nontrivialdispersive deformations. To be precise, these deformations are parametrised by two arbitraryfunctions, analogous to m and n above, which can be eliminated by a canonical transforma-tion. There are cases, however, where dispersive deformations are parametrised by two arbitraryfunctions and a constant. It is exactly this extra constant which gives rise to a non-trivial de-formation. We emphasize that canonical transformations can be used from the very beginningto bring the deformation to a ‘normal form’: since h ww = 0 one can set, say, f = f = 0 . Thisnormalisation simplifies all subsequent calculations. Our results can be summarised as follows.8 heorem 2 A Hamiltonian H = R h ( v, w ) dxdy of type II possesses a nontrivial integrabledeformation to the order ǫ if and only if, along with the integrability conditions (8), it satisfiesthe additional differential constraints h vvv h vww − h vvw = 0 , h vvv h − h vvw h vww = 0 , h h vvw − h vww = 0 , that is, rank (cid:18) h vvv h vvw h vww h vvw h vww h (cid:19) = 1 . (18) Modulo equivalence transformations, this gives two types of deformable densities: h ( v, w ) = w e v , h ( v, w ) = α v βvw + f ( w ) , where f ( w ) satisfies the integrability condition f ′′′′ f ′′ ( αf ′′ − β ) = f ′′′ (3 αf ′′ − β ) . Proof:
In contrast to the case of ǫ -corrections where all constraints were coming from deformationsof three-component reductions, at the order ǫ the main constraints appear at the level of one-component reductions already. Furthermore, it was verified by a direct calculation that multi-component reductions impose no extra conditions. Since the third order derivative h appearsas a factor in all deformation formulae, there are two cases to consider. Case 1: h = 0 . Then the integrability conditions imply h vww = 0 . The further analysisshows that one has to impose an extra condition, namely h vvw = 0 , otherwise all deformations aretrivial. Notice that conditions h = h vww = h vvw = 0 clearly imply (18). Modulo equivalencetransformations, this is the case of the Hamiltonian density h ( v, w ) = w + e v . Its dispersivedeformation is given in Example 1 of the Introduction. Case 1: h = 0 . In this case one gets a system of equations for the coefficients f , . . . , f which contains f as a factor. If f equals zero, all deformations are trivial. In the case f = 0 one can express f , f , f , f , f , f in terms of f , f , f . What is left will be a system of twocompatible first order PDEs for f , and a system of additional differential constraints for h ( v, w ) which coincides with (18). Solving equations for f we obtain a constant of integration whichis responsible for non-trivial dispersive deformations. To find integrable Hamiltonian densitiessatisfying (18) we set h = q, h vww = pq . Then the remaining third order derivatives of h can be parametrised as h = q, h vww = pq, h vvw = p q, h vvv = p q. Calculating the compatibility conditions we obtain p v = pp w , q v = ( pq ) w . With this ansatz theintegrability conditions (8) imply p =const so that q = F ( w + pv + c ) where f is a function ofone variable. Thus, h can be represented in the form h ( v, w ) = f ( w + pv + c ) + Q ( v, w ) , where Q ( v, w ) is an arbitrary quadratic form. Modulo the equivalence group any such density can bewritten in the form h ( v, w ) = α v + βvw + f ( w ) , and the substitution into (8) gives a fourthorder ODE for f . The dispersive deformation of this Hamiltonian is presented in Example 2.We believe that both Hamiltonians from Theorem 2 can be deformed to all orders in ǫ .9 Example 1: deformation of the Boyer-Finley equation
In this section we discuss the key example where dispersive deformations can be reconstructedexplicitly at all orders of the deformation parameter ǫ . Let us consider system (4) with theHamiltonian density h = w + e v , v t = w x , w t = e v v x + w y . On the elimination of w , it reduces to the Boyer-Finley equation [3], v tt − v ty = ( e v ) xx , (the left hand side can be put into the standard form v ty by a linear transformation of t and y ).An integrable dispersive deformation of this example is closely related to the 2D Toda equation,see [1] for an equivalent construction based on the central extension procedure. Let us introducethe auxiliary Hamiltonian system (cid:18) uw (cid:19) t = (cid:18) ǫ sinh( ǫd/dx ) ǫ sinh( ǫd/dx ) d/dy (cid:19) (cid:18) h u h w (cid:19) , (19)where the Hamiltonian density h is the same as above, h ( u, w ) = w + e u (the exact relationbetween u and v is specified below). Explicitly, this gives u t = 1 ǫ sinh( ǫd/dx ) w, w t = 1 ǫ sinh( ǫd/dx ) e u + w y , which, on elimination of w , leads to the integrable 2D Toda equation, u tt − u ty = 1 ǫ (sinh( ǫd/dx )) e u = 14 ǫ (cid:16) e u ( x +2 ǫ ) + e u ( x − ǫ ) − e u ( x ) (cid:17) . Introducing the change of variables u ↔ v by the formula u = 1 ǫ ( d/dx ) − sinh( ǫd/dx ) v = ( d/dx ) − (cid:18) v ( x + ǫ ) − v ( x − ǫ )2 ǫ (cid:19) = v + ǫ v xx + ǫ v xxxx + . . . , one can verify that the Hamiltonian operator in (19) transforms into the Hamiltonian operatorin (4), while the Hamiltonian density h ( u, w ) = w + e u takes the form h ( v, w ) = w exp (cid:18) ǫ ( d/dx ) − sinh( ǫd/dx ) v (cid:19) = w exp (cid:18) v + ǫ v xx + ǫ v xxxx + . . . (cid:19) = w e v (cid:18) ǫ v xx + ǫ
5! ( v xxxx + 53 v xx ) + . . . (cid:19) . This provides the required integrable deformation for the Hamiltonian density h = w + e v . The dimensional y -independent limit of this construction was discussed in [10].10 Example 2: deformation of the dKP equation
For β = 0 , α = 1 the Hamiltonian density from Example 2 takes the form h ( v, w ) = v + f ( w ) where f ′′′′ f ′′ = 3 f ′′′ . The corresponding deformation assumes the form H = Z v f ( w ) + ǫ f ′′′ (cid:18) − f ′′ v x + 2 w x + f ′′ w y (cid:19) + O ( ǫ ) dxdy. Without any loss of generality one can set f ( w ) = √ w / . In this case the dispersionless systemtakes the form v t = ( √ w ) x , w t = v x + ( √ w ) y . Introducing the new variable u = √ w one obtains v t = u x , uu t = v x + u y , which, on elimination of v , leads to the dKP equation ( u y − uu t ) t + u xx = 0 , with ‘non-standard’notation for the independent variables. The corresponding KP equation, ( u y − uu t ) t + u xx + ǫ u tttt = 0 , gives rise to the following integrable deformation of the original system: v t = ( √ w ) x , w t = v x + ( √ w ) y + ǫ ( √ w ) ttt . This, however, is clearly outside the class of Hamiltonian deformations.
In this section we summarise our results on deformations of Hamiltonian systems of the form (cid:18) vw (cid:19) t = (cid:18) d/dx d/dy (cid:19) (cid:18) h v h w (cid:19) , or, explicitly, v t = ( h v ) x , w t = ( h w ) y . The integrability conditions constitute a system of fourth order PDEs for the Hamiltonian density h ( v, w ) [14]: h vw ( h vw − h vv h ww ) h vvvv = 4 h vw h vvv ( h vw h vvw − h vv h vww )+3 h vv h vw h vvw − h vv h ww h vvv h vvw − h vw h ww h vvv ,h vw ( h vw − h vv h ww ) h vvvw = − h vw h vvv ( h vv h + h ww h vvw )+3 h vw h vvw − h vv h ww h vvv h vww + h vw h vvv h vww ,h vw ( h vw − h vv h ww ) h vvww = 4 h vw h vvw h vww − h vv h vvw ( h vw h + h ww h vww ) − h ww h vvv ( h vw h vww + h vv h ) ,h vw ( h vw − h vv h ww ) h = − h vw h ( h ww h vvv + h vv h vww )+3 h vw h vww − h vv h ww h h vvw + h vw h h vvw ,h vw ( h vw − h vv h ww ) h = 4 h vw h ( h vw h vww − h ww h vvw )+3 h ww h vw h vww − h vv h ww h h vww − h vw h vv h . (20)11his system is in involution, and is invariant under the -parameter group of Lie-point symme-tries, v → av + b,w → cw + d,h → αh + βv + γw + δ, which constitute the equivalence group of the problem. Furthermore, there is an obvious sym-metry corresponding to the interchange of v and w . Particular solutions include h ( v, w ) = vw + αv , h ( v, w ) = w √ v + αv / , h ( v, w ) = 12 ( v + w ) + e v , h ( v, w ) = ( w + a ( v )) , where a ( v ) solves the ODE a ′ a ′′ a ′′′′ = 2 a ′′ a ′′′ + a ′ a ′′′ , etc, see [15] for the general discussion. Asbefore, generic Hamiltonians are not deformable. Although Case I turns out to be considerablymore complicated from computational point of view, our calculations support the conjecture thatdeformable densities of type I are characterised by exactly the same additional constraints as incase II: Conjecture
A Hamiltonian H = R h ( v, w ) dxdy of type I possesses a nontrivial integrabledeformation to the order ǫ if and only if, along with the integrability conditions (20), it satisfiesthe additional differential constraints h vvv h vww − h vvw = 0 , h vvv h − h vvw h vww = 0 , h h vvw − h vww = 0 , or, equivalently, rank (cid:18) h vvv h vvw h vww h vvw h vww h (cid:19) = 1 . Modulo equivalence transformations, this gives three types of deformable densities: h ( v, w ) = vw + αv , h ( v, w ) = 12 ( v + w ) + e v , h ( v, w ) = α v + βvw + γ w + f ( v + w ) , where f satisfies the integrability condition ( β + f ′′ ) △ f ′′′′ = f ′′′ [3 △ + ( β − α )( β − γ )] , here △ = ( β + f ′′ ) − ( α + f ′′ )( γ + f ′′ ) . Dispersive deformations of these Hamiltonians are givenby the following formulae (we use the normalisation f = f = 0 which can always be achieved bya canonical transformation; furthermore, all ǫ -deformations are trivial, and have been set equalto zero): H = Z vw + αv + ǫ (6 αv x + v x w x ) + O ( ǫ ) dxdy,H = Z
12 ( v + w ) + e v + ǫ e v (cid:0) e v ) v x − w x − e v v x w x + v x w y + v y w x (cid:1) + O ( ǫ ) dxdy. The third case is somewhat more complicated: H = Z α v + βvw + γ w + f ( v + w ) + ǫ h + O ( ǫ ) dxdy, h = f v x + f v y + f w x + f w y + f v x w x + f ( v x w y + v y w x ) + f v y w y , and the coefficients f − f are defined as follows: f = ( α + f ′′ )( β + f ′′ ) △ , f = ( γ + f ′′ )( β + f ′′ ) △ ,f = (4 β − α + 3 f ′′ )( β + f ′′ ) △ , f = (4 β − γ + 3 f ′′ )( β + f ′′ ) △ ,f = ( β + f ′′ ) △ [ △ + 4( α + f ′′ )( β + f ′′ )] , f = ( β + f ′′ ) △ [ △ + 4( γ + f ′′ )( β + f ′′ )] ,f = 12 ( β + f ′′ ) △ [2 △ + (2 β − α − γ ) ] . We conjecture that Hamiltonians from Theorem 3 can be deformed to all orders in ǫ . In this section we consider Hamiltonian systems of the form (cid:18) vw (cid:19) t = " (cid:18) v ww (cid:19) ddx + (cid:18) vv w (cid:19) ddy + (cid:18) v x v y w x w y (cid:19) h v h w (cid:19) , or, explicitly, v t = (2 vh v + wh w − h ) x + ( vh w ) y , w t = ( wh v ) x + (2 wh w + vh v − h ) y . The integrability conditions constitute a system of fourth order PDEs for the Hamiltonian density h ( v, w ) which is not presented here due to its complexity. We have verified in [16] that this systemis in involution, and its solution space is -dimensional. It is invariant under an -dimensionalgroup of Lie-point symmetries, v → av + bw,w → cv + dw,h → αh + βv + γw + δ, which constitute the equivalence group of the problem. Particular integrable Hamiltonian den-sities include h ( v, w ) = wv + αv , h ( v, w ) = wv + ( v + c ) ln( v + c ) , h ( v, w ) = ( vw ) / . A more complicated example has the form h ( v, w ) = wf ( v/w ) where the function f ( y ) solvesthe ODE y f ′ f ′′ ( f ′ + 2 yf ′′ ) f ′′′′ = 2 y f ′ ( f ′ + 3 yf ′′ ) f ′′′ +2 yf ′′ (2 f ′ + 9 yf ′ f ′′ + 4 y f ′′ ) f ′′′ + 3 f ′′ (2 f ′ + 8 yf ′ f ′′ + 5 y f ′′ ) . However, calculations suggest that none of these examples are deformable:
Conjecture
For Hamiltonians of type III, all deformations of the order ǫ are trivial. In this paper we discuss, in the spirit of [8] – [13], the deformation theory of dimen-sional Hamiltonian systems of hydrodynamic type, defined by local Poisson brackets and localHamiltonians. Our results demonstrate that, already at the order ǫ , the requirement of theexistence of nontrivial dispersive deformations is very restrictive so that ‘generic’ integrableHamiltonians are not deformable. The main reason for this is apparently the assumption thatall higher order dispersive corrections are local expressions in the dependent variables v, w and x, y -derivatives thereof. It would be of interest to extend this scheme to the case of nonlocalbrackets/Hamiltonians, see [2] for particular examples obtained via Dirac reduction.Furthermore, to the best of our knowledge the theory of deformations of multi-dimensionalPoisson brackets of hydrodynamic type has not been constructed: is it true that all such defor-mations are trivial, as in the dimensional case?Finally, calculations leading to Example 1 of Sect. 2 show that any Hamiltonian of theform H = R w + f ( v ) dxdy , where the function f is arbitrary, possesses a unique dispersivedeformation of the form H = Z w f ( v ) + ǫ f ′′′ v x + O ( ǫ ) dxdy, which inherits all one-phase solutions to the order ǫ . Thus, one can speak of ‘partial integrability’of a certain kind. However, already the requirement of the inheritance of two-phase solutionsforces f to satisfy the integrability condition f ′′′′ f ′′ = f ′′′ . Acknowledgements
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