On a class of integrable Hamiltonian equations in 2+1 dimensions
aa r X i v : . [ n li n . S I] J a n On a class of integrable Hamiltonian equations in 2+1dimensions
B. Gormley , E.V. Ferapontov , , V.S. Novikov Department of Mathematical SciencesLoughborough UniversityLoughborough, Leicestershire LE11 3TUUnited Kingdom Institute of Mathematics, Ufa Federal Research Centre,Russian Academy of Sciences, 112, Chernyshevsky Street,Ufa 450077, Russiae-mails:
[email protected]@[email protected]
Abstract
We classify integrable Hamiltonian equations of the form u t = ∂ x (cid:18) δHδu (cid:19) , H = Z h ( u, w ) dxdy, where the Hamiltonian density h ( u, w ) is a function of two variables: dependent vari-able u and the non-locality w = ∂ − x ∂ y u . Based on the method of hydrodynamicreductions, the integrability conditions are derived (in the form of an involutive PDEsystem for the Hamiltonian density h ). We show that the generic integrable densityis expressed in terms of the Weierstrass σ -function: h ( u, w ) = σ ( u ) e w . DispersionlessLax pairs, commuting flows and dispersive deformations of the resulting equations arealso discussed.MSC: 35Q51, 37K10. Keywords:
Hamiltonian PDEs, hydrodynamic reductions, Einstein-Weyl geometry,dispersionless Lax pairs, commuting flows, dispersive deformations, Weierstrass ellipticfunctions. o Allan Fordy on the occasion of his 70th birthday Contents h = σ ( u ) e w In this paper we investigate Hamiltonian systems of the form u t = ∂ x (cid:18) δHδu (cid:19) , H = Z h ( u, w ) dxdy. (1)Here ∂ x is the Hamiltonian operator, and the Hamiltonian density h ( u, w ) depends on u and the nonlocal variable w = ∂ − x ∂ y u (equivalently, w x = u y ). Since δHδu = h u + ∂ − x ∂ y ( h w )we can rewrite equation (1) in the two-component first-order quasilinear form: u t = ( h u ) x + ( h w ) y , w x = u y . (2)Familiar examples of this type include the dispersionless KP equation ( h = w + u )and the dispersionless Toda (Boyer-Finley) equation ( h = e w ). Our main goal is to classifyintegrable systems within class (2) and to construct their dispersionless Lax pairs, com-muting flows and dispersive deformations. Before stating our main results, let us beginwith a brief description of the existing approaches to dispersionless integrability in 2+1dimensions. 2 .2 Equivalent approaches to dispersionless integrability Here we summarise three existing approaches to integrability of equations of type (2),namely, the method of hydrodynamic reductions, the geometric approach based on inte-grable conformal geometry (Einstein-Weyl geometry), and the method of dispersionless Laxpairs. Based on seemingly different ideas, these approaches lead to equivalent integrabilityconditions/classification results [13].
The method of hydrodynamic reductions , see e.g. [16, 9], consists of seeking multi-phase solutions to system (2) in the form u = u ( R , R , . . . , R n ) , w = w ( R , R , . . . , R n ) (3)where the phases R i ( x, y, t ) (also known as Riemann invariants; note that their number n can be arbitrary) satisfy a pair of commuting hydrodynamic-type systems: R iy = µ i ( R ) R ix , R it = λ i ( R ) R ix ; (4)we recall that the commutativity conditions are equivalent to the following constraints forthe characteristic speeds µ i , λ i [18, 19]: ∂ j µ i µ j − µ i = ∂ j λ i λ j − λ i , (5) i = j, ∂ j = ∂ R j . Substituting ansatz (3) into (2) and using (4), (5) one obtains anoverdetermined system for the unknowns u, w, µ i , λ i , viewed as functions of R , . . . , R n (the so-called generalised Gibbons-Tsarev system, or GT-system). System (2) is said tobe integrable by the method of hydrodynamic reductions if it possesses ‘sufficiently many’multi-phase solutions of type (3), in other words, if the corresponding GT-system is in-volutive. Note that the coefficients of GT-system will depend on the density h ( u, w ) andits partial derivatives. The requirement that GT-system is involutive imposes differentialconstraints for the Hamiltonian density h , the so-called integrability conditions. Details ofthis computation will be given in Section 2.1. Integrability via Einstein-Weyl geometry.
Let us first introduce a conformal structuredefined by the characteristic variety of system (2). Given a 2 × A ( v ) v x + B ( v ) v x + C ( v ) v x = 0where A, B, C are 2 × v = ( u, w ) T , the characteristic equationof this system, det( Ap + Bp + Cp ) = 0, defines a conic g ij p i p j = 0. This gives thecharacteristic conformal structure [ g ] = g ij dx i dx j where g ij is the inverse matrix of g ij .For system (2) direct calculation gives[ g ] = 4 h ww dxdt − dy − h uw dydt + 4( h ww h uu − h uw ) dt ; (6)3ere we set ( x , x , x ) = ( x, y, t ). Note that [ g ] depends upon a solution to the system (wewill assume [ g ] to be non-degenerate, which is equivalent to the condition h ww = 0). Itturns out that integrability of system (2) can be reformulated geometrically as the Einstein-Weyl property of the characteristic conformal structure [ g ]. We recall that Einstein-Weylgeometry is a triple ( D , [ g ] , ω ) where [ g ] is a conformal structure, D is a symmetric affineconnection and ω = ω k dx k is a 1-form such that D k g ij = ω k g ij , R ( ij ) = Λ g ij (7)for some function Λ [2, 3]; here R ( ij ) is the symmetrised Ricci tensor of D . Note thatthe first part of equations (7), known as the Weyl equations, uniquely determines D once[ g ] and ω are specified. It was observed in [13] that for broad classes of dispersionlessintegrable systems (in particular, for systems of type (2)), the one-form ω is given in termsof [ g ] by a universal explicit formula ω k = 2 g kj D s g js + D k ln (det g ij )where D k denotes the total derivative with respect to x k . Applied to [ g ] given by (6), thisformula implies ω = 0 ,ω = 2( h uuw v xx + 2 h uww v xy + h v yy ) h ww , (8) ω = 4( h uw ( h uuw v xx + 2 h uww v xy + h v yy ) h ww − h ww ( h v xx + 2 h uuw v xy + h uww v yy )) h ww . To summarise, integrability of system (2) is equivalent to the Einstein-Weyl property of[ g ] , ω given by (6), (8) on every solution of system (2). Note that in 3D, Einstein-Weylequations (7) are themselves integrable by the twistor construction [17], see also [8], andthus constitute ‘integrable conformal geometry’. Dispersionless Lax pair of system (2) consist of two Hamilton-Jacobi type equations foran auxiliary function S , S t = F ( S x , u, w ) , S y = G ( S x , u, w ) , whose compatibility condition, S ty = S yt , is equivalent to system (2). Dispersionless Laxpairs were introduced in [20] as quasiclassical limits of Lax pairs of integrable solitonequations in 2+1D. It is known that the existence of a dispersionless Lax representation isequivalent to hydrodynamic/geometric integrability discussed above [10, 13]. We refer toSection 2.3 for dispersionless Lax pairs of integrable systems (2).4 .3 Summary of the main results Our first result is the set of integrability conditions for the Hamiltonian density h . Theorem 1.
The following conditions are equivalent:(a) System (2) is integrable by the method of hydrodynamic reductions;(b) Characteristic conformal structure [g] and covector ω given by (6), (8) satisfy Einstein-Weyl equations (7) on every solution of system (2);(c) System (2) possesses a dispersionless Lax pair;(d) Hamiltonian density h ( u, w ) satisfies the following set of integrability conditions: h − h ww h = 0 ,h uww h − h ww h = 0 ,h uuw h − h ww h uuww = 0 , (9) h uuu h − h ww h uuuw = 0 , − h uuw + 4 h uww h uuu − h ww h uuuu = 0 . Theorem 1 is proved in Section 2.1. The system of integrability conditions (9) is in-volutive, and modulo natural equivalence transformations its solutions can be reduced toone of the six canonical forms.
Theorem 2.
Solutions h ( u, w ) of system (9) can be reduced to one of the six canonicalforms: h ( u, w ) = 12 w + 16 u ,h ( u, w ) = w + u w − u ,h ( u, w ) = uw + βu ,h ( u, w ) = e w ,h ( u, w ) = ue w ,h ( u, w ) = σ ( u ; 0 , g ) e w ; here β and g are constants, and σ ( u ; g , g ) denotes the Weierstrass sigma function. Theorem 2 is proved in Section 2.2. Dispersionless Lax pairs for the correspondingsystems (2) are constructed in Section 2.3.It turns out that every integrable system (2) possesses a higher commuting flow of theform u τ = a ( u, w, v ) u x + b ( u, w, v ) u y + c ( u, w, v ) w y + d ( u, w, v ) v y ,w x = u y , (10) v x = ( p ( u, w )) y , τ is the higher ‘time’, and v = ∂ − x ∂ y p ( u, w ) is an extra nonlocal variable (in contrastto the 1+1 dimensional case, higher commuting flows in 2+1 dimensions require highernonlocalities). Remarkably, the structure of higher nonlocalities is uniquely determined bythe original system (2), in particular, the function p ( u, w ) can be expressed in terms of h ( u, w ): p = h w . Furthermore, commuting flow (10) is automatically Hamiltonian. Theorem 3.
Every integrable system (2) possesses a higher commuting flow (10) with thenonlocality v x = ( h w ) y . Commuting flow (10) is Hamiltonian with the Hamiltonian density f ( u, w, v ) of the form f ( u, w, v ) = vh w ( u, w ) + g ( u, w ) , where g ( u, w ) can be recovered from the compatible equations g ww = 4 h uw h ww + αwh ww ,g uuu = 8 h uw h uuu + αwh uuu ,g uuw = 6 h uw h uuw + 2 h ww h uuu + αwh uuw . Here the constant α is defined by the relation α = 2 h ww ∂∂w (cid:0) h uw h ww (cid:1) which follows from inte-grability conditions (9). Theorem 3 is proved in Section 2.4. Dispersionless Lax pairs for commuting flows areconstructed in Section 2.5.
In this section we prove Theorems 1-3 and construct Lax pairs for integrable systems (2)and their commuting flows (10).
Equivalences (a) ⇔ (b) and (a) ⇔ (c) of Theorem 1 follow from the results of [13]and [10] which hold for general two-component systems of hydrodynamic type in 2+1dimensions. Equivalence (a) ⇔ (d) can be demonstrated as follows. Let us rewrite system (2) in theform u t = h uu u x + 2 h uw u y + h ww w y , w x = u y , and substitute the ansatz u = u ( R , R , . . . , R n ) , w = w ( R , R , . . . , R n ). Using equations(4) and collecting coefficients at R ix we obtain ∂ i w = µ i ∂ i u , along with the dispersionrelation λ i = h uu +2 h uw µ i + h ww ( µ i ) . Substituting the last formula into the commutativityconditions (5) we obtain ∂ j µ i = h uuu + h uuw ( µ j + 2 µ i ) + h uww (cid:0) µ i µ j + ( µ i ) (cid:1) + h µ j ( µ i ) h ww ( µ j − µ i ) ∂ j u. (11) ∂ i ∂ j w = ∂ j ∂ i w results in ∂ i ∂ j u = 2 h uuu + 3 h uuw ( µ j + µ i ) + h uww (( µ i ) + 4 µ i µ j + ( µ j ) ) + h ( µ j ( µ i ) + µ i ( µ j ) ) h ww ( µ j − µ i ) ∂ i u∂ j u. (12) Equations (11), (12) constitute the corresponding GT-system. As one can see, it containspartial derivatives of the Hamiltonian density h in the coefficients. Verifying involutivityof GT-system amounts to checking the compatibility conditions ∂ k ( ∂ j µ i ) = ∂ j ( ∂ k µ i ) and ∂ k ( ∂ i ∂ j u ) = ∂ j ( ∂ i ∂ k u ). Direct computation (performed in Mathematica) results in theintegrability conditions (9) for h ( u, w ). Note that without any loss of generality one canrestrict to the case when the number of Riemann invariants R i is equal to three, indeed,all compatibility conditions involve three distinct indices only. This finishes the proof ofTheorem 1. We have five integrability conditions, namely h − h ww h = 0 , (13) h uww h − h ww h = 0 , (14) h uuw h − h ww h uuww = 0 , (15) h uuu h − h ww h uuuw = 0 , (16) − h uuw + 4 h uww h uuu − h ww h uuuu = 0 . (17)The classification of solutions will be performed modulo equivalence transformations leavingsystem (2) form-invariant (and therefore preserving the integrability conditions). Theseinclude ˜ x = x − at, ˜ y = y − bt, ˜ h = h + au + buw + mu + nw + p, (18)as well as ˜ x = x − sy, ˜ w = w + su ; (19)(other variables remain unchanged). We will always assume h ww = 0 which is equivalentto the requirement of irreducibility of the dispersion relation. There are two main cases toconsider. Case 1: h = 0 . Then h ( u, w ) = α ( u ) w + β ( u ) w + γ ( u ) , and the integrability conditions imply α ′′ = 0 , β ′′′ = 0 , − β ′′ + 8 α ′ γ ′′′ − αγ ′′′′ = 0 . There are two further subcases: α = 1 and α = u .7he subcase α = 1 leads, modulo equivalence transformations (18), to densities of theform h ( u, w ) = w + β u w − β u + γ u ,β , γ = const . For β = 0 we obtain the first case of Theorem 2 (after a suitable rescaling).If β = 0 then we can eliminate the term u by a translation of u . This gives the secondcase of Theorem 2 (after rescaling of u and w ).The subcase α = u leads, modulo equivalence transformations (18), to densities of theform h ( u, w ) = uw + β u w + γ u + β u .β , γ = const . Note that we can set β = 0 using transformation (19) with s = β /
2. Thisgives the third case of Theorem 2.
Case 2: h = 0 . Then the first two integrability conditions (13) and (14) imply h = ch ww for some constant c (which can be set equal to 1). This gives h ( u, w ) = a ( u ) e w + p ( u ) w + q ( u ) . The next two integrability conditions (15) and (16) give p ′′ = 0 and q ′′′ = 0, respectively.Thus, modulo equivalence transformations (18) we can assume h ( u, w ) = a ( u ) e w . Finally,equation (17) implies aa ′′′′ − a ′ a ′′′ + 3 a ′′ = 0 , which is the classical equation for the Weierstrass sigma function (equianharmonic case g = 0). Setting ℘ = − (ln a ) ′′ we obtain ℘ ′′ = 6 ℘ , which integrates to ℘ ′ = 4 ℘ − g , (20) g = const . There are three subcases. Subcase g = 0 , ℘ = 0 . Then a ( u ) = e αu + β and modulo equivalence transformations (19)we obtain Case 4 of Theorem 2. Subcase g = 0 , ℘ = u . Then a ( u ) = ue αu + β and modulo equivalence transformations(19) we obtain Case 5 of Theorem 2. Subcase g = 0 . Then a ( u ) = σ ( u ; 0 , g ) e αu + β and modulo equivalence transformations(19) we obtain the last case of our classification. This finishes the proof of Theorem 2. Remark.
The paper [12] gives a classification of integrable two-component Hamiltoniansystems of the form U t W t = ∂ x ∂ x ∂ y δHδUδHδW (21)8here H = R F ( U, W ) dxdy . Explicitly, we have U t = ( F W ) x , W t = ( F U ) x + ( F W ) y . Let us introduce a contact change of variables (
U, W, F ) → ( u, w, f ) via partial Legendretransform: w = F W , u = U, f = F − W F W , f w = − W, f u = F U . In the new variables the system becomes w y = − ( f u ) x − ( f w ) t , u t = w x . Modulo relabelling u ↔ w, f → − h, y → t, t → x, x → y these equations coincide with(2). Thus, Hamiltonian formalisms (1) and (21) are equivalent. Examples of dKP andBoyer-Finley equations suggest however that Hamiltonian formalism (1) is more naturaland convenient, indeed, in the form (1) both equations arise directly in their ‘physical’variables. In this section we provide dispersionless Lax representations for all six canonical forms ofTheorem 2. The results are summarised in Table 1 below.
Table 1: Dispersionless Lax pairs for integrable systems (2) h ( u, w ) Dispersionless Lax pair h ( u, w ) = w + u System (2) : S t = S x + uS x + wu t = uu x + w y S y = S x + uw x = u y h ( u, w ) = w + u w − u System (2) : S t = (3 u + 2 w ) S x + 2 uS x + S x u t = (2 w − u ) u x + 4 uu y + 2 w y S y = uS x + S x w x = u y h ( u, w ) = uw + βu System (2) : S t = 4 u ℘ ( S x )( w + u ℘ ′ ( S x )) u t = 42 βu u x + 4 wu y + 2 uw y S y = u ℘ ( S x ) w x = u y here ℘ ′ = 4 ℘ − βh ( u, w ) = e w System (2) : S t = − e w S x + u u t = e w w y S y = − ln( S x + u ) w x = u y h ( u, w ) = ue w System (2) : S t = u e w S x u − S x u t = e w (2 u y + uw y ) S y = ln( S x − u )+ ε ln( S x − εu )+ ε ln( S x − ε u ) w x = u y here ε = exp (cid:0) πi (cid:1) h ( u, w ) = σ ( u ) e w System (2) : S t = σ ( u ) e w G u ( S x , u ) u t = e w ( σ ′′ u x + 2 σ ′ u y + σw y ) S y = G ( S x , u ) w x = u y here σ ( u ) = σ ( u ; 0 , g )In the last case the function G ( p, u ) is defined by the equations G p = G uu G u − ζ ( u ) , G uuu G u − G uu + 2 ℘ ( u ) G u = 0 (22)where ζ and ℘ are the Weierstrass functions (equianharmonic case g = 0). The generalsolution of these equations is given by the formula G ( p, u ) = ln σ ( λ ( p − u )) + ǫ ln σ ( λ ( p − ǫu )) + ǫ ln σ ( λ ( p − ǫ u )) (23)where ǫ = e πi/ = − + i √ and λ = i √ . Note that the degeneration g → , σ ( u ) → u takes the Lax pair corresponding to the Hamiltonian density h = σ ( u ) e w to the Lax pairfor the density h = ue w . We refer to the Appendix for a proof that formula (23) indeedsolves the equations (22): this requires some non-standard identities for equianharmonicelliptic functions. 10 .4 Commuting flows: proof of Theorem 3 Our aim is to show that every integrable system (2) possesses a commuting flow of theform (10), u τ = a ( u, w, v ) u x + b ( u, w, v ) u y + c ( u, w, v ) w y + d ( u, w, v ) v y ,w x = u y ,v x = ( p ( u, w )) y . Here τ is the higher ‘time’ variable and v = ∂ − x ∂ y p ( u, w ) is a new nonlocality (to bedetermined). Due to the presence of nonlocal variables, direct computation of compatibilitycondition u tτ = u τt is not straightforward. Therefore, we adopt a different approach andrequire that the combined system (2) ∪ (10), u t = ( h u ) x + ( h w ) y , (24) u τ = a ( u, w, v ) u x + b ( u, w, v ) u y + c ( u, w, v ) w y + d ( u, w, v ) v y , (25) w x = u y , (26) v x = ( p ( u, w )) y , (27)possesses hydrodynamic reductions. Thus, we seek multiphase solutions of the form u = u ( R , . . . , R n ), w = w ( R , . . . , R n ) and v = v ( R , . . . , R n ) where the Riemann invariants R i satisfy a triple of commuting systems of hydrodynamic type: R iy = µ i ( R ) R ix , R it = λ i ( R ) R ix , R iτ = η i ( R ) R ix . We recall that the commutativity conditions are equivalent to ∂ j µ i µ j − µ i = ∂ j λ i λ j − λ i = ∂ j η i η j − η i . (28)Following the same procedure as in Section 2.1, from equations (24) and (26) we obtainthe relations ∂ i w = µ i ∂ i u , the GT-system (11), (12), and the integrability conditions (9)for the Hamiltonian density h ( u, w ). Similarly, equation (27) implies ∂ i v = ( p u µ i + p w ( µ i ) ) ∂ i u, and the compatibility condition ∂ j ∂ i v = ∂ i ∂ j v results in the relations h uuw p w − h ww p uu = 0 , h uww p w − h ww p uw = 0 , h p w − h ww p ww = 0 . Modulo unessential constants of integration (which can be removed by equivalence trans-formations) these relations uniquely specify the nonlocality: p ( u, w ) = h w ( u, w ) . η i = a + bµ i + ( c + p u d )( µ i ) + p w d ( µ i ) . Substituting η i into the commutativity conditions (28) we obtain the following set of rela-tions: p w d v = 0 , (29) (cid:0) ( p ww d + p w d w ) + p w p u d v (cid:1) h ww = 2 p w dh , (30)( p uw d + p w d u ) h ww = 2 p w dh uww , (31) h ww ( c v + p u d v ) p w = p w dh , (32) h ww (cid:0) ( c w + p uw d + p u d w ) + ( c v + p u d v ) p u (cid:1) = 5 h uww dp w + ch + p u dh , (33) h ww b v p w = 2 p w dh uww , (34) h ww ( c u + p uu d + p u d u ) = 4 p w dh uuw + ch uww + p u dh uww , (35) h ww a v p w = p w dh uuw , (36) h ww ( b w + b v p u ) = 4 p w dh uuw + 2 ch uww + 2 p u dh uww , (37) h ww b u = 2 p w dh uuu + 2 ch uuw + 2 p u dh uuw , (38) h ww ( a w + a v p u ) = p w dh uuu + ch uuw + p u dh uuw , (39) h ww a u = ch uuu + p u dh uuu . (40)Using the fact that p = h w we solve these relations modulo the integrability conditions (9),recall that h ww = 0. Equation (29) gives d v = 0. Equations (30, 31) imply dh − h ww d w = 0 , dh uww − h ww d u = 0 , which can be solved for d : d = δh ww , for some constant δ (which will be set equal to 2 in what follows). Equation (32) gives c v = d h h ww . Setting c = d h h ww v + c for some c = c ( u, w ) and substituting into equation (35) we find c = 3 dh uw + c ( w ) h ww . Substituting c = d h h ww v + 3 dh uw + c ( w ) h ww into equation (33) we find( c ) w = d (cid:18) h uww h ww − h h uw h ww (cid:19) = d ∂∂w (cid:18) h uw h ww (cid:19) .
12t turns out that modulo the integrability conditions ( c ) w is a constant. If we set α = d ∂∂w (cid:18) h uw h ww (cid:19) , the final formula for c can be written as c = d h h ww v + 3 dh uw + αwh ww . The equations for the coefficients a and b cannot be integrated explicitly; rearranging theremaining equations gives the following final result: a u = h uuu h ww ( c + dh uw ) ,a w = h uuw h ww c + dh uuu ,a v = d h uuw h ww . b u = 2 (cid:0) dh uuu + h uuw h ww ( c + dh uw ) (cid:1) ,b w = 2 (cid:0) h uww h ww c + 2 dh uuw (cid:1) ,b v = 2 d h uww h ww .c = d h h ww v + 3 dh uw + αwh ww , d = δh ww , α = d ∂∂w (cid:18) h uw h ww (cid:19) . The equations for a and b are consistent modulo integrability conditions (9). This provesthe existence of commuting flows (10). Hamiltonian formulation of commuting flows.
Our next goal in to show that theobtained commuting flow can be cast into Hamiltonian form u τ = ∂ x (cid:18) δFδu (cid:19) , F = Z f ( u, w, v ) dxdy, (41)with the nonlocal variables w, v defined by w x = u y , v x = ( h w ) y . More precisely, we claimthat the commuting density f is given by the formula f ( u, w, v ) = vh w + g ( u, w )where the function g ( u, w ) is yet to be determined. We have δFδu = 2 vh uw + g u + ∂ − x ∂ y (2 vh ww + g w ) , so that equation (41) takes the form u τ = (2 vh uw + g u ) x + (2 vh ww + g w ) y , (42) w x = u y ,v x = ( h w ) y . u τ = (2 vh uuw + g uu ) u x + (4 vh uww + 2 g uw + 2( h uw ) ) u y + (2 h uw h ww + 2 vh + g ww ) w y + 2 h ww v y . Comparing this with (25) we thus require a = 2 vh uuw + g uu ,b = 4 vh uww + 2 g uw + 2 h uw ,c = 2 vh + 2 h uw h ww + g ww ,d = 2 h ww . Using the expressions for a, b, c, d calculated above we obtain the equations for g ( u, w ): g ww = 4 h uw h ww + αwh ww ,g uuu = 8 h uw h uuu + αwh uuu ,g uuw = 6 h uw h uuw + 2 h ww h uuu + αwh uuw ;note that these equations are consistent modulo the integrability conditions (9). Thisfinishes the proof of Theorem 3. In this section we calculate commuting flows of integrable systems (2) and construct theirdispersionless Lax pairs.
1. Hamiltonian density h ( u, w ) = u + w . The commuting density is f ( u, w, v ) = vw + u w. Commuting flow has the form (note that α = 0): u τ = 2 wu x + 4 uu y + 2 v y ,w x = u y ,v x = w y . Dispersionless Lax pair: S y = 12 S x + u,S τ = 12 S x + 2 uS x + 2 wS x + 2 u + 2 v. . Hamiltonian density h ( u, w ) = w + u w − u . The commuting density is f ( u, w, v ) = 2 wv + u v + 8 uw − u . Commuting flow has the form (note that α = 0): u τ = (4 v − u ) u x + (32 w + 8 u ) u y + 24 uw y + 4 v y ,w x = u y ,v x = (2 w + u ) y . Dispersionless Lax pair: S y = uS x + 14 S x ,S τ = (4 v + 32 uw + 16 u ) S x + (8 w + 24 w ) S x + 8 uS x + 45 S x .
3. Hamiltonian density h ( u, w ) = uw + βu . The commuting density is f ( u, w, v ) = 2 uwv + 4 uw + 20 βu w. Commuting flow has the form (note that α = 4): u τ = 840 βu wu x + (8 v + 32 w + 280 βu ) u y + 32 uww y + 4 uv y ,w x = u y ,v x = (2 uw ) y . Dispersionless Lax pair: S y = u ℘ ( S x ) ,S τ = (8 u v + 32 u w + 16 u w℘ ′ ( S x ) + 8 u ℘ ( S x )) ℘ ( S x ) , where ℘ ′ = 4 ℘ − β .
4. Hamiltonian density h ( u, w ) = e w . The commuting density is f ( u, w, v ) = ve w . Commuting flow has the form (note that α = 0): u τ = 2 ve w w y + 2 e w v y ,w x = u y ,v x = ( e w ) y . S y = − ln( S x + u ) ,S τ = − ve w S x + u + e w ( S x + u ) .
5. Hamiltonian density h ( u, w ) = ue w . The commuting density is f ( u, w, v ) = uve w + ue w . Commuting flow has the form (note that α = 0): u τ = (4 ve w + 6 e w ) u y + (2 uve w + 6 ue w ) w y + 2 ue w v y ,w x = u y ,v x = ( ue w ) y . Dispersionless Lax pair: S y = ln( S x − u ) + ε ln( S x − εu ) + ε ln( S x − ε u ) ,S τ = 3 u e w S x (2 u v − vS x − e w S x )( S x − u ) .
6. Hamiltonian density h ( u, w ) = σ ( u ) e w . The commuting density is f ( u, w, v ) = vσ ( u ) e w + σ ( u ) σ ′ ( u ) e w . Commuting flow has the form (note that α = 0): u τ = (2 vσ ′′ e w + ( σσ ′ ) ′′ e w ) u x + (4 vσ ′ e w + (4 σσ ′′ + 6 σ ′ ) e w ) u y + (2 vσe w + 6 σσ ′ e w ) w y + 2 σe w v y ,w x = u y ,v x = ( σe w ) y . Dispersionless Lax pair: S y = G ( S x , u ) S τ = 2[ ve w σ ( u ) + e w σ ( u ) σ ′ ( u )] G u ( S x , u ) − e w σ ( u ) G uu ( S x , u ) , here G ( S x , u ) is defined by equations (22). 16 Dispersive deformations
Dispersive deformations of hydrodynamic type systems in 1 + 1 dimensions were thor-oughly investigated in [4, 5, 6, 7] based on deformations of the corresponding hydrody-namic symmetries. In 2 + 1 dimensions, an alternative approach based on deformations ofhydrodynamic reductions was proposed in [14, 15].It still remains a challenging problem to construct dispersive deformations of all Hamil-tonian systems (2) obtained in this paper. In general, all three ingredients of the con-struction may need to be deformed, namely, the Hamiltonian operator ∂ x , the Hamiltoniandensity h ( u, w ) and the nonlocality w . Here we give just two examples. Example 1: dKP equation.
The Hamiltonian density h = w + u results in thedKP equation: u t = uu x + w y , w x = u y . It possesses an integrable dispersive deformation u t = uu x + w y − ǫ u xxx , w x = u y , which is the full KP equation (in this section ǫ denotes an arbitrary deformation parameter).The KP equation corresponds to the deformed Hamiltonian density h ( u, w ) = 12 w + 16 u + ǫ u x , while the Hamiltonian operator ∂ x and the nonlocality w = ∂ − x ∂ y u stay the same. Indeed,we have u t = ∂ x δHδu = ∂ x (cid:0) ∂ − x ∂ y w + 12 u − ǫ u xx (cid:1) = uu x + w y − ǫ u xxx . Example 2: Boyer-Finley equation.
The Hamiltonian density h = e w results in thedispersionless Toda (Boyer-Finley) equation: u t = e w w y , w x = u y . It possesses an integrable dispersive deformation u t = (cid:18) − T − ǫ (cid:19) e w , w x = (cid:18) T − ǫ (cid:19) u, which is the full Toda equation. Here T and T − denote the forward/backward ǫ -shifts inthe y -direction, so that T − ǫ and − T − ǫ are the forward/backward discrete y -derivatives.The Toda equation corresponds to the deformed nonlocality w = ∂ − x T − ǫ u , while the17amiltonian operator ∂ x and the Hamiltonian density h = e w stay the same. Indeed, wehave δHδu = ∂ − x (cid:18) − T − ǫ (cid:19) e w , so that u t = ∂ x δHδu = (cid:18) − T − ǫ (cid:19) e w , as required. h = σ ( u ) e w Here we prove that expression (23), G ( p, u ) = ln σ ( λ ( p − u )) + ǫ ln σ ( λ ( p − ǫu )) + ǫ ln σ ( λ ( p − ǫ u )) , where ǫ = e πi/ = − + i √ and λ = i √ , solves the equations (22), G p = G uu G u − ζ ( u ) , G uuu G u − G uu + 2 ℘ ( u ) G u = 0 . In what follows we will use the addition formula ζ ( u + v ) = ζ ( u ) + ζ ( v ) + 12 ℘ ′ ( u ) − ℘ ′ ( v ) ℘ ( u ) − ℘ ( v ) . (43)We will also need the following identity: Proposition 1.
In the equianharmonic case, the Weiesrtrass functions satisfy the identity λ ℘ ′ ( λu ) ℘ ( λu ) + 3 λζ ( λu ) − ζ ( u ) = 0 , λ = i √ . (44) Proof:
Using the standard expansions ζ ( z ) = 1 z − g z − . . . , ℘ ( z ) = 1 z + g z + . . . , one can show that formula (44) holds to high order in z for the specific parameter value λ = i √ . Therefore, it is sufficient to establish the differentiated (by u ) identity (44),namely, − λ ℘ ( λu ) + λ g ℘ ( λu ) + ℘ ( u ) = 0 , (45)18here we have used ℘ ′′ = 6 ℘ and ℘ ′ = 4 ℘ − g . Explicitly, (45) reads ℘ ( iu/ √ − g ℘ ( iu/ √
3) + 3 ℘ ( u ) = 0 . Setting u = i √ v we obtain ℘ ( v ) − g ℘ ( v ) + 3 ℘ ( i √ v ) = 0 . (46)Thus, it is sufficient to establish (46). Formulae of this kind appear in the context ofcomplex multiplication for elliptic curves with extra symmetry. Let us begin with thestandard invariance properties of the equianharmonic ζ -function: ζ ( ǫz ) = ǫ ζ ( z ) , ζ ( ǫ z ) = ǫζ ( z );here ǫ = e πi/ = − + i √ is the cubic root of unity. Setting z = 2 v this gives ζ ( − v + i √ v ) = ǫ ζ (2 v ) , ζ ( − v − i √ v ) = ǫζ (2 v ) . Using the addition formula (43) one can rewrite these relations in the form − ζ ( v ) + ζ ( i √ v ) + 12 − ℘ ′ ( v ) − ℘ ′ ( i √ v ) ℘ ( v ) − ℘ ( i √ v ) = ǫ ζ (2 v )and − ζ ( v ) − ζ ( i √ v ) + 12 − ℘ ′ ( v ) + ℘ ′ ( i √ v ) ℘ ( v ) − ℘ ( i √ v ) = ǫζ (2 v ) , respectively. Adding there relations together (and keeping in mind that 1 + ǫ + ǫ = 0) weobtain − ζ ( v ) − ℘ ′ ( v ) ℘ ( v ) − ℘ ( i √ v ) + ζ (2 v ) = 0 . Using the duplication formula ζ (2 v ) = 2 ζ ( v ) + ℘ ( v ) ℘ ′ ( v ) this simplifies to − ℘ ′ ( v ) ℘ ( v ) − ℘ ( i √ v ) + 3 ℘ ( v ) ℘ ′ ( v ) = 0 , which is equivalent to (46) via ℘ ′ = 4 ℘ − g . Proposition 2.
Expression (23) solves the equations (22).
Proof: G ( p, u ) gives G p = λζ ( λ ( p − u )) + λǫζ ( λ ( p − ǫu )) + λǫ ζ ( λ ( p − ǫ u )) ,G u = − λζ ( λ ( p − u )) − λǫ ζ ( λ ( p − ǫu )) − λǫζ ( λ ( p − ǫ u )) . Using the addition formula (43), the identity 1 + ǫ + ǫ = 0, and the invariance ζ ( ǫz ) = ǫ ζ ( z ) , ζ ( ǫ z ) = ǫζ ( z ) ,℘ ( ǫz ) = ǫ℘ ( z ) , ℘ ( ǫ z ) = ǫ ℘ ( z ) ,℘ ′ ( ǫz ) = ℘ ( z ) , ℘ ′ ( ǫ z ) = ℘ ( z ) , we obtain: λ G p = ζ ( λ ( p − u )) + ǫζ ( λ ( p − ǫu )) + ǫ ζ ( λ ( p − ǫ u ))= ζ ( λp ) − ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ℘ ( λu ) + ǫ (cid:16) ζ ( λp ) − ǫ ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ǫ℘ ( λu ) (cid:17) + ǫ (cid:16) ζ ( λp ) − ǫζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ǫ ℘ ( λu ) (cid:17) = − ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu )2 (cid:16) ℘ ( λp ) − ℘ ( λu ) + ǫ℘ ( λp ) − ǫ℘ ( λu ) + ǫ ℘ ( λp ) − ǫ ℘ ( λu ) (cid:17) = − ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu )2 3 ℘ ( λu ) ℘ ( λp ) − ℘ ( λu ) = − ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu )2 12 ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) = − ζ ( λu ) + ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) .
20 similar calculation gives: − λ G u = ζ ( λ ( p − u )) + ǫ ζ ( λ ( p − ǫu )) + ǫζ ( λ ( p − ǫ u ))= ζ ( λp ) − ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ℘ ( λu ) + ǫ (cid:16) ζ ( λp ) − ǫ ζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ǫ℘ ( λu ) (cid:17) + ǫ (cid:16) ζ ( λp ) − ǫζ ( λu ) + ℘ ′ ( λp )+ ℘ ′ ( λu ) ℘ ( λp ) − ǫ ℘ ( λu ) (cid:17) = ℘ ′ ( λp )+ ℘ ′ ( λu )2 (cid:16) ℘ ( λp ) − ℘ ( λu ) + ǫ ℘ ( λp ) − ǫ℘ ( λu ) + ǫ℘ ( λp ) − ǫ ℘ ( λu ) (cid:17) = ℘ ′ ( λp )+ ℘ ′ ( λu )2 3 ℘ ( λp ) ℘ ( λu ) ℘ ( λp ) − ℘ ( λu ) = ℘ ′ ( λp )+ ℘ ′ ( λu )2 12 ℘ ( λp ) ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) = ℘ ( λp ) ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) . To summarise, we have: G p = − λζ ( λu ) + 6 λ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) , G u = − λ℘ ( λp ) ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) . This gives G uu G u = (ln G u ) u = λ ℘ ′ ( λu ) ℘ ( λu ) + 6 λ℘ ( λu ) ℘ ′ ( λp ) − ℘ ′ ( λu ) , (47)and the first equation (22), G p = G uu G u − ζ ( u ), is satisfied identically due to (44). Finally,the second equation (22), G uuu G u − G uu + 2 ℘ ( u ) G u = 0, which can be written in theequivalent form − (cid:18) G uu G u (cid:19) u + (cid:18) G uu G u (cid:19) = 2 ℘ ( u ) , is satisfied identically due to (47) and (45). Acknowledgements
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