Second-order integrable Lagrangians and WDVV equations
SSecond-order integrable Lagrangians and WDVVequations
E.V. Ferapontov , M.V. Pavlov , Lingling Xue Department of Mathematical SciencesLoughborough UniversityLoughborough, Leicestershire LE11 3TU, UK Lebedev Physical InstituteRussian Academy of Sciences,Leninskij Prospekt 53, 119991 Moscow, Russia Department of MathematicsNingbo UniversityNingbo 315211, P.R. Chinae-mails:
[email protected]@[email protected]
Abstract
We investigate integrability of Euler-Lagrange equations associated with 2D second-orderLagrangians of the form (cid:90) f ( u xx , u xy , u yy ) dxdy. By deriving integrability conditions for the Lagrangian density f , examples of integrable La-grangians expressible via elementary functions, Jacobi theta functions and dilogarithms areconstructed. A link of second-order integrable Lagrangians to WDVV equations is established.Generalisations to 3D second-order integrable Lagrangians are also discussed. MSC: 35Q51, 37K05, 37K10, 37K20, 53D45.
Keywords:
Second-order Lagrangians, systems of hydrodynamic type, integrability (diagonalis-ability) conditions, Jacobi theta functions, Chazy equation, WDVV equations.1 a r X i v : . [ n li n . S I] J u l edicated to the memory of Boris Anatolievich Dubrovin Contents f = g ( u xx , u yy ) . . . . . . . . . . 112.7.2 Integrable Lagrangian densities of the form f = e u xx g ( u xy , u yy ) . . . . . . . . 112.7.3 Integrable Lagrangian densities polynomial in e u xx and e u yy . . . . . . . . . . 132.7.4 Integrable Lagrangian densities from WDVV prepotentials . . . . . . . . . . 142.7.5 WDVV prepotentials from integrable Lagrangian densities . . . . . . . . . . . 16 f = f ( u xy , u xt , u yt ) . . . . . . . . 203.4 2D densities as travelling wave reductions of 3D densities . . . . . . . . . . . . . . . 23 We investigate second-order Lagrangians (cid:90) f ( u xx , u xy , u yy ) dxdy, (1)such that the corresponding Euler-Lagrange equations are integrable (in the sense to be explainedbelow). Examples of integrable Lagrangians (1) have appeared in the mathematical physics litera-ture, thus, the Lagrangian density f = u xy ( u xx − u yy ) + α ( u xx − u yy ) + u xy ( βu xx + γu yy ) (2)governs integrable geodesic flows on a 2-torus which possess a fourth-order integral polynomial inthe momenta [3]. Similarly, the density f = u yy + u xx u yy + u xx u xy + 14 u xx (3)governs integrable Newtonian equations possessing a fifth-order polynomial integral. In Section 2we investigate the integrability aspects of 2D Lagrangians (1). Our main results can be summarisedas follows. 2 The Euler-Lagrange equation coming from Lagrangian (1) can be represented as a four-component Hamiltonian system of hydrodynamic type (Section 2.1). The requirement of itshydrodynamic integrability (which is equivalent to the vanishing of the corresponding Haantjestensor) leads to an involutive system of third-order PDEs for the Lagrangian density f (Section2.2). Analysis of the integrability conditions reveals that integrable Lagrangians (1) locallydepend on six arbitrary functions of one variable. Furthermore, the integrability conditionsare themselves integrable – a standard phenomenon in the theory of integrable systems. • The class of integrable Lagrangians (1) is invariant under the symplectic group Sp(4 , R ); underthis action the Lagrangian density f transforms as a genus two Siegel modular form of weight − • Potentials U ( x, t ) of classical Newtonian equations ¨ x = − U x that possess a fifth-order poly-nomial integral are governed by a Lagrangian (1) with density (3) (Section 2.5). • Integrable Lagrangians (1) are related to WDVV prepotentials of the form F ( t , t , t , t ) = 12 t t + t t t + W ( t , t , t );here W is a partial Legendre transform of the Lagrangian density f (Section 2.6). Thiscorrespondence works both ways: using known solutions of WDVV equations one can constructnew integrable Lagrangians (1). Conversely, integrable Lagrangian densities f give rise toWDVV prepotentials. Examples of this kind are given in Sections 2.7.4 and 2.7.5. • Further examples of integrable Lagrangians (1) expressible via elementary functions, Jacobitheta functions and dilogaritms are constructed in Section 2.7.In Section 3 we investigate 3D second-order Lagrangians of the form (cid:90) f ( u xx , u xy , u xt , u yy , u yt , u tt ) dxdydt. (4)Our results can be summarised as follows: • Integrable Lagrangians in 3D are governed by a third-order PDE system for the Lagrangiandensity f which comes from the requirement that all travelling wave reductions of 3D La-grangians to 2D are integrable in the sense of Section 2 (Section 3.1). • The class of integrable Lagrangians (4) is invariant under the symplectic group Sp(6 , R ); theLagrangian density f transforms as a genus three Siegel modular form of weight − • Examples of integrable Lagrangians (4) are constructed in Section 3.3. These include thedensities f = u yy − u xx u xt + u xx u yy + u xx u xy + 14 u xx ,f = ( u xy − u tt − u xx u xt + 13 u xx ) / ,f = u − xt ( u xt u yt − u xx u xt ) / , coming from the theory of dispersionless KP hierarchy (Section 3.3.1). • Classification of integrable densities of the form f = f ( u xy , u xt , u yt )is given in Section 3.3.2. Here the generic case is quite non-trivial, involving spherical trigonom-etry and Schl¨afly-type formulae, and is expressed in terms of the Lobachevky function L ( s ) = − (cid:82) s ln cos ξ dξ . 3n Section 4 we discuss examples of integrable dispersive deformations of integrable Lagrangiandensities (4). The general problem of constructing such deformations is largely open.Finally, we recall that paper [22] gives a characterisation of 3D first-order integrable Lagrangiansof the form (cid:90) f ( u x , u y , u t ) dxdydt. It was pointed out in [23] that the generic integrable Lagrangian density of this type is an auto-morphic function of its arguments. Note that 2D first-order Lagrangians densities f ( u x , u y ) lead tolinearisable Euler-Lagrange equations and, therefore, are automatically integrable. On the contrary,for second-order Lagrangian densities f ( u xx , u xy , u yy ), the 2D case (1) is already nontrivial. In this section we consider second-order integrable Lagrangians of type (1), (cid:90) f ( u xx , u xy , u yy ) dxdy. The Euler-Lagrange equation corresponding to Lagrangian (1) is a fourth-order PDE for u ( x, y ): (cid:18) ∂f∂u xx (cid:19) xx + (cid:18) ∂f∂u xy (cid:19) xy + (cid:18) ∂f∂u yy (cid:19) yy = 0 . (5)Setting a = u xx , b = u xy , c = u yy we can rewrite (5) in the form b x = a y , c x = b y , ( f a ) xx + ( f b ) xy + ( f c ) yy = 0 . (6)Introducing the auxiliary variable p via the relations p y = − ( f a ) x , p x = ( f b ) x + ( f c ) y , we can rewrite (6) as a first-order four-component conservative system a y = b x , b y = c x , ( f c ) y = ( p − f b ) x , p y = − ( f a ) x (7)or, in matrix form, R w y = S w x where w = ( a, b, c, p ) T and R = f ac f bc f cc
00 0 0 1 , S = − f ab − f bb − f bc − f aa − f ab − f ac . Assuming f cc (cid:54) = 0 we obtain a four-component system of hydrodynamic type, w y = V ( w ) w x , V ( w ) = R − S . (8) Remark 1.
System (7) can be put into a Hamiltonian form. For that purpose we introduce thenew dependent variables (
A, B, C ) which are related to ( a, b, c ) via partial Legendre transform, A = a, B = b, C = f c , h = cf c − f, h A = − f a , h B = − f b , h C = c.
4n the new variables, system (7) takes the form ( P = p ) A y = B x , B y = ( h C ) x , C y = ( P + h B ) x , P y = ( h A ) x , (9)which is manifestly Hamiltonian: ABCP y = ddx H A H B H C H P , with the Hamiltonian density H = h ( A, B, C ) + BP . Since hydrodynamic type system (7) is conservative, its integrability by the generalised hodographmethod [39] is equivalent to the diagonalisability of the corresponding matrix V ( w ) from (8). Thisis equivalent to the vanishing of the corresponding Haantjes tensor [24]. Recall that the Nijenhuistensor of the matrix V ( w ) = ( v ij ( w )) is defined as N ijk = v sj ∂ w s v ik − v sk ∂ w s v ij − v is ( ∂ w j v sk − ∂ w k v sj ) , where we adopt the notation w = ( a, b, c, p ) T = ( w , w , w , w ) T . The Haantjes tensor is definedas H ijk = N ipr v pj v rk + N pjk v ir v rp − N pjr v ip v rk − N prk v ip v rj . It is easy to see that both tensors are skew-symmetric in the low indices. The requirement of vanish-ing of the Haantjes tensor leads to a system of PDEs (integrability conditions) for the Lagrangiandensity f ( a, b, c ) which can be represented in symmetric conservative form:( f ab f cc − f ac f bc ) a = ( f bc f aa − f ab f ac ) c , ( f aa f cc − f ac ) a = ( f aa f bb − f ab ) c , ( f aa f cc − f ac ) c = ( f cc f bb − f bc ) a , ( f bb f cc − f bc ) b = 2( f ab f cc − f ac f bc ) c , ( f bb f aa − f ab ) b = 2( f bc f aa − f ac f ab ) a . (10)Integrability conditions (10) are invariant under the discrete symmetries a ↔ c and b → − b . Indeed,under the interchange of a and c equation (10) stays the same, while (10) , (10) and (10) , (10) get interchanged. Strictly speaking, the vanishing of the Haantjes tensor gives only the first four ofrelations (10), however, one can show that the fifth follows from the first four. We prefer to keep allof them for symmetry reasons.Our next goal is to show that the system of integrability conditions (10) is in involution, and itsgeneral solution depends on six arbitrary functions of one variable. Theorem 1
The general 2D integrable Lagrangian density f depends on six arbitrary functions ofone variable. Proof:
Let us introduce the new dependent variables s = ( s , s , s , s , s , s ) T = ( f aa , f ab , f ac , f bb , f bc , f cc ) T , s ) b = ( s ) a , ( s ) c = ( s ) a , ( s ) b = ( s ) a , ( s ) c = ( s ) b = ( s ) a , ( s ) c = ( s ) a , ( s ) c = ( s ) b , ( s ) c = ( s ) b . Taking these consistency conditions along with the four integrability conditions (10) (also rewrittenin terms of s -variables) we obtain a system of twelve first-order quasilinear equations for s i ( a, b, c )which can be represented in the form of two six-component systems of hydrodynamic type, s a = P ( s ) s c , s b = Q ( s ) s c , (11)where P , Q are the following 6 × P = 1 s s + s − s − s s s − s s s s s − s − s s s s , Q = 1 s s − s s s s − s − s s s s − s s − s − s s s s . Equations (11) possess six conserved densities s , s , s , s s − s s , s s − s s , s s + s s − s s − s which satisfy the equations( s ) a = ( s ) b , ( s ) a = ( s ) b , ( s ) a = ( s ) b , ( s ) c = ( s ) b , ( s ) c = ( s ) b , ( s ) c = ( s ) b , s s − s s ) a = ( s s − s ) b , s s − s s ) a = ( s s − s ) b , s s − s s ) c = ( s s − s ) b , s s − s s ) c = ( s s − s ) b , ( s s + s s − s s − s ) a = ( s s − s s ) b , ( s s + s s − s s − s ) c = ( s s − s s ) b . (12)Direct calculation shows that systems (11) commute, that is, s ab = s ba . Thus, equations (11) are ininvolution, and their general common solution depends on six arbitrary functions of one variable,namely, the Cauchy data s i (0 , , c ). This finishes the proof. Remark 2.
Relations (10) and (12) imply that there exists a potential ρ such that ρ aa = f aa f bb − f ab , ρ ac = f aa f cc − f ac , ρ cc = f cc f bb − f bc ,ρ ab = 2 ( f bc f aa − f ac f ab ) , ρ bc = 2 ( f ab f cc − f ac f bc ) ,ρ bb = 2( f ab f bc − f ac f bb + f aa f cc − f ac ) . (13) Remark 3.
System (10) possesses a Lax pair ψ a = λ K ψ, ψ b = λ L ψ, ψ c = λ M ψ, λ is a spectral parameter and the 4 × K , L , M are defined as K = f ac − f aa − f ab − ρ ab − ρ ac − f ab f ac − ρ aa − ρ ab − f aa , L = f bc − f ab − f bb − ρ bb − ρ bc − f bb f bc − ρ ab − ρ bb − f ab , M = f cc − f ac − f bc − ρ bc − ρ cc − f bc f cc − ρ ac − ρ bc − f ac . Remark 4.
We have verified that both systems (11) are linearly degenerate and non-diagonalisable(their Haantjes tensor does not vanish). This suggests that integrable Lagrangian densities (1) arerelated to the associativity (WDVV) equations where analogous commuting six-component systemswere obtained in [21], see also [34, 35] for related results. Such a link indeed exists, and is discussedin Section 2.6.
Let U be the 2 × u ( x, y ). Integrable Lagrangians of type (1) areinvariant under Sp(4 , R )-symmetry U → ( AU + B )( CU + D ) − , f → f det( CU + D ) , (14)where the matrix (cid:18) A BC D (cid:19) belongs to the symplectic group Sp(4 , R ) (here A , B , C , D are 2 × f transforms as a genus two Siegel mod-ular form of weight − u x , u y , x, y . Furthermore, integrable Lagrangians (1)are invariant under rescalings of f , as well as under the addition of a ‘null-Lagrangian’, namely,transformations of the form f → λ f + λ ( u xx u yy − u xy ) + λ u xx + λ u xy + λ u yy + λ , (15)which do not effect the Euler-Lagrange equations. Transformations (14 ) and (15) generate a groupof dimension 10 + 6 = 16 which preserves the class of integrable Lagrangians (1). These equivalencetransformations will be utilised to simplify the classification results in Section 2.7. For instance,modulo equivalence transformations the Lagrangian density (2) is equivalent to f = u xy ( u xx − u yy ). Integrability conditions (10) possess a compact formulation via higher genus Rankin-Cohen (Eholzer-Ibukiyama) brackets for Siegel modular forms [19]. This does not come as something unexpected,indeed, the integrability conditions possess Sp(4)-invariance (14) and, therefore, should be express-ible via Sp(4)-invariant operations. Here we mainly follow [31, 25], which specialised the generalresults of [19] to the genus two case. Let us introduce two matrix differential operators R = (cid:18) ∂ a ∂ b ∂ b ∂ c (cid:19) , S = (cid:18) ∂ ˜ a ∂ ˜ b ∂ ˜ b ∂ ˜ c (cid:19) , and define the operators P , P , P via the expansiondet( R + λS ) = P + λP + λ P . P = ∂ a ∂ c − ∂ b , P = ∂ a ∂ ˜ c + ∂ c ∂ ˜ a − ∂ b ∂ ˜ b , P = ∂ ˜ a ∂ ˜ c − ∂ b . Let us also define two operators Y , Y depending on the auxiliary parameters ξ = ( ξ , ξ ) by theformulae Y = ξRξ T = ξ ∂ a + ξ ξ ∂ b + ξ ∂ c , Y = ξSξ T = ξ ∂ ˜ a + ξ ξ ∂ ˜ b + ξ ∂ ˜ c . Finally, we introduce the ξ -dependent operator v = ( ∂ a ∂ ˜ b − ∂ b ∂ ˜ a ) ξ + 2( ∂ a ∂ ˜ c − ∂ c ∂ ˜ a ) ξ ξ + ( ∂ b ∂ ˜ c − ∂ c ∂ ˜ b ) ξ . Then integrability conditions (10) can be represented in the Hirota-type bilinear form( P Y v − P Y v )[ f ( a, b, c ) · f (˜ a, ˜ b, ˜ c )] (cid:12)(cid:12)(cid:12)(cid:12) ˜ a = a, ˜ b = b, ˜ c = c = 0 . (16)Here the left-hand side is a homogeneous quartic in ξ , ξ , with five nontrivial components. Equatingthem to zero we obtain all of the five integrability conditions (10). Remark 5.
It follows from [31], Proposition 2.3, that if f transforms as in (14), that is, as a weight − ⊗ det of GL(2 , C ). Remark 6.
The principal symbol of the Euler-Lagrange equation (5) is given by a compact formulain terms of the operator Y : Y [ f ] = f aa ξ + 2 f ab ξ ξ + (2 f ac + f bb ) ξ ξ + 2 f bc ξ ξ + f cc ξ . This expression transforms as a vector-valued Siegel modular form with values in the representationSym ⊗ det − . Here we sketch the derivation of the Lagrangian density (3). Consider a classical Newtonian equation¨ x = − U x where U ( x, t ) is the potential function, x = x ( t ), and dot denotes differentiation by t . This equationcan be written in the canonical Hamiltonian form˙ x = p, ˙ p = − U x . To be Liouville integrable, this Hamiltonian system should be equipped with an extra first integral F ( t, x, p ) such that dFdt ≡ ∂F∂t + ∂F∂x ˙ x + ∂F∂p ˙ p = F t + pF x − U x F p = 0 . (17)First integrals F polynomial in the momentum p were thoroughly investigated in [13], and later in[27, 33]. In particular, the following cases have been studied: F = p U p + V, F = p U p + V p + W, F = p U p + V p + W p + Q. In the last (fifth-order) case, equation (17) implies the following quasilinear system for the coeffi-cients: U t + V x = 0 , V t + W x = 3 U U x , W t + Q x = 2 V U x , Q t = W U x . u such that U = u xx , V = − u xt , W = u xx + u tt . Then the first twoequations will be satisfied identically, while the last two imply Q x = − u xt u xxx − u xx u xxt − u ttt , Q t = 32 u xx u xxx + u tt u xxx . The compatibility condition of these equations for Q leads to a fourth-order PDE for u , u tttt + 32 u xx u xxxx + 3 u xx u xxx + u tt u xxxx + 2 u xt u xxxt + 3 u xx u xxtt + 3 u xtt u xxx + 3 u xxt = 0 , which is nothing but the Euler-Lagrange equation for the second-order Lagrangian S = (cid:90) (cid:20) u tt + u xx u tt + u xx u xt + 14 u xx (cid:21) dxdt, whose density is identical to (3) up to relabelling t ↔ y . Let F ( t , . . . , t n ) be a function of n independent variables such that the symmetric matrix η ij = ∂ ∂ i ∂ j F is constant and non-degenerate (thus, t is a marked variable), and the coefficients c ijk = η is ∂ s ∂ j ∂ k F satisfy the associativity condition c sij c psk = c skj c psi ; here i, j, k ∈ { , . . . , n } . These requirementsimpose a nonlinear system of third-order PDEs for the prepotential F , the so-called associativity(WDVV) equations which were discovered in the beginning of 1990s by Witten, Dijkgraaf, Verlindeand Verlinde in the context of two-dimensional topological field theory. Geometry and integrabilityof WDVV equations has been thoroughly studied by Dubrovin, culminating in the remarkable theoryof Frobenius manifolds [15]. An important ingredient of this theory is an integrable hydrodynamichierarchy whose ‘primary’ part is defined by n − ∂ T α t i = c iαk ∂ X t k = ∂ X ( η is ∂ s ∂ α F ) (18)where T α are the higher ‘times’; here T = X . The flows (18) are manifestly Hamiltonian with theHamiltonian operator η is ddX and the Hamiltonian density ∂ α F . Note that WDVV equations areequivalent to the requirement of commutativity of these flows.We will need a particular case of the general construction when n = 4 and the matrix η isanti-diagonal, which corresponds to prepotentials F ( t , t , t , t ) = 12 t t + t t t + W ( t , t , t ) . (19)The corresponding primary flows (18) take the form ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) ,∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) ,∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , ∂ T t = ∂ X ( ∂ ∂ F ) , (20)which are Hamiltonian systems with the Hamiltonian densities ∂ F = t t + ∂ W, ∂ F = t t + ∂ W, ∂ F = 12 t + ∂ W, ∂ T α t i = ∂ X ( ∂ − i ∂ α F ) , i = 1 , , , , α = 2 , , . Setting ( t , t , t , t ) = ( P, B, C, A ) we obtain F = 12 P A + P BC + W ( B, C, A ) . In this case WDVV equations reduce to the following system of four PDEs for W : W AAA = W ABC + W ABB W ACC − W AAB W BCC − W AAC W BBC ,W AAB = W BBB W ACC − W ABB W BCC ,W AAC = W ABB W CCC − W ACC W BBC , W ABC = W BBB W CCC − W BBC W BCC . (21)The corresponding primary flows (20) take the form A T = C X , B T = ( P + W BC ) X , C T = ( W BB ) X , P T = ( W AB ) X ,A T = B X , B T = ( W CC ) X , C T = ( P + W BC ) X , P T = ( W AC ) X ,A T = P X , B T = ( W AC ) X , C T = ( W AB ) X , P T = ( W AA ) X . (22)Note that system (22) coincides with (9) under the identification h = W C , thus establishing a linkbetween WDVV equations and integrable Lagrangians. This link can be summarised as follows: • Take prepotential of type (19), set ( t , t , t ) = ( B, C, A ) and define h ( A, B, C ) = W C . • Reconstruct Lagrangian density f ( a, b, c ) by applying partial Legendre transform to h ( A, B, C ): a = A, b = B, c = h C , f = Ch C − h, f a = − h A , f b = − h B , f c = C. Examples of calculations of this kind will be given in Section 2.7.4.
Remark 7.
Conversely, given a Lagrangian density f ( a, b, c ), the corresponding prepotential W ( A, B, C ) can be reconstructed by the formulae W AA = − ρ a , W AB = − ρ b , W AC = − f a ,W BB = − ρ c , W BC = − f b , W CC = c.A = a, B = b, C = f c , where ρ is defined by formulae (13), see Section 2.7.5. In this section we present explicit examples of integrable Lagrangian densities f obtained by as-suming a suitable ansatz for f and computing the corresponding integrability conditions (10). Thisgives a whole range of integrable densities expressible via polynomials, elementary functions, Jacobitheta functions and dilogarithms. 10 .7.1 Integrable Lagrangian densities of the form f = g ( u xx , u yy )In this case the integrability conditions lead to the only constraint g aa g cc − g ac = k where k = const .Its solutions can be represented parametrically, thus, for k = 0 (parabolic case) and k = − a = p (cid:48) ( w ) v + q (cid:48) ( w ) , c = v, f = w [ p (cid:48) ( w ) v + q (cid:48) ( w )] − [ p ( w ) v + q ( w )] , and a = p (cid:48) ( w + v ) + q (cid:48) ( w − v ) , c = v, f = w [ p (cid:48) ( w + v ) + q (cid:48) ( w − v )] − [ p ( w + v ) + q ( w − v )] , respectively; here p and q are arbitrary functions and prime denotes differentiation. f = e u xx g ( u xy , u yy )We will show that the generic integrable density of this form corresponds to g ( b, c ) = [∆( ic/π )] − / θ ( b, ic/π )where ∆ is the modular discriminant and θ is the Jacobi theta function. The details are as follows.Substituting f = e a g ( b, c ) into the integrability conditions (10) one obtains gg bcc = 3 g cc g b − g bc g c , (23) gg bbb = g b g bb + 4 g bc g − g b g c , (24) gg ccc = g c g cc + 2 g cc g bb − g bc ) , (25) gg bbc = 2 g b g bc − g c g bb + 2 gg cc − g c ) . (26)This over-determined system for g is in involution, and can be solved as follows. First of all, equation(24) implies (cid:18) g bb g (cid:19) b = (cid:18) g c g (cid:19) b , so that one can set g c = 14 ( g bb − hg ) (27)where h is a function of c only. Using (27), both (23) and (26) reduce to g bbbb g − g b g bbb + 3 g bb = 4 h ( gg bb − g b ) − h (cid:48) g ; (28)here prime denotes differentiation by c . Modulo (27) and (28), equation (25) implies g bbb g + g bbb (4 g b − gg b g bb ) − g b g bb + 4 gg bb = 4 h ( g b − gg bb ) + 8 h (cid:48) g ( g b − gg bb ) + 83 h (cid:48)(cid:48) g . (29)Note that (29) can be obtained from (28) by differentiating it with respect to c , and using (27),(28). Similarly, differentiating (29) with respect to c we obtain the Chazy equation [8] for h : h (cid:48)(cid:48)(cid:48) = 2 hh (cid:48)(cid:48) − h (cid:48) . (30)Equations (28) and (29) can be simplified by the substitution v = − (ln g ) bb , which implies v bb = 6 v + 4 hv + 4 h (cid:48) (31)and v b = 4 v + 4 hv + 8 h (cid:48) v + 83 h (cid:48)(cid:48) , (32)11espectively. Since (31) follows from (32) via differentiating with respect to b , we end up with thefollowing compact form of integrability conditions (23)-(26): g c = 14 ( g bb − hg ) , v = − (ln g ) bb , v b = 4 v + 4 hv + 8 h (cid:48) v + 83 h (cid:48)(cid:48) ; (33)here h solves the Chazy equation (30). We recall that modulo the natural SL(2 , R )-symmetry [9],the Chazy equation possesses three non-equivalent solutions: h = 0 , h = 1 and h =
12 ∆ (cid:48) ∆ where ∆ isthe modular discriminant. These three solutions (which correspond to rational, trigonometric andelliptic cases of the Weierstrass ℘ -function equation in (33)) are considered separately below. Notethat both the rational and trigonometric cases lead to degenerate Lagrangians, so only the ellipticcase is of interest. Rational case h = 0. In this case equations (33) simplify to g c = 14 g bb , v = − (ln g ) bb , v b = 4 v , which are straightforward to solve. Modulo unessential constants the generic solution of theseequations is g = e µb + µ c ( b + µc ) where µ = const . The corresponding Lagrangian density f takesthe form f = e u xx +2 µu xy + µ u yy ( u xy + µu yy ) . Note that the change of independent variables x = ˜ x, y = ˜ y + µ ˜ x brings this Lagrangian to thedegenerate form ˜ f = e u ˜ x ˜ x u ˜ x ˜ y (the order of the corresponding Euler-Lagrange equation can bereduced by two by setting v = u ˜ x ). Trigonometric case h = 1. In this case equations (33) simplify to g c = 14 ( g bb − g ) , v = − (ln g ) bb , v b = 4 v + 4 v , which are also straightforward to solve. Modulo unessential constants the generic solution of theseequations is g = e µb + µ c sinh( b + µc ) where µ = const . The corresponding Lagrangian density f takes the form f = e u xx +2 µu xy + µ u yy sinh( u xy + µu yy ) . Note that the same change of variables as in the rational case brings this Lagrangian to the degen-erate form ˜ f = e u ˜ x ˜ x sinh u ˜ x ˜ y . Elliptic case h =
12 ∆ (cid:48) ∆ , see e.g. [38]. Here the modular discriminant ∆ is given by the formula∆( c ) = (2 π ) q ∞ (cid:89) (1 − q n ) , q = e πic , recall that h has the q -expansion h ( c ) = πiE = πi (cid:32) − ∞ (cid:88) n =1 σ ( n ) q n (cid:33) where E is the Eisenstein series (here σ ( n ) is the divisor function). Setting g ( b, c ) = [∆( c )] − / w ( b, c )we see that the first equation (33) becomes the heat equation for w : w c = 14 w bb , v = − (ln w ) bb , v b = 4 v + 4 hv + 8 h (cid:48) v + 83 h (cid:48)(cid:48) . (34)The general solution of system (34) was constructed in [7]: w ( b, c ) = ∆ / σ ( b, g , g ) e b h/ σ is the Weierstrass sigma function with the invariants g = h − h (cid:48) , g = − h + hh (cid:48) − h (cid:48)(cid:48) . Note that ∆ = π ( g − g ). Thus, g ( b, c ) = σ ( b, g , g ) e b h/ . Remark 8.
An alternative (real-valued) representation of the general solution of system (34) interms of the Jacobi theta function θ is as follows: w ( b, c ) = θ ( b, ic/π ) = 2 ∞ (cid:88) n =0 ( − n e − ( n +1 / c sin[(2 n + 1) b ];here for h in the last equation (34) one has to use iπ h ( ic/π ) = − ∞ (cid:88) n =1 σ ( n ) e − nc = − e − c + 3 e − c + 4 e − c + 7 e − c + . . . ) , which is another (real-valued) solution of the Chazy equation (note that the Chazy equation isinvariant under the scaling symmetry h ( c ) → λh ( λc ). Thus, g ( b, c ) = [∆( ic/π )] − / θ ( b, ic/π ) . Note that the function (cid:52) − / ( τ ) θ ( z, τ ) appears in the theory of weak Jacobi forms (it is a holo-morphic weak Jacobi form of weight − / e u xx and e u yy Here we describe integrable Lagrangian densities f that are linear/quadratic in e u xx and e u yy , thecoefficients being functions of u xy only. Linear case: f = p + p e a + p e c . Substituting this ansatz into the integrability conditions (and assuming p , p to be nonzero) weobtain a system of ODEs for the coefficients p i ( b ) which, modulo equivalence transformations, canbe simplified to p = p = p, p (cid:48)(cid:48) = p, p (cid:48)(cid:48) = α/p ;here α = const (which can be set equal to 1 if nonzero) and prime denotes differentiation by b .Modulo equivalence transformations, these equations possess two essentially different solutions: f = αe − b + ( e a + e c ) e b and f = αq ( b ) + ( e a + e c ) sinh b, where the function q ( b ) satisfies q (cid:48)(cid:48) = b . This implies q (cid:48) = ln − e b e b , and another integration gives q ( b ) = Li ( − e b ) − Li ( e b )where Li is the dilogarithm function: ( Li ( x )) (cid:48) = − ln(1 − x ) x . Quadratic case: f = p + p e a + p e c + p e a + p e a + c + p e c . Substituting this ansatz into the integrability conditions we obtain a large system of ODEs forthe coefficients p i ( b ) which, modulo equivalence transformations, lead to the following integrabledensities (here we only present those examples that are not reducible to the linear case by a changeof variables): f = e kb + a + c , f = e √ b + a + c + e √ b +2 c , f = αe − √ b + αe √ b + a + e √ b + c + e √ b + a + c ,f = pe a + 2 p e a + c + pe c , p = cosh (cid:18) √ b (cid:19) . .7.4 Integrable Lagrangian densities from WDVV prepotentials In this section we discuss polynomial prepotentials F of type (19) associated with finite Coxetergroups W as given in [16], p. 107. Applying the procedure outlined at the end of Section 2.6 wecompute the corresponding integrable Lagrangian densities f which, in general, will be algebraicfunctions of a, b, c (presented below up to appropriate scaling factors). Group W ( A ) : F = 12 t t + t t t + 12 t + 13 t + 6 t t t + 9 t t + 24 t t + 2165 t ; f = (cid:0) c − a − ab (cid:1) / . Swapping t and t (which is an obvious symmetry of WDVV equations) and following the sameprocedure gives a polynomial density f : F = 12 t t + t t t + 12 t + 13 t + 6 t t t + 9 t t + 24 t t + 2165 t ; f = 54 a − a c + 16 c − b a. Group W ( B ) : F = 12 t t + t t t + t + t t t t t + t t t t t +6 t t + t t + 18 t t t f = 2 aC + (cid:0) a + b (cid:1) C − ab , where C is defined by the quadratic equation 3 aC + (6 a + 2 b ) C + a (6 a + 5 b ) = c. Swapping t and t gives a polynomial density f : F = 12 t t + t t t + t + t t t t t + t t t t t + 6 t t + t t + 18 t t t f = 12 a + 12 a b − a c − bac + 112 c − b . Group W ( D ) : F = 12 t t + t t t + t t + t t + 6 t t t + 5435 t ; f = c a − ba . Group W ( F ) : F = 12 t t + t t t + t t
18 + 3 t t t t t t t
60 + t t
28 + t · · ·
13 ; f = 1 √ a (cid:0) a + 14 ba − c (cid:1) / . Swapping t and t gives a rational density f : F = 12 t t + t t t + t t
18 + 3 t t t t t t t
60 + t t
28 + t · · ·
13 ; f = a − ca − b a + 3 c a . Group W ( H ) : = 12 t t + t t t + 2 t t t t
240 + t t t
18 + t t t
15 + t t · · t t t · · t t · ·
11 + t t · · + 2 t t · ·
19 + 32 t · · ·
31 ; f = a C + a C + a b C + a · · C − a b , where C is defined by the cubic equation a C + a C + a b C + a · · C + a b · + a · · = c. Swapping t and t gives a rational density f : F = 12 t t + t t t + 2 t t t t
240 + t t t
18 + t t t
15 + t t · · t t t · · t t · ·
11 + t t · · + 2 t t · ·
19 + 32 t · · ·
31 ; f = 32 a · · + 8 ba · · − ca · ·
11 + 2 b a · − bca − b a + c a . Non-polynomial prepotentials (19) associated with extended affine Weyl groups can be foundin [14]: F = 12 t t + t t t − t t + 1720 t t − t + 2 t t e t + 16 t e t + 12 e t + 16 t t ; f = − C
756 + 148 bC + 43 e a C − b e a + b C , where − C + bC + 2 C e a − b + b C = c. Swapping t and t gives: F = 12 t t + t t t − t t + 1720 t t − t + 2 t t e t + 16 t e t + 12 e t + 16 t t ; f = b
80 + 16 cb + 12 c b − a b. Modular prepotentials [2, 32] give rise to modular Lagrangian densities (as an example we tookprepotential 4.2.2. from [32]): F = 12 t t + t t t − t t γ ( t ) + t g ( t ) + t t g ( t ) + t g ( t ); f = 112 g ( a ) [ c + 12 b γ ( a )] − g ( a ) b . Swapping t and t gives: F = 12 t t + t t t − t t γ ( t ) + t g ( t ) + t t g ( t ) + t g ( t ); f = 24 C g ( a ) + 8 bC g ( a ) , where C is defined by the algebraic equation30 C g + 12 bC g − b γ = c. Here g = Kg , g = Kg ( g (cid:48) − g γ ) where the functions of γ ( a ) and g ( a ) satisfy the ODEs γ (cid:48) = 12 γ − Kg , g (cid:48)(cid:48) = 2 γg (cid:48) − g γ (cid:48) ,K = const . The above ODE system falls within Bureau’s class and its solutions are given in termsof the Schwarzian triangle functions [32]. 15 .7.5 WDVV prepotentials from integrable Lagrangian densities In view of the correspondence between integrable Lagrangians and WDVV prepotentials F ( t , t , t , t ) = 12 t t + t t t + W ( t , t , t )described in Section 2.6, integrable Lagrangian densities f ( a, b, c ) constructed in this paper give riseto prepotentials some of which are apparently new. Here we list some examples (omitting details ofcalculations; we will only present the corresponding function W ). Example 1.
The polynomial Lagrangian density from Section 2, f = b ( a − c ) , gives rise to the prepotential W = 115 t − t t t + 13 t t − t t . Example 2.
Lagrangian densities from Section 2.7.3 (linear case): the density f = αe − b + ( e a + e c ) e b gives rise to the prepotential W = − αe t − αe − t t − e t e t t − t t + t t ;the density f = αq ( b ) + ( e a + e c ) sinh b gives rise to the prepotential W = 18 e t − e t t sinh t − αe t − αq ( t ) t + 12 t ln t sinh t − t . Here q ( t ) = Li ( − e t ) − Li ( e t )where Li is the dilogarithm function. Example 3.
Lagrangian densities from Section 2.7.3 (quadratic case): the density f = e kb + a + c gives rise to the prepotential W = − t t − k t t + t t ;the density f = e √ b + a + c + e √ b +2 c gives rise to the prepotential W = t t − t t − √ t t − γe √ t +2 t t ;the density f = αe − √ b + αe √ b + a + e √ b + c + e √ b + a + c W = t t e √ t + e t + √ t − α e t − αt e t + √ t + e t +2 √ t e √ t + e t + √ t . Example 4.
The Lagrangian density f = e c g ( b, a ) from Section 2.7.2 gives rise to the prepotential(recall that system (10) is invariant under the interchange a ↔ c ; for our convenience we choose f = e c g ( b, a ) instead of f = e a g ( b, c )): W = t t g ( t , t ) . Here g ( t , t ) = [∆( it /π )] − / θ ( t , it /π )where ∆ is the modular discriminant and θ is the Jacobi theta function. Note the formula∆ / ( it /π ) = √ π θ (cid:48) (0 , it /π ) where prime denotes derivative by t . The corresponding solutionof WDVV equations is related to Whitham averaged one-phase solutions of NLS/Toda equations[17], see also [1, 11]. Example 5.
The Lagrangian density f = (cid:112) b ( αa + βb )( αb + βc ) from Section 3.4 gives rise to theprepotential W = − β t ( αt + βt ) ln t − α β t t + β t t + 18 ( αt + βt ) ln( αt + βt ) . In this section we consider second-order integrable Lagrangians of the form (4), (cid:90) f ( u , u , u , u , u , u ) dx dx dx , here u ij = u x i x j . Let us require that all travelling wave reductions of a 3D Lagrangian density to two dimensions areintegrable in the sense of Sections 2.2 and 2.4. This gives the necessary conditions for integrabilitywhich, in our particular case, prove to be also sufficient. The computational details are as follows.Consider a traveling wave reduction of a 3D Lagrangian density f ( u , u , u , u , u , u ) ob-tained by setting u ( x , x , x ) = v ( x, y ) + Q where x = s x + s x , y = s x + s x , s i = const ,and Q is an arbitrary homogeneous quadratic form in x , x , x . We have u = s v xx + ζ , u = s s v xy + ζ , u = s v yy + ζ ,u = s s ( v xx + v xy ) + ζ , u = s s ( v xy + v yy ) + ζ , u = s ( v xx + 2 v xy + v yy ) + ζ , where ζ i are the coefficients of the quadratic form Q . Setting v xx = a, v xy = b, v yy = c we obtainthe reduced 2D Lagrangian density f in the form f ( a, b, c ) = f ( u , u , u , u , u , u )= f ( s a + ζ , s s b + ζ , s c + ζ , s s ( a + b ) + ζ , s s ( b + c ) + ζ , s ( a + 2 b + c ) + ζ ) . We have the following differentiation rules: ∂ a = s ∂ u + s s ∂ u + s ∂ u ,∂ b = s s ∂ u + s s ∂ u + s s ∂ u + 2 s ∂ u ,∂ c = s ∂ u + s s ∂ u + s ∂ u . (35)17tc. Substituting partial derivatives of the reduced density f ( a, b, c ) into the 2D integrability con-ditions (10) we obtain homogeneous polynomials of degree ten in s , s , s whose coefficients areexpressed in terms of partial derivatives of the original 3D density f ( u ij ). Equating to zero thecoefficients of these polynomials we obtain 3D integrability conditions for f (note that due to thepresence of arbitrary constants ζ i the arguments of f can be viewed as independent of s , s , s ).The integrability conditions can be represented in compact Hirota-type form analogous to (16):( P Y v − P Y v )[ f ( u ij ) · f (˜ u ij )] (cid:12)(cid:12)(cid:12)(cid:12) ˜ u ij = u ij = 0 . (36)Here the operators on the left-hand side of (36) are identical to that from Section 2.4, with the onlydifference that we substitute expressions (35) (and their tilded versions) for ∂ a , ∂ b , ∂ c and ∂ ˜ a , ∂ ˜ b , ∂ ˜ c .Thus, ∂ a = s ∂ u + s s ∂ u + s ∂ u , ∂ ˜ a = s ∂ ˜ u + s s ∂ ˜ u + s ∂ ˜ u , etc. The left-hand side of (36) is an Sp(6)-invariant operation which transforms a function f definedon the space of 3 × u ij into a homogeneous form of degree four in ξ , ξ anddegree ten in s , s , s . Let U be the 3 × u ( x , x , x ). Integrable Lagrangians of type (4)are invariant under Sp(6)-symmetry U → ( AU + B )( CU + D ) − , f → f det( CU + D ) , (37)where the matrix (cid:18) A BC D (cid:19) belongs to the symplectic group Sp(6 , R ) (here A , B , C , D are 3 × f , as well as under the addition of a ‘null-Lagrangian’, namely,transformations of the form f → λ f + (cid:88) λ σ U σ (38)where U σ denote all possible minors of the Hessian matrix U . Transformations (37) and (38)generate a group of dimension 21 + 15 = 36 which preserves the class of integrable Lagrangians (4). In this section we give some explicit examples of 3D integrable Lagrangian densities f . Here we construct three explicit integrable Lagrangian densities arising in the context of the dKPhierarchy: f = u yy − u xx u xt + u xx u yy + u xx u xy + 14 u xx , (39) f = ( u xy − u tt − u xx u xt + 13 u xx ) / , (40) f = u − xt ( u xt u yt − u xx u xt ) / . (41)These examples come from the following dKP flows.18 ase 1. The fifth-order flow of the dKP hierarchy comes from the dispersionless Lax representation p y = (cid:18) p w (cid:19) x , p t = (cid:18) p wp + vp + bp + c (cid:19) x , which gives rise to the equations w y = v x , b x = v y + 3 ww x , c x = b y + 2 vw x , w t = bw x + c y . Setting w = u xx , v = u xy and b = u yy + u xx we obtain two equations for c , c x = u yyy + 3 u xx u xxy + 2 u xy u xxx , c y = u xxt − u yy u xxx − u xx u xxx , whose compatibility condition results in the following fourth-order PDE for u : u yyyy − u xxxt + 3 u xx u xxyy + 2 u xy u xxxy + 3 u xxy + 3 u xxx u xyy + u yy u xxxx + 32 u xx u xxxx + 3 u xx u xxx = 0 . This is the Euler-Lagrange equation corresponding to the polynomial Lagrangian density (39). Notethat the two-dimensional density (3) is just the stationary ( t -independent) reduction of (39). Case 2.
Another flow of the dKP hierarchy is associated with the Lax representation p t = (cid:18) p wp + v (cid:19) x , p y = (cid:18) p wp + vp + bp + c (cid:19) x , which gives rise to the equations b t = − bw x + wb x − vv x + w y , c t = wc x − bv x + v y ,w t = − ww x + b x , v t = − vw x − wv x + c x . Setting w = u xx , b = u xt + u xx and v = u xy − u tt − u xx u xt + u xx we obtain two equations for c , c t = v y + u xx v t + 2 vu xx u xxx + ( u xx − u xt ) v x , c x = v t + 2 vu xxx + 2 u xx v x , whose compatibility condition yields v tt + ( vu xx ) xt = v xy + ( v ( u xx − u xt )) xx . This PDE is the Euler-Lagrange equation corresponding to the density (40).
Case 3.
This example comes from the dispersionless Lax pair p t = (cid:18) rp − q (cid:19) x , p y = (cid:18) p wp + v (cid:19) x , which gives rise to the equations q y = q q x + qw x + wq x + v x , r y = q r x + 2 qrq x + rw x + wr x , w t = − r x , v t = − qr x − rq x . Setting w = u xx , r = − u xt , q = u yt u xt − u xx , the second and the third equations will be satisfiedidentically, while the first and the fourth imply v x = q y − (cid:18) q + qu xx (cid:19) x , v t = ( u xt q ) x , whose consistency condition gives q yt − (cid:20) q (cid:18) u yt u xt + 2 u xx (cid:19)(cid:21) xt = ( u xt q ) xx . This is the Euler-Lagrange equation corresponding to the density (41).19 .3.2 Integrable Lagrangian densities of the form f = f ( u xy , u xt , u yt )We will show that the general integrable density of this form is expressible in terms of the Lobachevskyfunction L ( s ) = − (cid:82) s ln cos ξ dξ , a special function which features in Lobachevsky’s formulae for hy-perbolic volumes [28].Using the notation u xy = v , u xt = v , u yt = v and f ij = f v i v j one can show that theintegrability conditions (which we do not present here explicitly) can be rewritten as simple relationsfor the 2 × F = Hessf . Namely, they are equivalent to the conditionsthat the minors f f − f = a , f f − f = a , f f − f = a , and f f − f f = p , f f − f f = p , f f − f f = p , are such that a i = const and p i is a function of the argument v i only. This gives the inverse of theHessian matrix F in the form f f f f f f f f f − = 1det F a p p p a p p p a . (42)Taking the determinant of both sides we obtaindet F = (cid:113) a a a − a p − a p − a p + 2 p p p . (43)Inverting the matrix identity (42) gives f f f f f f f f f = 1det F a a − p p p − a p p p − a p p p − a p a a − p p p − a p p p − a p p p − a p a a − p (44)where we use (43) for det F . The consistency conditions of equations (44) lead to simple ODEs forthe functions p i ( v i ): p (cid:48) = c ( p − a a ) , p (cid:48) = c ( p − a a ) , p (cid:48) = c ( p − a a ) , where c is yet another arbitrary constant. The further analysis depends on how many constantsamong a i are equal to zero. All constants are zero.
In this case without any loss of generality one can set p i = 1 /v i whichleads to the integrable Lagrangian density f = √ u xy u xt u yt . Two constants are zero.
Then one can also set p i = 1 /v i . Modulo (complex) rescalings this leadsto the Lagrangian density f = (cid:112) u xy u xt (2 u yt − u xy u xt ) − u yt arctan (cid:113) u yt u xy u xt − . One constant is zero.
This leads to the Lagrangian density f = ( u xt − u xy ) arctan √ u xt u xy coth u yt − u xt − u xy u xt − u xy − ( u xt + u xy ) arctan √ u xt u xy coth u yt − u xt − u xy u xt + u xy . All constants are nonzero.
This case is more interesting. Setting a i = 1 , c = − p (cid:48) i = 1 − p i so that p i = tanh v i . Equations (44) can be integrated once to yield f = arcsin p − p p (cid:112) − p (cid:112) − p , f = arcsin p − p p (cid:112) − p (cid:112) − p , f = arcsin p − p p (cid:112) − p (cid:112) − p . p i as the new independent variables we obtain f p = − p arcsin p − p p √ − p √ − p ,f p = − p arcsin p − p p √ − p √ − p ,f p = − p arcsin p − p p √ − p √ − p , or, in differentials, df = dp − p arcsin p − p p √ − p √ − p + dp − p arcsin p − p p √ − p √ − p + dp − p arcsin p − p p √ − p √ − p . (45)In the original variables v , v , v relation (45) takes the form df = arcsin(cosh v cosh v tanh v − sinh v sinh v ) dv + arcsin(cosh v cosh v tanh v − sinh v sinh v ) dv + arcsin(cosh v cosh v tanh v − sinh v sinh v ) dv . Remark 9.
Relation (45) has an unexpected link to spherical trigonometry. On the unit sphere S , consider a spherical triangle (cid:52) ABC with interior angles
A, B, C and side lengths a, b, c (so thatside a lies opposite the angle A , etc). The spherical laws of cosines arecos a = cos b cos c + sin b sin c cos A, cos b = cos a cos c + sin a sin c cos B, cos c = cos a cos b + sin a sin b cos C, (46)and cos A = − cos B cos C + sin B sin C cos a, cos B = − cos A cos C + sin A sin C cos b, cos C = − cos A cos B + sin A sin B cos c, (47)respectively. Note that the map ( A, B, C ) → ( a, b, c ) sending angles of a spherical triangle to itsside lengths is integrable in the sense of multidimensional consistency [36], and is closely related tothe discrete Darboux system [5, 26]. Setting p = cos a, p = cos b, p = cos c (48)and using (46) we can rewrite (45) in the following Schl¨afly-type form: df = ( A − π/ da sin a + ( B − π/ db sin b + ( C − π/ dc sin c . (49)Recall that the classical Schl¨afly formula expresses the differential of the volume of a sphericalpolyhedron in terms of its side lengths and dihedral angles. Expression (49), which can be viewedas a two-dimensional Schl¨afly formula, has appeared in [12, 29] as a special case of a one-parameterfamily of closed Schl¨afly-type forms associated with spherical triangles (case h = 0 of Theorem 3.2(b)in [29]). Note that the function f of a spherical triangle defined by (49) is essentially the volumeof the ideal hyperbolic octahedron which is the convex hull of the six intersection points of thethree circles on the sphere at infinity bounding the spherical triangle (cid:52) ABC ([29], Appendix C, seealso [30]). This function f is related to the ‘capacity’ function of a spherical triangle. Expressionssimilar to (49) have appeared before in the context of variational principles for circle packings andtriangulated surfaces [10, 4, 6, 12]. 21ntegration of (45) is quite non-trivial, the answer is given in terms of the Lobachevsky function.The computations below are based on formula (49) and the spherical cosine laws (46), (47). Using da sin a = d ln − cos a a we can rewrite df in the form df = − π d (cid:16) ln − cos a a + ln − cos b b + ln − cos c c (cid:17) + A d ln − cos a a + B d ln − cos b b + C d ln − cos c c . Equivalently, df = − π d (cid:16) ln − cos a a + ln − cos b b + ln − cos c c (cid:17) + d (cid:16) A ln − cos a a + B ln − cos b b + C ln − cos c c (cid:17) − (cid:16) ln − cos a a dA + ln − cos b b dB + ln − cos c c dC (cid:17) . Let us rewrite the last term of this expression as a total differential. Note that using (47) we have1 − cos a a = sin B sin C − cos A − cos B cos C sin B sin C + cos A + cos B cos C = − cos A − cos( B + C )cos A + cos( B − C ) = cos π − A − B − C cos B + C − A cos A + B − C cos A + C − B . With similar formulae for − cos b b and − cos c c we obtain: − (cid:16) ln − cos a a dA + ln − cos b b dB + ln − cos c c dC (cid:17) = − ln cos π − A − B − C cos B + C − A cos A + B − C cos A + C − B dA − ln cos π − A − B − C cos A + C − B cos A + B − C cos B + C − A dB − ln cos π − A − B − C cos A + B − C cos B + C − A cos A + C − B dC = ln cos π − A − B − C d (cid:0) π − A − B − C (cid:1) + ln cos A + B − C d (cid:0) A + B − C (cid:1) + ln cos A + C − B d (cid:0) A + C − B (cid:1) + ln cos B + C − A d (cid:0) B + C − A (cid:1) . On integration, we obtain the final formula for f : f = − π (cid:16) ln − cos a a + ln − cos b b + ln − cos c c (cid:17) + A ln − cos a a + B ln − cos b b + C ln − cos c c − L (cid:0) π − A − B − C (cid:1) − L (cid:0) A + B − C (cid:1) − L (cid:0) A + C − B (cid:1) − L (cid:0) B + C − A (cid:1) , where L ( s ) = − (cid:82) s ln cos ξ dξ is the Lobachevsky function. In the original variables v , v , v definedas p = cos a = tanh v , p = cos b = tanh v , p = cos c = tanh v this gives: f = π ( v + v + v ) − ( Av + Bv + Cv ) − L (cid:0) π − A − B − C (cid:1) − L (cid:0) A + B − C (cid:1) − L (cid:0) A + C − B (cid:1) − L (cid:0) B + C − A (cid:1) ;here A, B, C are defined, as functions of v , v , v , via the spherical cosine laws:cos A = cos a − cos b cos c sin b sin c = cosh v cosh v tanh v − sinh v sinh v , cos B = cos b − cos a cos c sin a sin c = cosh v cosh v tanh v − sinh v sinh v , cos C = cos c − cos a cos b sin a sin b = cosh v cosh v tanh v − sinh v sinh v . π ( v + v + v ) can be ignored as it does not effect the Euler-Lagrangeequations corresponding to the density f . Euler-Lagrange equation.
In terms of the side lengths a, b, c and angles
A, B, C of a sphericaltriangle, the Euler-Lagrange equation corresponding to the density f takes the form A yt + B xt + C xy = 0 , a x sin a = b y sin b = c t sin c , (50)where we keep in mind the spherical cosine laws (46), (47). Indeed, the first relation is equivalent to( f ) yt +( f ) xt +( f ) xy = 0, while the second set of relations comes from the consistency conditions ofrelations (48) rewritten in the form u xy = arctanh(cos c ), u xt = arctanh(cos b ), u yt = arctanh(cos a ). Remark 10.
Similar analysis of the Lagrangian densities f = f ( u xx , u yy , u tt ) gives no interestingexamples: one can show that in this case the integrability conditions imply that all × f must necessarily be constant, thus leading to quadratic densities f withlinear Euler-Lagrange equations. Given a 3D integrable Lagrangian density f ( u xx , u xy , u yy , u xt , u yt , u tt ) one can apply a travellingwave ansatz, u ( x, y, t ) = u ( ξ, η ) where ξ = a x + a y + a t, η = b x + b y + b t , to obtain an integrable2D Lagrangian density of the form f ( u ξξ , u ξη , u ηη ). In fact, modulo linear transformations of ξ and η it is sufficient to assume ξ = x + αt, η = y + βt . Applying this construction to the density f = √ u xy u xt u yt found in Section 3.3.2 one obtains 2D integrable densities of the form f = (cid:113) u ξη ( αu ξξ + βu ξη )( αu ξη + βu ηη ) . Some integrable Lagrangian densities possess integrable dispersive deformations (both in 2D and3D). Here we give three examples (a complete classification of integrable dispersive deformations isa non-trivial open problem).
Example 1.
The Lagrangian density (39), f = u yy − u xx u xt + u xx u yy + u xx u xy + 14 u xx , (Section 3.3.1, case 1) possesses integrable dispersive deformation f (cid:15) = u yy − u xx u xt + u xx u yy + u xx u xy + 14 u xx − (cid:15) u xx u xxx − (cid:15) u xxy + (cid:15) u xxxx , here (cid:15) is a deformation parameter. The corresponding (dispersive) Euler-Lagrange equation has theLax pair (cid:15)ψ y = (cid:15) ψ xx + aψ, (cid:15)ψ t = (cid:15) ψ xxxxx + (cid:15) aψ xxx + (cid:15) bψ xx + (cid:15)cψ x + wψ, where a = u xx , b = u xy + (cid:15) u xxx , c = u yy + u xx + (cid:15)u xxy + (cid:15) u xxxx , and the variable w is definedby the equations w x = u yyy + 3 u xx u xxy + 2 u xy u xxx + 3 (cid:15) u xxx + 3 (cid:15) u xx u xxxx + (cid:15) u xxyy + 3 (cid:15) u xxxxy + 3 (cid:15) u xxxxxx ,w y = u xxt + (cid:15) u xyyy + (cid:15) u xxxyy + 3 (cid:15) u xxxxxy − (cid:15) u xxxxxxx . Example 2.
The Lagrangian density (40), f = (cid:18) u xy − u tt − u xx u xt + 13 u xx (cid:19) / , (Section 3.3.1, case 2) possesses integrable dispersive deformation f (cid:15) = (cid:18) u xy − u tt − u xx u xt + 13 u xx + (cid:15)
12 (4 u xx u xxxx + 3 u xxx − u xxxt ) + (cid:15) u xxxxxx (cid:19) / . The corresponding dispersive Euler-Lagrange equation has the Lax pair (cid:15)ψ t = (cid:15) ψ xxx + (cid:15)wψ x + vψ, (cid:15)ψ y = (cid:15) ψ xxxxx + (cid:15) wψ xxx + (cid:15) ( v + (cid:15)w x ) ψ xx + (cid:15)bψ x + cψ, where w = u xx , v = f / (cid:15) + (cid:15) u xxx , b = u xt + u xx + 2 (cid:15) u xxxx + (cid:15)v x , and the function c is determined by the equations c x = v t + 2( u xx v ) x + 23 (cid:15) v xxx ,c t = v y + ( u xx v ) t + (cid:18) vu xx − vu xt − (cid:15)vv x − (cid:15) u xx v xx − (cid:15) v x u xxx + (cid:15) c xx − (cid:15) v xxxx (cid:19) x . Example 3.
The Lagrangian density (41), f = u − xt (cid:0) u xt u yt − u xx u xt (cid:1) / , (Section 3.3.1, case 3) possesses integrable dispersive deformation f (cid:15) = u − xt (cid:18) u xt u yt − u xx u xt + (cid:15) u xxt − (cid:15) u xt u xxxt (cid:19) / , The corresponding dispersive Euler-Lagrange equation comes from the Lax pair (cid:15)ψ y = (cid:15) ψ xxx + (cid:15)wψ x + vψ, (cid:15) ψ xt = (cid:15)qψ t + rψ, where w = u xx , r = − u xt , q = (cid:18) f (cid:15) u xt (cid:19) / + (cid:15) u xxt u xt , and the variable v is defined by the equations v t = ( u xt q ) x , v x = q y − (cid:18) u xx q + 13 q + (cid:15)qq x + (cid:15) q xx (cid:19) x . Here we list some problems for further study. 24
Multi-dimensional Lagrangians.
It would be of interest to describe multi-dimensionalversions of second-order integrable Lagrangians. Thus, anti-self-dual four-manifolds with aparallel real spinor are described by the integrable 4D Dunajski system [18] a xt + a yz + u xx a yy + u yy a xx − u xy a xy = 0 ,u xt + u yz + u yy u xx − u xy = a, which can be written as a single fourth-order PDE for the function u . This PDE comes fromthe second-order Lagrangian (cid:90) ( u xt + u yz + u yy u xx − u xy ) dxdydzdt. Similarly, anti-self-dual scalar flat four-manifolds (Flaherty-Park spaces, see [37] and referencestherein) are governed by the equations u xz (ln F ) yt − u xy (ln F ) zt − u zt (ln F ) xy + u yt (ln F ) xz = 0 ,u xz u yt − u xy u zt = F, which are equivalent to a single fourth-order PDE for u . The corresponding Lagrangian is S = (cid:90) [ F ln F − F ] dxdydzdt, where one has to substitute F = u xz u yt − u xy u zt . • Multi-component Lagrangians.
Our approach can be generalised in a straightforward wayto describe 2-field integrable Lagrangians of the form (cid:90) f ( u x , u y , v x , v y ) dxdy, as well as their 3D analogues, (cid:90) f ( u x , u y , u t , v x , v y , v t ) dxdydt. • Higher-order quasilinear PDEs.
Similarly, one can classify third-order integrable PDEsof the form a u xxx + a u xxy + a u xyy + a u yyy = 0where the coefficients a i are functions of the second-order derivatives u xx , u xy , u yy only. Thisproblem also has a natural 3D analogue. Acknowledgments
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