Integrable Lagrangians and modular forms

In tegrable Lagrangians and mo dular forms E.V. F erap onto v and A.V. Odesskii Departmen t of Mathematical Sciences Lough b orough Univ ersit y Lough b orough, Leicestershire LE11 3TU, UK and Sc ho ol of Mathematics Univ ersit y of Manc hester Oxford Road, Manc hester M13 9PL, UK e-mails: E.V.Ferapon tov@lboro.ac.uk Alexander.O desskii@manchester.ac.uk Abstract W e inv estigate non-d ege n erate Lagrangia n s of the form Z f ( u x , u y , u t ) dx dy dt suc h that the corresp onding Euler-Lagrange equ ati on s ( f u x ) x + ( f u y ) y + ( f u t ) t = 0 are in tegrable by the method o f h yd ro d ynamic reductions. W e demonstrate that the in tegra- bilit y conditions, whic h constitute an inv olutiv e o v er-d etermined system of fourth order PDEs for the Lagrangian densit y f , are in v arian t un der a 20-parameter group of Lie-p oint symmetries wh ose actio n on the mo duli space of integrable Lagrangians h as an op en orbit. The density of the ‘master-Lagrangian’ corresp onding to this orb it is sh o wn to b e a mo d- ular form in three v ariables defined on a complex h yp erb olic ball. W e d emonstrate how the kno wledge of the symmetry group allo ws one to linearise the inte grability conditions. MSC: 35Q58, 37K05 , 37K10, 37K25 . Keyw ord s : Integrable Lagrangians, Symmetries, Mod ular F orms. 1 1 In tro d uction In this pap er w e inv estigate in tegrable three-dimensional Euler-Lagrange equations, ( f u x ) x + ( f u y ) y + ( f u t ) t = 0 , (1.1) corresp onding to Lagrangian densities o f the form f ( u x , u y , u t ). F amiliar examples include the disp ersionles s KP equation u xt − u x u xx = u y y with the Lagrangian densit y f = 1 3 u 3 x + u 2 y − u x u t ; this equation, also kno wn a s the Khokhlo v- Z ab olotsk a ya equation, a rises in non-linear acoustics [19]. Another example, u xx + u y y = e u t u tt , is kno wn as the Bo yer-Finley equation [3]: it app ears as a s ymmetry reduction of the self-dualit y equations, and corresp o nds to the Lagrangian densit y f = u 2 x + u 2 y − 2 e u t . The pap er [8] pro vides a system of partial differen tial equations for the Lagrangian densit y f ( a, b, c ) (w e set a = u x , b = u y , c = u t ) whic h are neces sary and sufficien t for the in tegrabilit y of the equation (1.1) b y the method of h ydro dynamic reductions a s prop osed in [7]. These conditions can b e r epresen ted in a remark able compact form: Theorem 1 [8]. F or a no n-de gene r ate L agr angian, the Euler-L agr ange e quation (1.1) is inte g r able by the me tho d of hydr o dynamic r e ductions if and only if the d e n sity f satisfie s the r e l a tion d 4 f = d 3 f dH H + 3 H det ( dM ); (1.2) here d 3 f and d 4 f are the symmetric differen t ia ls of f . The Hessian H and the 4 × 4 matrix M are defined as follo ws: H = d et   f aa f ab f ac f ab f bb f bc f ac f bc f cc   , M =     0 f a f b f c f a f aa f ab f ac f b f ab f bb f bc f c f ac f bc f cc     . (1.3) The differen tia l dM = M a da + M b db + M c dc is a matrix-v alued form     0 f aa f ab f ac f aa f aaa f aab f aac f ab f aab f abb f abc f ac f aac f abc f acc     da +     0 f ab f bb f bc f ab f aab f abb f abc f bb f abb f bbb f bbc f bc f abc f bbc f bcc     db +     0 f ac f bc f cc f ac f aac f abc f acc f bc f abc f bbc f bcc f cc f acc f bcc f ccc     dc. A Lagrang ian is said to b e non-degenerate iff H 6 = 0 (w e p oint out that the equations H = 0 and detM = 0 hav e b een discuss ed in the literature, see [6] and r eferences therein). Both sides of the relation (1.2) ar e homogeneous symmetric quartics in da, d b, dc . Equating similar terms w e obtain expres sions for a l l fourth order pa rtial deriv at ives of the densit y f in terms of its second and third order deriv atives (1 5 equations altogether). The resulting o v er- determined system for f is in inv o lution, and its solution space is 20-dimensional: indeed, the v alues of par t ia l deriv ativ es of f up to o rder 3 at a p oin t ( a 0 , b 0 , c 0 ) amoun t to 20 arbitrary constan ts. Thus , w e are dealing with a 20-dimensional mo duli space of in tegra ble La grangians. 2 In Sect. 2 we pro ve that the inte g rabilit y conditions (1.2) a re in v arian t under a 20 -parameter group of Lie-p oin t symmetries whose action on the mo duli space o f inte g rable Lagrangia ns p ossess es a n op en orbit. Explicit formulae for inte g rable Lagrang ians in terms of mo dular forms are constructed in Sect. 3. W e first consider Lagrangian densities of the for m f = u x u y g ( u t ), whic h can be view ed as a deformation of the in tegr a ble densit y f = u x u y u t found in [8]. By virtue o f the in tegrability conditions (1 .2 ), the f unction g has t o satisfy the fo ur t h order ODE g ′′′′ ( g 2 g ′′ − 2 g ( g ′ ) 2 ) − 9( g ′ ) 2 ( g ′′ ) 2 + 2 g g ′ g ′′ g ′′′ + 8( g ′ ) 3 g ′′′ − g 2 ( g ′′′ ) 2 = 0 , whic h inherits a remark able Gl (2 , R )-inv ariance. W e prov e that he ‘generic’ solution of t his ODE is giv en b y the series g ( u t ) = X ( α,β ) ∈ Z 2 e ( α 2 − αβ + β 2 ) u t = 1 + 6 e u t + 6 e 3 u t + 6 e 4 u t + 12 e 7 u t + .... ; notice tha t under the substitution u t = 2 π iz the rig h t hand side of t his for m ula b ecomes a sp ecial mo dular fo r m of w eigh t one and lev el three, know n as the Eisenstein series E 1 , 3 ( z ). W e p oin t out t ha t mo dular forms and non-linear ODEs related to them app ear in a v a riet y of problems in mathematical ph ysics, see e.g. [1, 2, 4, 5, 9, 10, 14, 1 5, 18] and references therein. Lagrangian densities of the form g ( u x , u y ) u t and the general case f ( u x , u y , u t ) are discussed in Sect. 3.2 and 3.3, resp ectiv ely . Here the ‘generic’ solution is an a uto morphic form of tw o (three) v ariables. 2 Symmetry gro up of the prob lem The first main observ ation, o ve rlo o k ed in [8], is the inv ariance of the integrabilit y conditions (1.2) under pro jectiv e transformations of the form ˜ a = l 1 ( a, b, c ) l ( a, b, c ) , ˜ b = l 2 ( a, b, c ) l ( a, b, c ) , ˜ c = l 3 ( a, b, c ) l ( a, b, c ) , ˜ f = f l ( a, b, c ) ; (2.4) here l , l 1 , l 2 , l 3 are arbitrar y (inhomog eneous ) linear forms in a, b, c . Introducing the quart ic form F = H d 4 f − d 3 f dH − 3 det ( dM ) , one can ve r if y that ˜ F = l 4 F , whic h establishes the S L (4 , R )-in v ariance of t he integrabilit y conditions (1 .2 ). Com bined with ob vious symmetries of the form ˜ f = sf + αa + β b + γ c + δ, (2.5) this provides a 20-dimensional symmetry group of the problem. Remark. The class of Euler-Lagra ng e equations (1.1) is form-inv arian t under a p oin t group generated by arbitra r y linear tr a nsformations of the v ariables x, y , t and u . Ob viously , p oint 3 transformations preserv e the in tegrability . Since the prolongation of these transformations to the v ariables a, b, c and f is giv en by (2 .4), this explains the S L (4 , R ) -in v a riance of the in tegrability conditions (1.2). The main result of this section is the follo wing Theorem 2. The action of the symmetry gr oup on the 20 -d i m ensional mo duli s p ac e of inte g r able L agr angians p ossesses an op en orbit. Pro of: The infinitesimal generators of the symmetry group ( 2 .4), (2.5) are the follow ing v ector fields: 3 translatio ns in a, b, c : ∂ ∂ a , ∂ ∂ b , ∂ ∂ c ; 9 linear transformations of a, b, c : a ∂ ∂ a , b ∂ ∂ a , c ∂ ∂ a , a ∂ ∂ b , b ∂ ∂ b , c ∂ ∂ b , a ∂ ∂ c , b ∂ ∂ c , c ∂ ∂ c ; 3 pro jectiv e transformations of a, b, c, f : a 2 ∂ ∂ a + ab ∂ ∂ b + ac ∂ ∂ c + af ∂ ∂ f , ab ∂ ∂ a + b 2 ∂ ∂ b + bc ∂ ∂ c + bf ∂ ∂ f , ac ∂ ∂ a + bc ∂ ∂ b + c 2 ∂ ∂ c + cf ∂ ∂ f ; moreo v er, we ha ve 5 extra generators corresp onding to the transformations (2.5): ∂ ∂ f , a ∂ ∂ f , b ∂ ∂ f , c ∂ ∂ f , f ∂ ∂ f . The main idea of the pro of is to pro long these infinitesimal generators to the 20-dimensional mo duli space of solutions of the in v olutive system (1.2). W e p oin t out that, since all fourth order deriv ativ es of f are explicitly kno wn, this mo duli space can b e iden tified with the v alues of f and its partial deriv atives f i , f ij , f ij k up to order three (20 par ameters altog ether). The prolongation can b e calculated as follow s: (1) F ollo wing the standard notation adopted in the symmetry analysis of differential equations [11, 16], w e in tro duce the v ariables x 1 = a, x 2 = b, x 3 = c and represen t eac h of the ab o ve generators in the f o rm ξ i ∂ ∂ x i + η ∂ ∂ f ; here ξ i and η are functions o f x i and f . (2) Prolong infinites imal g enerators to the third order jet space with co ordinates x i , f , f i , f ij , f ij k , ξ i ∂ ∂ x i + η ∂ ∂ f + ζ i ∂ ∂ f i + ζ ij ∂ ∂ f ij + ζ ij k ∂ ∂ f ij k , where ζ i , ζ ij and ζ ij k are calculated according to the standard prolongation form ulae ζ i = D i ( η ) − f s D i ( ξ s ) , ζ ij = D j ζ i − f is D j ( ξ s ) , ζ ij k = D k ζ ij − f ij s D k ( ξ s ); (2.6) 4 here D i denotes the op erator of total differen tiation with respect to x i . (3) T o eliminate the ∂ ∂ x i -terms we subtract the linear com binat io n of tota l deriv ativ es ξ i D i from the pro lo nged op erators where, in D i , it is sufficien t to ke ep only the f ollo wing terms: D i = ∂ ∂ x i + f i ∂ ∂ f + f ij ∂ ∂ f j + f ij k ∂ ∂ f j k + f ij k l ∂ ∂ f j k l ; notice that , since f ij k l are explicit functions of lo wer order deriv atives , the resulting o pera- tors will b e w ell-defined ve ctor fields on the 20-dimensional mo duli space with co ordinates f , f i , f ij , f ij k . Although t hese o p erato r s will dep end on the v ariables x i as on parameters (in- deed, the isomorphism of the mo duli space with the space f , f i , f ij , f ij k dep ends on the c hoice of a p oin t in t he x -space), all algebraic prop erties o f these op erators will b e x -indep enden t. (4) Finally , the dimension of the maximal orbit equals the rank of the 20 × 20 matrix of co efficien ts of these op erators. It remains to p oin t out that this rank equals 20 for an y ‘random’ n umerical c hoice of the v alues for x i , f , f i , f ij , f ij k . 3 Lagrangian den sities in terms o f mo dular forms In this section w e pro vide explicit fo rm ulae fo r inte g rable La grangians in terms of mo dular forms. W e start with the case f ( u x , u y , u t ) = u x u y g ( u t ) ( Sect. 3.1), where g is shown to b e an Eisenstein series E 1 , 3 : a sp ecial mo dular for m o f w eight 1 and lev el three. The case f ( u x , u y , u t ) = g ( u x , u y ) u t and the g eneral case f ( u x , u y , u t ) are discussed in Sect. 3.2 and 3.3, resp ectiv ely . W e demonstrate ho w the know ledge of the symmetry group of the problem allo ws one to linearize the complicated nonlinear equations for Lagrangian densities resulting from the in tegrablity conditions. 3.1 Lagrangian densities of the form f = u x u y g ( u t ) The integrabilit y conditions (1.2) imply a single fourth order OD E for g ( z ), g ′′′′ ( g 2 g ′′ − 2 g ( g ′ ) 2 ) − 9 ( g ′ ) 2 ( g ′′ ) 2 + 2 g g ′ g ′′ g ′′′ + 8( g ′ ) 3 g ′′′ − g 2 ( g ′′′ ) 2 = 0; (3.7) to comply with the standard notation, the argument of g is now denoted b y z . This equation enjo ys a remark able S L (2 , R )-inv ariance inherited from (2.4): ˜ z = αz + β γ z + δ , ˜ g = ( γ z + δ ) g ; (3.8) here α , β , γ , δ are arbitrary constan ts suc h that αδ − β γ = 1. Moreo v er, there is an obvious scaling symmetry g → λg . The equation ( 3 .7) can b e linearized as follo ws. In tro ducing h = g ′ /g , w e first rewrite it in the f orm h ′′′ ( h ′ − h 2 ) = h 6 − 3 h 4 h ′ + 9 h 2 ( h ′ ) 2 − 3( h ′ ) 3 − 4 h 3 h ′′ + ( h ′′ ) 2 ; (3.9) 5 the corresp onding symmetry group mo difies to ˜ z = αz + β γ z + δ , ˜ h = ( γ z + δ ) 2 h + γ ( γ z + δ ) . (3.10) W e p oint out that the same symmetry o ccurs in the case of the Chazy equation ( [1 ], p. 342), as w ell as its analogue discussed recen tly in [2]. The prese nce of the S L (2 , R )-symmetry of this t yp e implies the linearizabilit y of the equation under study . One can formulate the following general stat ement whic h is, in fact, con tained in [5]: Theorem 3. A ny thir d or der ODE of the form F ( z , h, h ′ , h ′′ , h ′′′ ) = 0 , which i s invariant under the action of S L (2 , R ) as sp e cifi e d by (3.10), c an b e line arize d by a substitution z = w 1 w 2 , h = d dz ln w 2 (3.11) wher e w 1 ( t ) and w 2 ( t ) ar e two li n e arly indep endent solutions of a li n e ar e quation d 2 w /dt 2 = V ( t ) w with the Wr onskian W n o rmalize d as W = w 2 dw 1 /dt − w 1 dw 2 /dt = 1 (the p otential V ( t ) dep ends on the g i v en thir d or der O D E, and c an b e effe ctively r e c onstructe d). In p articular, the gener al solution of the e quation (3.9) is gi v en by p ar ametric form ulae (3.11) wher e w 1 ( t ) and w 2 ( t ) ar e two line arly indep enden t solutions o f the line ar e quation d 2 w /dt 2 = 2 9 (cosh − 2 t ) w with W = 1 . Pro of: T o establish the first part of the theorem we esse ntially repro duce the calculation from Sect. 5 in [5]. Let us consider a linear OD E d 2 w /dt 2 = V ( t ) w , tak e t w o linearly indep enden t solutions w 1 ( t ), w 2 ( t ) with W = 1, and in tro duce the new dependen t and indep enden t v ariables h, z by parametric relations z = w 1 w 2 , h = d dz ln w 2 . Using the readily v erifiable formu la e dt/dz = w 2 2 and h = w 2 dw 2 /dt , one obta ins the iden tities h ′ − h 2 = w 4 2 V , h ′′ − 6 hh ′ + 4 h 3 = w 6 2 dV /dt and h ′′′ − 12 hh ′′ − 6( h ′ ) 2 + 48 h 2 h ′ − 24 h 4 = w 8 2 d 2 V /dt 2 ; here prime denotes differen tiation with respect to z . Thus , one arrive s at the relat io ns I 1 = ( h ′′ − 6 hh ′ +4 h 3 ) 2 ( h ′ − h 2 ) 3 = ( dV /dt ) 2 V 3 , I 2 = h ′′′ − 12 hh ′′ − 6( h ′ ) 2 +48 h 2 h ′ − 24 h 4 ( h ′ − h 2 ) 2 = d 2 V /dt 2 V 2 . 6 W e p oint out that I 1 and I 2 are the simplest second- and third-order differen tial inv arian ts of the a ctio n (3.10) whose infinitesimal generators, prolo nged to the t hir d jets z , h, h ′ , h ′′ , h ′′′ , are of the form X 1 = ∂ z , X 2 = z ∂ z − h∂ h − 2 h ′ ∂ h ′ − 3 h ′′ ∂ h ′′ − 4 h ′′′ ∂ h ′′′ , X 3 = z 2 ∂ z − (2 z h + 1) ∂ h − (2 h + 4 z h ′ ) ∂ h ′ − (6 h ′ + 6 z h ′′ ) ∂ h ′′ − (12 h ′′ + 8 z h ′′′ ) ∂ h ′′′ ; notice the standard commu t a tion relations [ X 1 , X 2 ] = X 1 , [ X 1 , X 3 ] = 2 X 2 , [ X 2 , X 3 ] = X 3 . One can v erify that the Lie deriv ativ es of I 1 , I 2 with resp ect to X 1 , X 2 , X 3 are indeed zero. Th us, an y third order ODE whic h is in v arian t under the S L (2 , R )- action (3.10), can b e represen ted in the form I 2 = F ( I 1 ) where F is an arbitrary function of one v ariable. The corresp onding p oten tial V ( t ) has to satisfy the relation d 2 V /dt 2 V 2 = F  ( dV /dt ) 2 V 3  . This simple sc heme pro duces some of the w ell-kno wn equations, for instance, the relation I 2 = − 24 implies the Chazy equation fo r h , that is, h ′′′ − 12 hh ′′ + 18 ( h ′ ) 2 = 0. The corresp o nding p oten tial satisfies the equation d 2 V /dt 2 = − 24 V 2 . Similarly , the choice I 2 = I 1 − 8 results in the ODE h ′′′ = 4 hh ′′ − 2( h ′ ) 2 + ( h ′′ − 2 hh ′ ) 2 h ′ − h 2 whic h, under the substitution h = y / 2, coincides with the equation (4.7) from [2]. The p oten tial V satisfies the equation V d 2 V /dt 2 = ( dV / d t ) 2 − 8 V 3 . Finally , the relation I 2 = I 1 − 9 coincides with (3.9). The corresp onding p oten tial V satisfies the equation V d 2 V /dt 2 = ( dV /dt ) 2 − 9 V 3 . It r emains to p oint out that, up to elemen tary equiv alence transformatio ns, the general solution of the last equation for V is giv en by V = 2 9 cosh − 2 t . Since the equation d 2 w /dt 2 = 2 9 cosh − 2 t w is r elat ed to the h yp ergeometric equation s (1 − s ) w ss + (1 − 2 s ) w s − 2 9 w = 0, corresp onding to the parameter v alues a = 1 / 3 , b = 2 / 3 , c = 1, b y a c hang e of v ar iables s/ (1 − s ) = e 2 t , w e can reform ulat e the ab o v e Theorem as follows: Prop osition 1. The gener al solution of the e quation (3.9) is g iven by p ar am etric formulae (3.11) wher e w 1 ( s ) and w 2 ( s ) ar e two line arly indep endent solutions of the hyp er ge ometric e quation s (1 − s ) w ss + (1 − 2 s ) w s − 2 9 w = 0 with the Wr onskian normalize d as w 2 dw 1 /ds − w 1 dw 2 /ds = 1 / (2 s (1 − s )) . As h = g ′ /g , this immediately implies the follow ing for m ula for the general solution of (3.7): Prop osition 2. T he gener a l solution of the e quation (3.7) is given by p ar ametric formulae z = w 1 w 2 , g = w 2 , wher e w 1 ( s ) and w 2 ( s ) ar e two line arly i n dep enden t solutions to the h yp er ge ometric e quation s (1 − s ) w ss + (1 − 2 s ) w s − 2 9 w = 0 with the Wr onsk i a n normal i z e d as w 2 dw 1 /ds − w 1 dw 2 /ds = 1 / (2 s (1 − s )) . 7 One can construct the following explicit solution o f the equation (3 .7), g ( z ) = X ( α,β ) ∈ Z 2 q ( α 2 − αβ + β 2 ) = 1 + 6 q + 6 q 3 + 6 q 4 + 12 q 7 + .... ; (3.12) here q = e 2 π iz . T o get a real-v alued solution, one has t o restrict z to the imaginary axis. This function is kno wn as the Eisenstein series E 1 , 3 ( z ). Equiv alen tly , it can b e defined b y the formula g ( z ) = E 1 , 3 ( z ) = 1 + 6 ∞ X n =1   X d | n χ 3 ( d )   q n where χ 3 denotes the Legendre sym b ol mo d 3 (that is, χ 3 ( d ) = 0 if d ≡ 0 mo d 3, χ 3 ( d ) = 1 if d ≡ 1 mo d 3, and χ 3 ( d ) = − 1 if d ≡ 2 mo d 3). The Eisenstein series transforms as g ( αz + β γ z + δ ) = χ 3 ( δ )( γ z + δ ) g ( z ) under the Hec ke congruence subgroup Γ 0 (3) defined as  α β γ δ  ∈ Γ 0 (3) ⊂ S L (2 , Z ) if  α β γ δ  ≡  α β 0 δ  mo d 3 . It follow s tha t g ( z ) is a mo dular form of weigh t one and lev el 3, namely , g ( αz + β γ z + δ ) = ( γ z + δ ) g ( z ) where  α β γ δ  ∈ S L (2 , Z ) and  α β γ δ  ≡  1 β 0 1  mo d 3 . The function g ( z ) can a lso b e written in the for m in v olving summation ov er N only , g ( z ) = 1 − 6 X k ∈ N  q 3 k − 1 1 − q 3 k − 1 − q 3 k − 2 1 − q 3 k − 2  . Theorem 4. The function g ( z ) is a s olution of the differ e n tial e quation (3.7). Pro of: Recall that, give n a mo dular form g ( z ) of we ight k , its R ankin-Cohen br a c k ets [ g , g ] 2 and [ g , g ] 4 are defined as follo ws: [ g , g ] 2 = ( k + 1)  k g g ′′ − ( k + 1)( g ′ ) 2  , and [ g , g ] 4 = ( k + 2)( k + 3)  k ( k + 1) 12 g g ′′′′ − ( k + 1)( k + 3) 3 g ′ g ′′′ + ( k + 2)( k + 3) 4 ( g ′′ ) 2  ; these are kno wn to b e mo dular forms of w eights 2 k + 4 and 2 k + 8, resp ectiv ely (w e use the normalization o f [20]). In our case k = 1, so that w e get G = [ g , g ] 2 = 2( g g ′′ − 2( g ′ ) 2 ) , [ g , g ] 4 = 2 ( g g ′′′′ − 16 g ′ g ′′′ + 18( g ′′ ) 2 ) , 8 w eigh ts 6 and 10, resp ectiv ely . One can v erify that, up to a constan t m ultiple, the left hand side of the equation (3.7) can b e represen ted in the form 7[ g , g ] 4 [ g , g ] 2 + [ G, G ] 2 , whic h sho ws that it is a mo dular form (in fact, a cusp form) of we ig ht 16 with resp ect to the same group. T o sho w that this form v a nishes iden tically w e recall that the dimension of the space of cusp forms of w eigh t 16 and lev el 3 equals 4, and the order of zero cannot exceed 5. Th us, it is sufficien t to v erify the v anishing of the first fiv e co efficien ts in t he decomp osition of this fo rm as a p o w er series in q = exp ( 2 π iz ). This can b e done by a direct calculation. Remark. The relation b et wee n the mo dular fo rm (3.12) and the hy p ergeometric equation from the Prop osition 2 can b e summarized as fo llo ws. Cho osing a basis of solutions o f the h yp ergeometric equation in the form w 2 = 1 + 2 9 s + 10 81 s 2 + ..., w 1 = w 2 ln s + 5 9 s + 57 162 s 2 + ..., one obtains parametric equations z = w 1 w 2 = ln s + 5 9 s + 37 162 s 2 + ..., g = w 2 = 1 + 2 9 s + 10 81 s 2 + .... Solving the first equation for s in the form s = e z + ae 2 z + be 3 z + ... one arrives at s = e z − 5 9 e 2 z + 19 81 e 3 z + ... . The substitution in to the second equation implies the expression for g ( z ) in the form g ( z ) = 1 + pe z + q e 3 z + ... whic h, up to an appropriate affine transformation o f z , coincides with (3.12). 3.2 Lagrangian densities of the form f = g ( u x , u y ) u t As follows from [8], the in tegrability conditions (1.2) result in a system of fiv e equations ex- pressing all fourth order partial deriv ativ es of g ( a, b ) in terms of its low er order deriv a t ives . In sym b olic form, this system can b e represen t ed as follows: d 4 g = d 3 g dh h + 6 dg h det ( dm ) + 3 ( dg ) 2 h det ( dn ) . (3.13) Here d s g are symmetric differen tials of g , the matrices m and n are defined as m =   0 g a g b g a g aa g ab g b g ab g bb   , n =  g aa g ab g ab g bb  , and h = − det ( m ) = g 2 b g aa − 2 g a g b g ab + g 2 a g bb . The non-degeneracy of the Lag rangian densit y f ( a, b, c ) = g ( a, b ) c is equiv alen t to the con- dition h 6 = 0. One can show that the ov er- determine d system (3.13) is in inv o lution, and 9 its solution space is 10-dimensional (indeed, the v alues of partial deriv a tiv es of g up to or- der 3 a t a p oint ( a 0 , b 0 ) amount to 1 0 arbitrary constants). The system (3 .1 3) is in v ariant under a 10- dimens iona l group of Lie-p oin t symmetries whic h consists of arbitrar y pro jectiv e transformations of a and b , isomorphic to S L (3 , R ), along with transformations of the form g → αg + β , α, β = const. The correspo nding infinitesimal generators include 2 translatio ns : ∂ ∂ a , ∂ ∂ b ; 4 linear transformations : a ∂ ∂ a , b ∂ ∂ a , a ∂ ∂ b , b ∂ ∂ b ; 2 pro jectiv e transforma t ions : a 2 ∂ ∂ a + ab ∂ ∂ b , ab ∂ ∂ a + b 2 ∂ ∂ b ; 2 affine transformations of g : ∂ ∂ g , g ∂ ∂ g . (3.14) W e will demonstrate that the existence of this symmetry group allows one to linearize the in tegrability conditions (3.13). F or this purp ose w e consider a linear system of the form z xx = M z x − I z y + Az , z xy = − N z x − M z y + B z , (3.15) z y y = − J z x + N z y + C z , where the co efficien t s I , J, M , N , A, B , C are certain functions of x, y whic h ha ve to satisfy the compatibilit y conditions resulting from the requiremen t of consiste ncy of the equations (3.15): A = 2( M 2 + I N ) + I y − M x , B = M y + N x + I J − M N , C = 2 ( N 2 + J M ) + J x − N y , along with t wo extra relations inv olving I , J, M , N only: J xx = 2 N xy + M y y − 3( N 2 + J M ) x − 3 N M y + 2 J I y + I J y , I y y = 2 M xy + N xx − 3( M 2 + I N ) y − 3 M N x + 2 I J x + J I x . These relations imply that the space of solutio ns of the linear system (3.15) is three-dimensional. Notice that an y in v olutive second order linear system of the form z ij = Γ k ij z k + g ij z with t w o indep enden t v ariables x, y can b e reduced to the form (3.15) b y a gauge transformation z → ϕ ( x, y ) z . This is a standard normalizat io n in the theory of m ulti-dimensional Sch w arzian deriv atives [17]. It implies that the W ronskian W of any three linearly indep enden t solutions, W = d et   z 1 z 2 z 3 ( z 1 ) x ( z 2 ) x ( z 3 ) x ( z 1 ) y ( z 2 ) y ( z 3 ) y   , is constan t: W x = W y = 0. Let us c ho ose three linearly indep enden t solutions z i ( x, y ) with the W ronskian normalized as W = 1 , and in tro duce the new indep enden t v aria bles a = z 1 z 3 , b = z 2 z 3 . 10 Let us consider x and y as functions of a, b : x = x ( a, b ) , y = y ( a, b ) . A direct calculation sho ws that these functions satisfy the follow ing nonlinear system: x 2 a x bb − 2 x a x b x ab + x 2 b x aa = ( x a y b − y a x b ) 2 J, y 2 a y bb − 2 y a y b y ab + y 2 b y aa = ( x a y b − y a x b ) 2 I , y a x aa − x a y aa = 3 N x a y 2 a + J y 3 a − I x 3 a − 3 M x 2 a y a , x b y bb − y b x bb = 3 M x 2 b y b + I x 3 b − J y 3 b − 3 N x b y 2 b ; (3.16) here I , J, M , N are the same functions of x, y as in (3.15). The system (3.16) is in in v olutio n, and its solutio n space, whic h is 8-dimensional, p ossesses a transitiv e a ction of S L (3 , R ): indeed, there is an S L (3 , R )-freedom in the c hoice of a basis z 1 , z 2 , z 3 . Con v ersely , one can show that an y inv olutiv e system of four second order PDEs for tw o functions x ( a, b ) and y ( a, b ) whic h is in v arian t under a tra nsitiv e pro jectiv e action of S L (3 , R ), has the form (3.16), and comes from a linear system (3.15). This immediately follows fro m the equiv alen t represen tation of the system (3.1 6 ), x 2 a x bb − 2 x a x b x ab + x 2 b x aa ( x a y b − y a x b ) 2 = J ( x, y ) , y 2 a y bb − 2 y a y b y ab + y 2 b y aa ( x a y b − y a x b ) 2 = I ( x, y ) , y 2 b x aa + y 2 a x bb + 2 x b y b y aa + 2 x a y a y bb − 2 y a y b x ab − 2( x a y b + x b y a ) y ab ( x a y b − y a x b ) 2 = − 3 M ( x, y ) , x 2 a y bb + x 2 b y aa + 2 x a y a x bb + 2 x b y b x aa − 2 x a x b y ab − 2( x a y b + x b y a ) x ab ( x a y b − y a x b ) 2 = − 3 N ( x, y ) . (3.17) The v ar ia bles x, y and the left hand sides of (3.17) form a basis of differen tial inv arian ts of the S L (3 , R )-action extended to second order jet space with co ordinates a, b, x, y , x a , x b , y a , y b , x aa , x ab , x bb , y aa , y ab , y bb . Th us, an y system with the required symme try pro p erties can b e ob- tained b y expressing the four second order differen tial in v arian ts a s functions of x and y . The expressions in the left hand sides of (3.17) a r e related to the tw o-dimensional Sc hw ar zian deriv a- tiv es [17]. Let us return to the integrabilit y conditions (3.1 3). Extending the action of the symme- try generators (3.1 4) to t he third order jet space with co ordinates a, b, g , g a , g b , g aa , g ab , g bb , g aaa , g aab , g abb , g bbb according to the prolongation form ulae (2.6), one obtains 10 v ector fields on a 12- dimens io nal space; thus , there exist t w o differential inv ariants , whic h w e will denote b y 11 x and y , resp ectiv ely (the explicit formulae for x and y in terms of g and its deriv ativ es ar e pro vided b elo w). W e claim that x and y , view ed as functions of a and b , satisfy a system of the form (3.16) (equiv alen tly , (3.17)). Indeed, the action o f the S L (3 , R ) part of the symmetry group on the v ariables a, b, x, y is exactly the same as for the syste m (3 .16). The passage from a, b, g to a, b, x, y can b e view ed as a factorizatio n of the in tegrability equations (3.13) b y the t w o -dimensional affine gro up corresp onding to the last t w o g enerators (3 .14), whic h resp ects the action of S L (3 , R ). T o write dow n the express io ns for the differen tial in v a rian ts x and y we in tro duce the following notation: Z 1 = g aaa g a − 3 2 g 2 aa g 2 a ( g 2 a g bb − 2 g a g b g ab + g 2 b g aa ) 2 g 2 a g 4 b , Z 2 = g bbb g b − 3 2 g 2 bb g 2 b ( g 2 a g bb − 2 g a g b g ab + g 2 b g aa ) 2 g 4 a g 2 b , V 1 = g aab g a − 2 g aa g ab g 2 a + 1 2 g b g 3 a g 2 aa ( g 2 a g bb − 2 g a g b g ab + g 2 b g aa ) 2 g 3 a g 3 b , V 2 = g abb g b − 2 g bb g ab g 2 b + 1 2 g a g 3 b g 2 bb ( g 2 a g bb − 2 g a g b g ab + g 2 b g aa ) 2 g 3 a g 3 b . Moreo v er, let Σ = Z 1 − 3 V 1 − ( Z 2 − 3 V 2 ) , S = Z 1 + Z 2 − V 1 − V 2 . Then the in v arian ts x and y can b e c hosen in the fo llo wing form: x = [Σ 3 + 9Σ S + 18( Z 1 − Z 2 )] 2 [Σ 2 + 6 S + 3] 3 , y = Σ 2 − S 2 + 4Σ( V 1 − V 2 ) + 4( V 1 + V 2 ) − 1 [Σ 2 + 6 S + 3] 2 ; notice that the expressions for x and y are manifestly symmetric under the in t erchange of indices 1 ↔ 2. The functions x ( a, b ) a nd y ( a, b ) satisfy a system of the fo rm (3.16) whic h is in v arian t under t he action of S L (3 , R ) as sp ecified ab ov e. The explicit formulae for the co efficien ts I , J, M , N , as we ll as the prop erties of the corr esponding linear syste m (3.15), will b e discussed elsewhere . 3.3 General case: Lagrangian densities of the form f ( u x , u y , u t ) Let us b egin by in tro ducing sp ecial functions whic h will app ear in the general formula fo r f . The first one is a t heta-function of order one [12] (with mo dular parameter τ = ε ), θ ( z ) = X k ∈ Z ( − 1) k exp  2 π i ( k z + k ( k − 1) 2 ε )  , (3.18) whic h is known to satisfy the relations θ ( z + 1) = θ ( z ) , θ ( z + ε ) = − exp( − 2 π iz ) θ ( z ); here ε = 1 2 (1 + √ 3 i ). Note that ε is a primitiv e 6th ro ot of unity . In particular, ε 3 = − 1 and ε 2 = ε − 1. It is known that θ (0) = 0, and this is the only zero of t he f unction θ ( z ) mo dulo 1, ε . 12 Moreo v er, θ ′ (0) 6 = 0. Next w e define a function ˜ θ ( z ) whic h differs from θ ( z ) b y an exp o nen tial factor: ˜ θ ( z ) = 1 θ ′ (0) exp  − 2 π i ( ε 3 − 1 6 ) z 2 − π iz  θ ( z ) . (3.19) It can b e readily v erified that ˜ θ ( z + 1) = exp( − 2 π i 3 ((2 ε − 1) z + ε − 2)) ˜ θ ( z ) , ˜ θ ( z + ε ) = exp( − 2 π i 3 (( ε + 1 ) z + ε − 2) )) ˜ θ ( z ) . (3.20) These relations imply ˜ θ ( z + α + β ε ) = exp  − 2 π i (( 2 ε − 1 3 α + ε + 1 3 β ) z + ε + 1 3 α 2 + ε + 1 3 αβ + ε + 1 3 β 2 )  ˜ θ ( z ) (3.21) where α, β ∈ Z . One can also sho w that ˜ θ ( εz ) = ε ˜ θ ( z ) , whic h implies ˜ θ ( z ) = X j ≥ 0 a j z 6 j +1 ; (3.22) note that a 0 = ˜ θ ′ (0) = 1. Finally , let us define the ‘master-densit y’ f by the form ula f ( x, y , z ) = xy + X ( k, l ) ∈ Z 2 \ 0 ˜ θ (( k − εl ) x ) ˜ θ (( k − εl ) y ) ( k − εl ) 2 exp  2 π i 3 ( k 2 − k l + l 2 ) z  , (3.23) whic h w e claim to b e a ‘g eneric ’ solution o f (1.2) 1 . F or con ve nience, the arg umen t s of f are no w denoted by x, y , z . F rom (3.21) one can deriv e the following mo dular pro perties of f : f ( x + 1 , y , z + (2 ε − 1) x + ε + 1) = f ( x, y , z ) + y , f ( x + ε, y , z + ( ε + 1) x + ε + 1) = f ( x, y , z ) + εy , (3.24) f ( x, y + 1 , z + (2 ε − 1) y + ε + 1 ) = f ( x, y , z ) + x, f ( x, y + ε, z + ( ε + 1) y + ε + 1) = f ( x, y , z ) + εx. Substituting (3 .22) in to (3.23) w e obtain an alternative represen tation for f , f ( x, y , z ) = xy + X ( k, l ) ∈ Z 2 \ 0 , m,n ≥ 0 a m a n x 6 m +1 y 6 n +1  ( k − εl ) 6( m + n ) exp  2 π i 3 ( k 2 − k l + l 2 ) z  = 1 W e did not prove that f sa tisfies the system (1.2), althoug h computer calculatio ns supp ort this claim. 13 X m,n ≥ 0 a m a n x 6 m +1 y 6 n +1 g m + n ( z ) where g n ( z ) = X k ,l ∈ Z 2 1 2 (( k − εl ) 6 n + ( εk − l ) 6 n ) exp  2 π i 3 ( k 2 − k l + l 2 ) z  . It is kno wn [13] that g n ( z ) is a mo dular form, namely , g n ( αz + β γ z + δ ) = ( γ z + δ ) 6 n +1 g n ( z ) where  α β γ δ  ∈ S L (2 , Z ) and  α β γ δ  ≡  1 0 γ 1  mo d 3. Indeed, co efficien ts at the exponents are harmonic p olynomials with resp ect to the quadratic form k 2 − k l + l 2 . This giv es us the follo wing mo dular pro p erty of f ( x, y , z ) with resp ect to the same group: f  x γ z + δ , y γ z + δ , αz + β γ z + δ  = f ( x, y , z ) 1 γ z + δ . (3.25) Note that g ( z ) = g 0 ( z ) is a solution of the differen tial equation (3.7) and is giv en b y (3.12) (up to a rescaling of z ). F unctions g n ( z ) can b e represe nted as rational differen tial functions in g ( z ), for example, g 1 ( z ) = g ( z ) 2 g ′′ ( z ) − 2 g ( z ) g ′ ( z ) 2 = 1 2 g [ g , g ] 2 where [ g , g ] 2 is the Rankin-Cohen brac k et (see the pro of of Theorem 4). T ransformations (3.24) and (3.25) generate a discrete subgroup Γ ⊂ S L (4 , C ) whic h pla ys the role of the mo dular group for f ( x, y , z ). The subgroup Γ has the f o llo wing algebraic description. Let D ⊂ C 3 b e a domain in C 3 defined by D = { ( x, y , z ) ∈ C 3 ; | x | 2 + | y | 2 < 2 √ 3 ℑ z } . Note that D is a complex h yp erb olic ball. One can c hec k that the series (3.23) con verges exactly in the domain D . Let G ⊂ S L (4 , C ) b e a group defined b y G = { A ∈ S L (4 , C ); AJ A ∗ = J } where J =     1 0 0 0 0 1 0 0 0 0 0 − √ 3 i 0 0 √ 3 i 0     , and A ∗ stands for the Hermitian conjugate of A . One can c hec k that the complex h yp erb olic ball D is an orbit o f G under its standar d pro jectiv e action on C P 3 : if A = ( a ij ) ∈ G and ( x, y , z ) = ( x : y : z : 1) ∈ C P 3 , then A ( x, y , z ) =  a 11 x + a 12 y + a 13 z + a 14 a 41 x + a 42 y + a 43 z + a 44 , a 21 x + a 22 y + a 23 z + a 24 a 41 x + a 42 y + a 43 z + a 44 , a 31 x + a 32 y + a 33 z + a 34 a 41 x + a 42 y + a 43 z + a 44  . In our contex t, the domain D and the group G play a role similar to that of the upp er half plane and its a utomorphism gr o up S L (2 , R ) in the classical t heory of mo dular forms. L et Γ ⊂ G b e 14 a discrete subgroup of G consisting of matrices A = ( a ij ) ∈ G with the follow ing prop erties: a ij ∈ Z [ ε ] and A ≡     1 0 a 13 a 14 0 1 a 23 a 24 0 0 1 0 0 0 a 43 1     mo d (1 + ε ) . Here Z [ ε ] = { m + εn ; n, m ∈ Z } , and a ≡ b mo d(1 + ε ) for a, b ∈ Z [ ε ] means a − b 1+ ε ∈ Z [ ε ]. Note that all matrices corresponding to the transformations (3.24) and (3.25) b elong to Γ. W e conjecture that the group Γ is generated b y these transformations. Remark 1: limiting cases. La grangian densities of the form xg ( y , z ) can b e obtained as lim t → 0 f ( tx,y, z ) t from f ( x, y , z ) as defined b y (3.23). This giv es the function g ( y , z ) in the form g ( y , z ) = y + X ( k, l ) ∈ Z 2 \ 0 ˜ θ (( k − εl ) y ) k − εl exp  2 π i 3 ( k 2 − k l + l 2 ) z  = X n ≥ 0 a n y 6 n +1 g n ( z ) . (3.26) This function is defined o n the domain { ( y , z ) ∈ C 2 ; | y | 2 < 2 √ 3 ℑ z } and satisfies the follo wing mo dular prop erties: g ( y + 1 , z + (2 ε − 1) y + ε + 1 ) = g ( y , z ) + 1 , g ( y + ε, z + ( ε + 1) y + ε + 1) = g ( y , z ) + ε, (3.27) g  y γ z + δ , αz + β γ z + δ  = g ( y , z ) . Similarly , Lagrangia n densities of the form xy g ( z ) can b e obtained as lim t → 0 f ( tx,ty,z ) t 2 . This brings us bac k to the mo dular form g ( z ) discussed in Sect. 3 .1. Remark 2. Computer exp erimen ts sho w tha t solutions of the system (1.2 ) can also b e sough t in the form of a p o w er series, f ( x, y , z ) = X i,j,k ≥ 0 c ij k x 6 i +1 y 6 j +1 z 6 k + 1 . Moreo v er, c ij k = a i a j a k b i + j + k where a i are the same a s in (3.22), and b i is yet another sequence of complex n umbers. T aking a limit w e obtain g ( y , z ) = X j,k ≥ 0 a j a k b j + k y 6 j +1 z 6 k + 1 for densities of the form f ( x, y , z ) = xg ( y , z ), and g ( z ) = X k ≥ 0 a k b k z 6 k + 1 for densities of the form f ( x, y , z ) = xy g ( z ). W e p oin t out that in tegrable densities of this ty p e differ from the solutions constructed ab ov e by appropriat e transformations from the equiv alence group. 15 Remark 3. The a ction o f the 20-dimensional equiv alence group o n the 23-dimensional space of third or der jets of the function f ( a, b, c ) p ossesses three differen tial in v arian ts whic h w e denote by x, y , z . View ed as functions o f a, b, c , these in v arian ts satisfy a nonlinear system whic h p ossess es a transitiv e action of S L (4 , R ). This system can b e linearised f ollo wing the pro cedure outlined in Sect. 3.2. W e pla n to repo rt the details elsewhere. Remark 4. There exist a n umber of examples of in tegr a ble Lagrangia n densitie s expressible in terms of elemen tary f unctions. One can mention, e.g., the four p olynomial Lagrangians classified in [7]: f = u x u y u t , f = u 2 x u y + u y u t , f = u 3 x / 3 + u 2 y − u x u t and f = u 4 x + 2 u 2 x u t − u x u y − u 2 t . It w ould b e in teresting to explicitly demonstrate ho w these (and o t her) examples can b e ob- tained a s degenerations of the ‘master-Lagrangia n’ constructed in Section 3.2, and to describ e singular orbits of lo w er dimensions. Ac kno wl edgemen ts W e thank R. Halburd, J. Harnad, M. Pa vlo v, V. Sok olov and A. V eselo v for n umerous helpful discussions . The researc h of EVF was partially supp orted by the EPSR C grant EP/D03 6178/1, the Europ ean Union through the FP6 Marie Curie R TN pro ject ENIGMA (Con tract n um b er MR TN-CT-2004-5652), and the ESF programme MISGAM. References [1] M.J. Ablo witz and P .A. 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