Second order quasilinear PDEs and conformal structures in projective space

Second order quasilinear PDEs and conformal structures i n pro jectiv e space P .A. Buro vskiy , E.V. F erap o n to v and S.P . Tsarev ∗ Departmen t of Mathematica l S ciences Lough b orough Univ ersit y Lough b orough, L eicestershire LE11 3TU, UK and Institute of Mathematics Sib erian F ederal Unive rsit y 79 Svob o dny Prosp ect Krasno y arsk 660041 Russia e-mails: P.A.Buro vskiy@lb oro.ac.uk E.V.Fera pontov@l boro.ac.uk tsarev@n ewmail.r u Abstract W e inv estigate second order quasilinear eq uations of the for m f ij u x i x j = 0 , where u is a function of n indep endent v ariables x 1 , ..., x n , and the co eﬃcients f ij depe nd on the ﬁrst o rder deriv atives p 1 = u x 1 , ..., p n = u x n only . W e demonstr a te that the natura l equiv alence gr oup of the problem is iso morphic to S L ( n + 1 , R ), which ac ts b y pro jective transformatio ns on the space P n with co ordinates p 1 , ..., p n . The co eﬃcient ma trix f ij de- ﬁnes on P n a conforma l structur e f ij ( p ) dp i dp j . In this pap er w e conc e ntrate o n the cas e n = 3, although some results hold in any dimensio n. The necessary and suﬃcient conditions for the in tegrability of suc h eq ua tions by the metho d of hydrodynamic reductions are de- rived. These co nditions co nstitute an over-determined system of PDEs for the co eﬃcients f ij , whic h is in inv olution. W e prov e that the moduli space o f integrable equa tions is 20- dimensional. Based on these r e sults, we show that any equation s atisfying the integrabilit y conditions is necessarily cons erv ativ e, a nd posses s es a disp ers io nless Lax pair. Refo rmulated in diﬀerential-geometric terms, the integrabilit y conditions imply that the conformal struc- ture f ij ( p ) dp i dp j is conformally ﬂat, and pos s esses a n inﬁnity o f 3-co njugate null co o rdinate systems parametrized by three arbitrar y functions of one v ariable. Int egra ble equations provide an abundance of explicit examples of suc h co nformal structur e s parametrized by elementary functions, elliptic functions and mo dula r for ms . MSC: 3 5Q58, 3 7 K05, 37K1 0, 37K25. Keywords: Multi-dimensional Disp ersionless Integrable Systems, Hydro dynamic Reduc- tions, Integrability , Conforma l Structures, Dispe r sionless Lax Pairs, Conserv ation Laws. ∗ SPT ac kno wledges p artial ﬁnancial support from th e RFBR gran t 06-01-00814-a and the Russian–T aiw anese gran t 06-01-89507-HHC (95WFE03000 07). 1 1 In tro duc tion The m ain ob ject of our s tu dy are s econd order quasilinear equations of the form f 11 u xx + f 22 u y y + f 33 u tt + 2 f 12 u xy + 2 f 13 u xt + 2 f 23 u y t = 0 , (1) where u is a function of three indep endent v ariables x, y , t , and the coeﬃcients f ij dep end on the ﬁrst order deriv ativ es u x , u y , u t only . Equations of this type arise in a wide range of applications in mec hanics, general relativit y , diﬀeren tial geometry and the theory of in tegrable systems. Among the most familiar examples one s h ould mentio n the Bo y er-Finley equation, u xx + u y y − e u t u tt = 0 , whic h is descriptiv e of a class of self-dual 4-manifolds [10], as w ell as the disp ersionless Kadom tsev- P etviash vili (dKP) equation, u xt − u x u xx − u y y = 0 , also kno wn as the Khokhlo v-Zab olotsk a y a equation, which arises in non-linear acoustics [45] and the theory of Einstein-W eyl str u ctures [14]. The in tegrabilit y asp ects of equ ations of the f orm (1) ha v e b een inv estigated by a whole v ariet y of mo der n tec hniques includin g sym metry analysis, diﬀerentia l-geometric and algebro- geometric metho ds, disp ersionless ¯ ∂ -dr essing, factoriza tion tec hniques, Virasoro constrain ts, hy- dro dyn amic reductions, et c. Ho wev er, until recen tly there wa s no intrinsic ap p roac h wh ic h would allo w a un iﬁed treatmen t of these and other examples. Moreo v er, there was no satisfactory deﬁ- nition of the integrabilit y which w ould (a) b e algorithmica lly veriﬁable, (b) lead to classiﬁcation results and (c) provide a sc heme for the construction of exact solutions. W e emphasize that equations of the f orm (1) diﬀer essentia lly from their solitonic coun terparts, an d r equire an alternativ e approac h. Suc h appr oac h, based on the metho d of h ydro dyn amic red u ctions and, primarily , on the work [25, 26], see also [11, 31, 32, 15, 16, 37], etc, w as p rop osed in [17, 20]. It w as suggested to deﬁn e the inte grabilit y of a multi-dimensional disp er s ionless system by re- quiring the existence of ‘suﬃcien tly many’ hydro dynamic reductions wh ic h p ro vide multi-phase solutions playing a role similar to that of algebro-geo metric solutions of soliton equ ations. In Sect. 2 we brieﬂy review the metho d of h ydro dy n amic reductions, and apply it to equations of the form (1 ). Th is leads to a system of diﬀerenti al constraint s for the co eﬃcient s f ij , which are necessary an d suﬃcien t f or the integ rabilit y . W e demonstrate that the system of constrain ts is in inv olution, and p ro v e ou r ﬁ rst m ain result (Theorem 1 of Sect. 2): • The mo duli space of in tegrable equations of the form (1) is 20 -dimensional. In Sect. 3 we p oint out that the class of equations (1) is form-in v arian t under the action of S L (4 , R ), wh ic h constitutes the equiv alence group of th e p roblem. This action corresp onds to linear transformations of the v ariables x, y , t and u . Since equiv alence transformations pr e- serv e the inte grabilit y , any t w o S L (4 , R )-related equations are regarded as ‘the s ame’. All our classiﬁcation r esults are obtained m o dulo this equiv alence. Based on the inte grabilit y conditions, in S ect. 4 we classify integ rable equations of the form (1) un der v arious simplifying assumptions. This leads to a wide class of non-trivial examples, b oth kno wn and new, which are expressible in terms of elemen tary functions, elliptic fu nctions and mo d ular form s. Just to men tion a few of them, we hav e found an inte grable equation α ℘ ′ ( u x ) − ℘ ′ ( u y ) ℘ ( u x ) ℘ ( u y ) u xy + β ℘ ′ ( u t ) − ℘ ′ ( u x ) ℘ ( u x ) ℘ ( u t ) u xt + γ ℘ ′ ( u y ) − ℘ ′ ( u t ) ℘ ( u y ) ℘ ( u t ) u y t = 0 , 2 here ℘ is the W eierstrass ℘ -fun ction, ( ℘ ′ ) 2 = 4 ℘ 3 − g 3 , and α, β , γ are arb itrary constants. Another interesting example comes from the class u xy + ( u x u y r ( u t )) t = 0; for equations of this form the in tegrabilit y cond itions result in a single third order ODE for r , r ′′′ ( r ′ − r 2 ) − r ′′ 2 + 4 r 3 r ′′ + 2 r ′ 3 − 6 r 2 r ′ 2 = 0 , whic h app eared recen tly in the conte xt of m o dular form s of level tw o [1]. Its generic solution is giv en by the Eisenstein series E ( q ) = 1 − 8 ∞ X n =1 ( − 1) n nq n 1 − q n , q = e 4 u t , whic h is associated with the congruence subgroup Γ 0 (2) of the mo d ular group . F urther exam- ples of integ rable equations expressib le in terms of mo d ular form s can b e found in [22] in the classiﬁcation of integ rable Lagrangian equations of the f orm (1) corresp onding to ﬁrst order Lagrangian densities g ( u u , u y , u t ). A generic Lagrangia n d ensit y turns out to be a mo du lar form of its arguments. F urther generalizations of the ab o v e example includ e the equation u xy + 2  u x (log θ ) ′  t = 0 , where θ  u y 2 π , − u t π i  is the Jacobi theta-function, θ ( z , τ ) = 1 + 2 ∞ X n =1 e π in 2 τ cos(2 π nz ) , and prim e denotes diﬀerentia tion by u y . In Sect. 5 we stud y ﬁ rst ord er conserv ation laws, that is, relations of the form g 1 ( u x , u y , u t ) x + g 2 ( u x , u y , u t ) y + g 3 ( u x , u y , u t ) t = 0 , (2) whic h hold id en tically m o dulo (1). Ou r second result (Theorem 2 of sect. 5) states that • Any in tegrable equation of the form (1) p ossesses exactly four ﬁrst order con- serv ation laws. In Sect. 6 w e in v estigate the existence of d isp ersionless Lax pairs (scalar pseud o-p oten tials), S t = F ( S x , u x , u y , u t ) , S y = G ( S x , u x , u y , u t ) , (3) whic h imply Eq. (1) via th e consistency condition S ty = S y t (w e p oint out that the dep endence of F and G on S x is generally n on -linear). Lax pairs of th is t y p e ﬁ rst app eared in the constru ction of the un iv ersal Wh itham hierarc h y , see [32, 33] and referen ces th er ein. It was observed in [47] that consisten t Hamilt on-Jacobi-t y p e relations of the form (3 ) arise fr om the u sual ‘solitonic’ Lax pairs in th e disp ersionless limit. Disp ersionless Lax p airs constitute a k ey ingredient of the disp ers ionless ¯ ∂ -metho d, and a n ov el v ersion of the inv erse scattering transform [7, 30, 35]. It w as demonstrated in [18, 20] that, for a n umber of particularly in teresting cla sses of sy s tems, the existence of d isp ersionless Lax pairs is e quivalent to the existence of hydrod ynamic reductions and, thus, to the integ rabilit y . Our third main result (Th eorem 3 of Sect. 6) can b e form ulated as follo ws: 3 • Any integrable equation of the form (1) p ossesses a disp ersionless L ax pair. F urthermore, the existence of a disp ersionless Lax pair is equiv alen t to the existence of an inﬁnity of h ydro dynamic reductions and, th us, is necessary and suﬃcien t for the in t egrabilit y . Diﬀeren tial-g eometric asp ects of in tegrable equations of the form (1) are discuss ed in Sect. 7. Our main observ ation is that the equiv alence group S L (4 , R ) acts by p ro jectiv e transformations on the space of ﬁrst order deriv ativ es p 1 = u x , p 2 = u y , p 3 = u t , whic h is th us id en tiﬁed with the p ro jectiv e space P 3 , and th e co eﬃcient matrix f ij ( p ) su pplies P 3 with a well- deﬁned conformal structur e f ij dp i dp j . The classiﬁcation of integ rable equations of the form (1) up to the action of the equiv alence group S L (4 , R ) is therefore equiv alen t to the classiﬁcation of conformal s tructures in P 3 up to the pr o jectiv e equiv alence. Our ﬁrst r esult in this connection is a c haracterization of linearizable equations (Theorem 4 of Sect. 7.1): • Equation of the form (1) is linearizable b y a transformation from the equiv a- lence group if and only if the corresp onding conformal structure p ossesses a quadratic complex of n ull lines wit h the Segre sym b ol [(222)]. A tensorial c haracterizati on of linearizable and Lagrangian equations is pr o vided in Sect. 7.2. The integrabilit y conditions imp ose s trong restrictions on the conformal structure f ij dp i dp j (Theorem 5 of Sect. 7.4): • Conformal structures asso ciated with integrable equa t ions of t he form (1) are conformally ﬂat. Th us, the theory of int egrable equations of the f orm (1) has t w o ‘ﬂat’ coun terparts: one is the standard pro jectiv e sp ace P 3 with co ordinates p 1 , p 2 , p 3 , and the pr o jectiv e action of S L (4 , R ); alternativ ely , one can sp eak of a pro j ectiv ely ﬂat connection. Another is a ﬂat conformal stru c- ture f ij dp i dp j . Although, viewed separately , b oth stru ctures are trivial, this is no longer true when they are imp osed simultaneously: t heir ‘ﬂat co ord in ate sys tems’ ma y not coincide (in fact, they do coincide for lin earizable equations only). Our ﬁnal r esult pro vides a diﬀeren tial- geometric charact erization of the integrabilit y conditions (Sect. 7.4): • Integrable equations of the form ( 1) corresp ond to conformal structures in P 3 whic h p ossess an inﬁnity of three-conjugate n ull co ordinate systems parametrized b y t hree arbitrary functions of one v ariable. Notice that we are in the realm of tw o diﬀeren t geometries, namely , conformal geometry (re- sp onsib le f or the prop ert y of b eing null), and pro j ective geometry (resp onsible for the prop ert y of b eing conju gate). It is quite remark able th at the mo duli space of suc h stru ctures is only 20-dimensional! 2 Deriv ation of the in tegrabilit y conditions: t he metho d of h y- dro d ynamic reductions As p rop osed in [17 ], the metho d of hydrod ynamic reductions applies to quasilinear equations of the follo wing general form: A ( u ) u x + B ( u ) u y + C ( u ) u t = 0; (4) here u = ( u 1 , ..., u m ) t is an m -comp onen t column v ector of the dep endent v ariables, and A, B , C are l × m matrices where l , the num b er of equations, is allo wed to exceed the num b er of the 4 unknowns, m . The metho d of hydrod ynamic r ed uctions consists of seeking m ulti-phase s olutions in the form u = u ( R 1 , ..., R N ) (5) where the ‘p h ases’ R i ( x, y , t ) are required to satisfy a pair of consisten t equations of h ydro d y- namic type, R i y = µ i ( R ) R i x , R i t = λ i ( R ) R i x . (6) W e recall that the consistency (or comm utativit y ) conditions, R i y t = R i ty , imply the follo wing restrictions for the characte ristic sp eeds µ i and λ i : ∂ j µ i µ j − µ i = ∂ j λ i λ j − λ i , (7) i 6 = j, ∂ i = ∂ /∂ R i , see [44]. Deﬁnition. [17] A system (4) is said to b e inte gr able if, for any numb e r of phases N , it p ossesses i nﬁnitely many N -phase solutions p ar ametrize d by 2 N arbitr ary functions of one variable . In tegrable equations of the form (4) arise in a whole range of ph ysical and d iﬀeren tial- geometric applicatio ns, and con tain man y particularly imp ortant classes of systems. Thus, in the case of square matrices, m = l , on e arr iv es at sys tems of hydro dynamic t yp e, u t + A ( u ) u x + B ( u ) u y = 0; w e r ecall th at n -phase s olutions for hydro dyn amic typ e systems, also kno wn as solutions with a degenerate h o dograph (when n = 1 or n = 2), h a v e b een extensivel y inv estigated in gas dynamics [43]. S econd order quasilinear PDEs of th e form (1) can b e cast in to the form (4) by setting u = ( u x , u y , u t ); this giv es a system of four equations in the three unkn o wns: m = 3 , l = 4. Another interesting su b class corresp onds to equations of the disp ersionless Hirota typ e, F ( u xx , u xy , u y y , u xt , u y t , u tt ) = 0 , see [21] and references therein: one can c ho ose any ﬁve second order partial d eriv ativ es of the function u ( x, y , t ) as the d ep endent v ariables u . This corresp onds to the case m = 5 , l = 8. L et us illustrate the m etho d of hydro dynamic redu ctions using the dKP equation as an example. Example. Rewriting the dKP equation, u xt − u x u xx − u y y = 0 , in the n ew v ariables a = u x , b = u y , c = u t , one obtains a quasilinear system of f our equations in the three un kno wns, a y = b x , a t = c x , b t = c y , a t − aa x − b y = 0 . (8) Lo oking for N -phase solutions in the form a = a ( R 1 , ..., R N ) , b = b ( R 1 , ..., R N ) , c = c ( R 1 , ..., R N ), where th e ph ases R i satisfy Eqs. (6), one readily obtains the relations ∂ i b = µ i ∂ i a, ∂ i c = λ i ∂ i a, λ i = a + ( µ i ) 2 . (9) The compatibilit y cond itions ∂ i ∂ j b = ∂ j ∂ i b and ∂ i ∂ j c = ∂ j ∂ i c imply ∂ i ∂ j a = ∂ j µ i µ j − µ i ∂ i a + ∂ i µ j µ i − µ j ∂ j a, (10) 5 while the commutati vit y conditions (7) result in ∂ j µ i = ∂ j a µ j − µ i . (11) Ultimately , the s ubstitution of (11) in to (10 ) imp lies th e follo wing system f or a ( R ) an d µ i ( R ), ∂ j µ i = ∂ j a µ j − µ i , ∂ i ∂ j a = 2 ∂ i a∂ j a ( µ j − µ i ) 2 , (12) i 6 = j , which was ﬁr st deriv ed in [25, 26] in the con text of h ydro d ynamic r eductions of Benney’s momen t equations. F or an y solution µ i , a of the s y s tem (12) one can reconstruct λ i and b, c b y virtue of (9). A remark ab le feature of the system (12) is its m ulti-dimensional consistency: ∂ k ∂ j µ i = ∂ j ∂ k µ i and ∂ k ∂ i ∂ j a = ∂ j ∂ i ∂ k a . The general solution of the system (12) dep en d s, mo dulo reparametrizations R i → f i ( R i ), on N arbitrary functions of a single v ariable. T aking in to accoun t that N extra arbitrary functions come from the general solution of Eqs. (6), wh ic h can b e obtained by the generalised h o dograph m etho d [44], one ends up with an inﬁn ity of N -phase solutions dep en d ing on 2 N arbitrary fun ctions of a single v ariable. W e p oin t out that the compatibilit y conditions, ∂ k ∂ j µ i = ∂ j ∂ k µ i and ∂ k ∂ i ∂ j a = ∂ j ∂ i ∂ k a , in v olv e triples of ind ices i 6 = j 6 = k only . T hus, the consistency of the s y s tem (12) for N = 3 imp lies its consistency for arbitrary N . Th is tur n s out to b e a general phenomenon. The m ain resu lt of this section is the follo w ing Theorem 1 The mo duli sp ac e of inte gr able e quations of the form (1) is 20 -dimensional. Pro of: Here we only ske tc h the pro of: full details are pro vided in the App end ix. Our strategy is to deriv e a set of constraints wh ic h are n ecessary and suﬃcien t for the existence of an inﬁ nit y of n -phase solutions. T hese constrain ts constitute a complicated system of second order PDEs for the coeﬃcients f ij , whic h is in in v olution; after that, the calculation of the dimension of the mo duli space reduces to a simple parameter count. The main steps of the der iv ation of the inte grabilit y conditions can b e summarized as follo ws. First we in trod ucing the v ariables a = u x , b = u y , c = u t , whic h transform Eq. (1) in to the quasilinear form (4), a y = b x , a t = c x , b t = c y , f 11 a x + f 22 b y + f 33 c t + 2 f 12 a y + 2 f 13 a t + 2 f 23 b t = 0 . F ollo win g the metho d of h ydro dynamic r eductions, we seek multi- phase solutions in the form a = a ( R 1 , . . . , R N ) , b = b ( R 1 , . . . , R N ) , c = c ( R 1 , . . . , R N ) , where the phases R 1 ( x, y , t ), . . . , R N ( x, y , t ) are arbitr ary solutions of a p air of comm uting h ydro d ynamic typ e ﬂo ws (6). The sub stitution of this ansatz int o the ab ov e quasilinear system leads to the equations ∂ i b = µ i ∂ i a, ∂ i c = λ i ∂ i a, along with the disp ersion r elation D ( λ i , µ i ) = f 11 + f 22 ( µ i ) 2 + f 33 ( λ i ) 2 + 2 f 12 µ i + 2 f 13 λ i + 2 f 23 µ i λ i = 0 . The consistency of the equations for ∂ i b and ∂ i c implies ∂ i ∂ j a = ∂ j λ i λ j − λ i ∂ i a + ∂ i λ j λ i − λ j ∂ j a. 6 Diﬀeren tiating the disp ersion relation with resp ect to R j , j 6 = i, and k eeping in mind Eqs. (7), one obtains explicit expr essions for ∂ j λ i and ∂ j µ i in the f orm ∂ j λ i = ( λ i − λ j ) B ij ∂ j a, ∂ j µ i = ( µ i − µ j ) B ij ∂ j a, where B ij are certain r ational expressions in λ i , λ j , µ i , µ j , wh ose co eﬃcients d ep end on f ij ( a, b, c ) and ﬁrs t order deriv ativ es thereof (see th e App endix for explicit formulae). Thus, ∂ i ∂ j a = − ( B ij + B j i ) ∂ i a∂ j a. Calculating the consistency conditions ∂ j ∂ k λ i = ∂ k ∂ j λ i , ∂ j ∂ k µ i = ∂ k ∂ j µ i and ∂ i ∂ j ∂ k a = ∂ i ∂ k ∂ j a , one arr iv es at 3 0 diﬀeren tial relatio ns for the coeﬃcien ts f ij , whic h are linear in the sec- ond order deriv ativ es thereof. T hese relations are manifestly conformally in v arian t, and without an y loss of generalit y one can set, say , f 22 = 1. S olving for th e s econd ord er deriv ative s of the remaining co eﬃcient s f 11 , f 12 , f 13 , f 23 , f 33 , one obtains 30 r elations wh ic h can b e r epresen ted in the symb olic form d 2 f ij = 1 F R ( f k l , d f k l ); (13) here F = det { f ij } , and R is a p olynomial expression quadratic in eac h of its argum en ts. I n other words, an y second order der iv ativ e of any co eﬃcien t f ij is a certain exp licit expression in terms of f k l and ﬁ rst order deriv ativ es thereof. A direct computatio n shows that the system (13) is in in v olution: all compatibilit y conditions are satisﬁed ident ically . Since the v alues of the ﬁve functions f 11 , f 12 , f 13 , f 23 , f 33 , and ﬁrst order deriv ativ es thereof, are n ot r estricted by an y additional constraints, we obtain a 5 + 3 · 5 = 20-dimensional mo d uli space of integrable equations. This ﬁn ishes the pro of of Theorem 1. The inv arian t tensorial formulati on of the in tegrabilit y conditions (13) is pro vided in Sect. 7.4. Remark. Notice that, in t w o dimensions, any equation of the f orm f 11 ( u x , u y ) u xx + 2 f 12 ( u x , u y ) u xy + f 22 ( u x , u y ) u y y = 0 is automatically int egrable. Ind eed, in the n ew v ariables a = u x , b = u y it tak es the form of a t w o-component qu asilinear system, a y = b x , f 11 ( a, b ) a x + 2 f 12 ( a, b ) a y + f 22 ( a, b ) b y = 0, whic h linearises u n der the ho dograph transf orm ation inte rc hanging dep end ent and indep en d en t v ariables. T h is trick, ho we v er, d o es n ot work in higher dimensions. 3 The equiv alence group The resu lts of this section are n ot restricted to the dimension three, and hold in an y d imension. Let u s consider a m ulti-dimensional second order equation of the form X f ij u x i x j = 0 (14) where u = u ( x 1 , ..., x n ) is a function of n ind ep endent v ariables, and the co eﬃcient s f ij ( p ) = f ij ( p 1 , ..., p n ) dep end on the ﬁrst ord er d eriv ativ es p i = u x i only . In matrix f orm, Eq. (14 ) can b e repr esen ted as tr F U = 0 , where F = ( f ij ) is the (symmetric) matrix of co eﬃcients, deﬁned u p to a scalar multiple, and U is the Hessian matrix of the fun ction u . Our main observ ation is that the space P n with co ordinates p 1 , ..., p n admits a natural pr o jectiv e action of the group S L ( n + 1 , R ), and th e matrix f ij endo ws P n with a well -deﬁned n on-degenerate conformal str ucture f ij dp i dp j . This 7 can b e seen as follo ws. The class of equations (14) is in v arian t und er linear p oin t transformations of the form ˜ x = C x + bu, ˜ u = c x + β u, (15) where x = ( x 1 , ..., x n ) t is a (column) v ector of th e in dep end en t v ariables, C is a n × n matrix, b and c are n -comp onen t column and r o w v ectors, resp ectiv ely , and β is a constan t. The requirement that the transform ation (15) b elongs to the sp ecial linear group S L ( n + 1 , R ) is equiv alen t to the condition detC ( β − cC − 1 b ) = 1. Th e induced transf orm ation la w f or the (ro w) v ector of ﬁr st ord er der iv ativ es p = ( p 1 , ..., p n ) is manifestly p ro jectiv e, p = ( ˜ p C − c ) / ( β − ˜ p b ) , with th e inv erse transformation give n by ˜ p = κ p C − 1 + cC − 1 ; (16) here the scalar κ is deﬁned as κ = β − cC − 1 b 1+ p C − 1 b = 1 det ( C + b p ) . A direct calculation giv es d ˜ p = κd p  C − 1 − C − 1 b p C − 1 1 + p C − 1 b  = κd p ( C + b p ) − 1 . (17) Note also that d ˜ x = ( C + b p ) d x . Th e transformation la w for the Hessian matrix U can b e obtained by su b stituting d ˜ p , d ˜ x in to the equalit y d ˜ p = d ˜ x t ˜ U : U = 1 κ ( C + b p ) t ˜ U ( C + b p ) . Using this exp ression for U one obtains tr F U = 1 κ tr [ F ( C + b p ) t ˜ U ( C + b p )] = 1 κ tr [( C + b p ) F ( C + b p ) t ˜ U ] = tr ˜ F ˜ U where th e new co eﬃcien t matrix ˜ F is giv en by ˜ F = 1 κ ( C + b p ) F ( C + b p ) t . (18) Using the formulae (17) and (18 ) one r eadily veriﬁes the identi t y d ˜ p ˜ F d ˜ p t = κd p F d p t , whic h sho ws that the conformal class of the quadratic form d p F d p t = f ij dp i dp j is deﬁned in an inv arian t wa y . Th us, the co eﬃcien t matrix F ( p ) can b e view ed as deﬁn ing a conformal structure in the pro jectiv e space P n with co ord inates p 1 , ..., p n and the standard pro jectiv e action of S L ( n + 1 , R ). T h us, the study of e quations of the form (14) up to line ar tr ansformations of the dep endent and inde- p endent variables is e qu ivalent to the study of c onformal structur es in P n up to the pr oje ctive e quivalenc e . The glob al c ounterp art of Eq. (14) would b e a c onformal structur e on a pr oje ctive manifold M n (a manifold is said to b e pr oje ctive, or endowe d with a ﬂat pr oje c tive structur e, if it p ossesses an atlas with pr oje ctive tr ansition maps). The integ rabilit y conditions constitute a sys tem of s econd order PDEs for the co eﬃcient matrix F ( p ). Since p oint transformations preserve the integrabilit y , th e group generated b y Eqs. (16) and (18) constitutes a p oin t sy m metry group of the inte grabilit y conditions. Th is S L ( n + 1 , R )- in v ariance p la ys a cru cial role in the f urther analysis. In particular, all classiﬁcation r esults present ed b elo w are form ulated mod ulo this equiv alence: t wo S L ( n + 1 , R )-related equations are regarded as th e ‘same’. Returning to the case n = 3, we ha ve a 20-dimensional mo duli space of in tegrable equations, with the action of the equiv alence group S L (4 , R ). O n e can prov e that this action is locally free, that is, its generic orbits are 15-dimensional. Thus, up to the action of S L (4 , R ), a generic in tegrable equation is exp ected to d ep end on 5 essentia l p arameters. 8 4 Examples and c lassiﬁcation results Here we p resen t some further examples (b oth kno wn and new) and p artial classiﬁcation results of in tegrable PDEs of the form (1) based on the in tegrabilit y conditions derived in Sect. 2. Although these cond itions are quite complicated in general, they can b e written do wn and solv ed explicitly under v arious simplifyin g assu mptions. A computer program wh ic h calculates the integ rabilit y conditions can b e obtained from http://w ww-staff .lboro.ac.uk/~maevf/bft-supplementary-materials-2008.tar.gz W e construct a wh ole v ariet y of new inte grable examples exp ressible in elemen tary fun ctions, elliptic functions and mo d ular f orms, thereby manifesting the richness of th e p roblem. 4.1 List of kno wn examples In this su bsection w e bring together some of the known examples wh ic h arise, p rimarily , in the con text of the disp ersionless K P /T o da hierarchies. The simp lest generalizations of th e dKP equation are th e mo diﬁ ed dKP equation, u xt + 1 2 u 2 x u xx − u y y − u y u xx = 0 , and the deformed dK P equation, u xt + ǫ 2 2 u 2 x u xx − ǫu y u xx − u y y − u x u xx = 0 , see e.g. [34]. F urther examples includ e disp ersionless limits of the KP and T o da singular manifold equations, u t u x − 3 8  u y u x  2 ! x = 3 4  u y u x  y and u xy − u x u y ( e u t − 1)(1 − e − u t ) u tt = 0 , (19) resp ectiv ely [8]. V arious sy m metric forms of the disp ersionless KP , BKP and T o da h ierarc hies w ere obtained in [8, 9]: ( u y − u z ) u y z + ( u z − u x ) u xz + ( u x − u y ) u xy = 0 , u x ( u y − u z ) u y z + u y ( u z − u x ) u xz + u z ( u x − u y ) u xy = 0 , u x ( u 2 y − u 2 z ) u y z + u y ( u 2 z − u 2 x ) u xz + u z ( u 2 x − u 2 y ) u xy = 0 , ( e u x − 1)( e u y − e u z ) u y z + ( e u y − 1)( e u z − e u x ) u xz + ( e u z − 1)( e u x − e u y ) u xy = 0 . (20) A mo diﬁcation of the standard R-matrix sc heme leads to the so-called r -th d isp ersionless m o d- iﬁed K P and r -Dym equations, u xy − 3 − r (2 − r ) 2 u tt + (3 − r )(1 − r ) 2 − r u t u xx + (3 − r ) r 2 − r u x u xt + (3 − r )(1 − r ) 2 u 2 x u xx = 0 and u y t − 3 − r 2 − r  1 2 − r u y u xx − 1 1 − r u x u xy  = 0 , 9 see [5], [6], [36]. F urther examples arise in the con text of the so-called ‘unive rsal hierarc hy’: u y y = u y u xz − u xy u z , u xz = u x u y y − u xy u y ,  u t u x  t =  u y u x  x , see [38, 40]. A class of inte grable Eu ler-Lagrange equations of the form ( g u x ) x + ( g u y ) y + ( g u t ) t = 0 w as inv estigated in [20]; here the Lagrangian density g is a function of the ﬁrst order deriv ativ es only , g = g ( u x , u y , u t ). Among the apparently n ew examples f ound in [20] one should men tion the equation u t u xy + u x u y t + u y u xt = 0 , whic h corresp onds to th e L agrangian densit y g = u x u y u t . It was demonstrated in [22] that the generic integrable Lagrangian density g is a mo du lar form of its argument s. F urther examples of integ rable equati ons of the form (1) result, via appr opriate sub s titutions, from 2D in tegrable systems of hydro dynamic t yp e. F or ins tance, the p ap er [23] pro vides a complete list of integ rable Hamiltonian s y s tems of the form v t + ( H v ) y = 0 , w x + ( H w ) y = 0 (21) where the Hamiltonian densit y H is a fu nction of v , w . Eq . (21) 2 implies the existence of a p oten tial u such th at u y = w , u x = − H w . Expressing v and w in terms of u x and u y and substituting th ese expressions in to Eq. (21) 1 , one obtains an equation of the form (1). T h us, the simplest integ rable example H = 1 2 v w 2 results in the equation u x u y t − u y u xt + u 3 y u y y = 0, etc. Another p ossible construction exploits th e fact that an y t w o-comp onen t integ rable sy s tem of hydro dynamic t yp e p ossesses a disp ers ionless Lax pair of the form S t = F ( S x , v , w ) , S y = G ( S x , v , w ) , see [18]. Here v and w are the d ep endent v ariables whic h satisfy a system of hydro dynamic t yp e resulting from the compatibilit y conditions S y t = S ty . Ex p ressing v and w in terms of S x , S y and S t , and sub s tituting th ese expr essions in to the equations for v and w , one obtains a single second order equ ation of the f orm (1) for S . This construction is analogous to the ‘eigenfunction equations’ in soliton theory [29]. W e h a v e v eriﬁed that all examples listed in this section indeed satisfy the int egrabilit y con- ditions of S ect. 2. 4.2 Equations of the form u tt = f ( u x , u y , u t )( u xx + u y y ) F or these equations, whic h can b e view ed as nonlinear analog ues of the (2 + 1)-dimensional w a v e equation, the integrabilit y conditions tak e the form 2 f f bb + f 2 a − 3 f 2 b = 0 , 2 f f aa − 3 f 2 a + f 2 b = 0 , 2 f 2 f cc − 2 f f 2 c + f 2 a + f 2 b = 0 , f a f c − f ac f = 0 , 2 f a f b − f ab f = 0 , f b f c − f bc f = 0 , 10 recall that a = u x , b = u y , c = u t . T hese relations are straigh tforw ard to solv e. The case f a = f b = 0 leads, up to equiv alence tr ansformations, to a unique solution f = e c , wh ic h corresp onds to th e Bo y er-Finley equ ation u tt = e u t ( u xx + u y y ) . Assuming f a and f b to b e n onzero (notice that f a and f b can only b e zero or nonzero s im ulta- neously), the last three equations imply f ( a, b, c ) = p ( c ) ϕ ( a ) + ψ ( b ) , and the sub stitution into the remaining equations giv es f ( a, b, c ) = αe β c + γ e − β c + δ a 2 + b 2 + ξ , where th e constan ts α, β , γ , δ, ξ s atisfy a single quadratic constraint 4 αγ − δ 2 = 0. Th is corre- sp ond s to equations of the form u tt = αe β u t + γ e − β u t + δ u 2 x + u 2 y + ξ ( u xx + u y y ) , whic h are analogous to the singular manifold dT o d a equation (19). 4.3 Equations of the form u tt = f ( u x , u y , u t ) x These equations are related to the Hirota-t yp e equations Ω tt = f (Ω xx , Ω xy , Ω xt ) via a substitu- tion u = Ω x . In this form they were inv estigated in [41, 19]. T he in tegrabilit y cond itions tak e the form f ccc = 2 f 2 cc f c , f acc = 2 f ac f cc f c , f bcc = 2 f bc f cc f c , f aac = 2 f 2 ac f c , f abc = 2 f ac f bc f c , f bbc = 2 f 2 bc f c , f bbb = 2 f 2 c  f b f 2 bc + f bc ( f c f bb + 2 f ac ) − f cc ( f b f bb + 2 f ab )  , f abb = 2 f 2 c  f a f 2 bc + f ac ( f c f bb + f ac ) − f cc ( f a f bb + f aa )  , f aab = 2 f 2 c ( f cc ( f b f aa − 2 f a f ab ) − f ac ( f b f ac − 2 f c f ab ) − f bc ( f c f aa − 2 f a f ac )) , f aaa = 2 f 2 c  ( f a + f 2 b ) f 2 ac + f 2 a f 2 bc + f 2 c ( f 2 ab − f aa f bb ) − f bb f cc f 2 a + f ac f c ( f aa + 2( f a f bb − f b f ab )) + 2 f bc ( f b ( f c f aa − f a f ac ) − f a f c f ab ) − f cc (( f a + f 2 b ) f aa − 2 f a f b f ab )  . 11 This s ystem is in inv olution, and its general solution dep end s on 10 arbitrary constan ts. T he in tegration leads to the f our essent ially diﬀeren t canonical form s , f = u y + 1 4 A ( Au t + 2 B u x ) 2 + C e − Au x , f = u y u x +  1 u x + A 4 u 2 x  u 2 t + B u 2 x u t + B 2 Au 2 x + C e A/u x , f = u y u t + 1 6 η ( u x ) u 2 t , f = l n u y − ln θ 1 ( u t , u x ) − 1 4 u x Z η ( τ ) dτ , see [41]. Here A, B , C are arbitrary constan ts, η is a solution of the Chazy equation [12], η ′′′ + 2 η η ′′ = 3( η ′ ) 2 , and θ 1 is the Jacobi theta-function. 4.4 Equations of the form f ( u x , u y , u t ) u xy + g ( u x , u y , u t ) u xt + h ( u x , u y , u t ) u y t = 0 Equations of this t yp e arise in the con text of the disp ersionless K P , BKP and T o da h ierarc hies, see (20). F u rther examples were constructed in [2] in the searc h for consisten t trip les of second order equations. First of all, the int egrabilit y conditions imply that any equation within this class can b e redu ced to a simpliﬁed form, f ( u x , u y ) u xy + g ( u x , u t ) u xt + h ( u y , u t ) u y t = 0 , via a multiplica tion by an app r opriate scalar factor. In terms of the co eﬃcien ts f ( a, b ) , g ( a, c ) and h ( b, c ), the integrabilit y conditions tak e the form f ab = h b h f a + g a g f b − g a g h b h f , g ac = h c h g a + f a f g c − f a f h c h g , h bc = g c g h b + f b f h c − f b f g c g h, f aa = g f a + f g a f g f a + f g a − g f a f h f b + g h b − f h c h 2 f a + f g a h c − g g a h b g h 2 f , f bb = f h b + hf b f h f b + f h b − hf b f g f a + hg a − f g c g 2 f b + f h b g c − hg a h b g 2 h f , g cc = g h c + hg c g h g c + g h c − hg c f g g a + hf a − g f b f 2 g c + g f b h c − hf a h c f 2 h g , g aa = f g a + g f a f g g a + g f a − f g a g h g c + f h c − gh b h 2 g a + g f a h b − f f a h c f h 2 g , h cc = g h c + hg c g h h c + hg c − gh c f h h b + g f b − hf a f 2 h c + hf a g c − g f b g c f 2 g h, h bb = g h b + hf b f h h b + hf b − f h b g h h c + f g c − hg a g 2 h b + hg a f b − f f b g c f g 2 h. Remark ably , the ﬁrst three equations app eared previously in [23] in the problem of classiﬁcation of integrable Hamilt onian systems of h y d ro dyn amic t yp e. It was observ ed that their general 12 solution is rep r esen table in the form f = p ( a ) − q ( b ) P ( a ) Q ( b ) , g = r ( c ) − p ( a ) P ( a ) R ( c ) , h = q ( b ) − r ( c ) Q ( b ) R ( c ) , (22) where p, q , r and P , Q , R are arb itrary fu nctions of the indicated argumen ts. The further a nalysis splits int o four diﬀerent cases dep en ding on ho w many fu nctions among p, q , r are constan t. In the simplest case when all three of them are constan t, the integ rabilit y conditions reduce to P ′′ = Q ′′ = R ′′ = 0. Thus, an y equation of the form a 1 u x u y t + a 2 u y u xt + a 3 u t u xy = 0 is automatically int egrable. Let us concen trate on the generic case w hen none of p, q , r are constan t (intermediate cases can b e considered in a similar w a y). Under the substitution (22 ), the ﬁ r st three integ rabilit y conditions will b e satisﬁed iden tically , while the last six imp ly a system of second order ODEs for p, q , r and P , Q, R : P ′′ p ′ = P ′ − Q ′ p − q + P ′ − R ′ p − r − Q ′ − R ′ q − r , R ′′ r ′ = R ′ − Q ′ r − q + R ′ − P ′ r − p − Q ′ − P ′ q − p , (23) Q ′′ q ′ = Q ′ − P ′ q − p + Q ′ − R ′ q − r − P ′ − R ′ p − r , p ′′ p ′ P = q − p q − r  Rr ′ r − p − R ′  + r − p r − q  Qq ′ q − p − Q ′  + p ′ P ( q − 2 p + r ) ( p − r )( p − q ) , r ′′ R ′ R = q − r q − p  P p ′ p − r − P ′  + p − r p − q  Qq ′ q − r − Q ′  + r ′ R ( p − 2 r + q ) ( r − p )( r − q ) , (24) q ′′ q ′ Q = p − q p − r  Rr ′ r − q − R ′  + r − q r − p  P q ′ p − q − P ′  + q ′ Q ( p − 2 q + r ) ( q − p )( q − r ) . The s ep aration of v ariables in Eqs. (23) implies P ′ = F ( p ) , Q ′ = F ( q ) , R ′ = F ( r ) , where F ( · ) is an arbitrary qu adratic p olynomial, F ( x ) = ǫx 2 + m 1 x + m 0 ; here the parameters ǫ, m 1 , m 0 pla y th e role of separation constan ts. A similar separation of v ariables in Eqs. (24) results in p ′ P = S ( p ) , q ′ Q = S ( q ) , r ′ R = S ( r ) , where S ( · ) is a cubic p olynomial, S ( x ) = ǫx 3 + n 2 x 2 + n 1 x + n 0 . These equations lead to P ′ /P = p ′ F ( p ) /S ( p ) , Q ′ /Q = q ′ F ( q ) /S ( q ) , R ′ /R = r ′ F ( r ) /S ( r ), and the in tegration yields P = c 1 ( p − α ) µ ( p − β ) ν ( p − γ ) η , Q = c 2 ( q − α ) µ ( q − β ) ν ( q − γ ) η , R = c 3 ( r − α ) µ ( r − β ) ν ( r − γ ) η ; here c i are th ree extra inte gration constants, α, β , γ are the ro ots of the p olynomial S , and the exp onents µ, ν, η , wh ic h are related to the co eﬃcient s m 1 , m 0 of the quadratic p olynomial F , satisfy a single relation µ + ν + η = 1. Thus, setting ǫ = 1, we hav e p ′ = 1 c 1 ( p − α ) 1 − µ ( p − β ) 1 − ν ( p − γ ) 1 − η , q ′ = 1 c 2 ( q − α ) 1 − µ ( q − β ) 1 − ν ( q − γ ) 1 − η , 13 r ′ = 1 c 3 ( r − α ) 1 − µ ( r − β ) 1 − ν ( r − γ ) 1 − η ; recall that quadratur es of this t yp e arise in the con text of the S c h w arz-Christoﬀel mapp ings of triangular domains. P articularly inte resting examples corresp ond to the sy m metric c h oice µ = ν = η = 1 / 3. In this case the ODEs f or p, q , r tak e the form p ′ 3 = 1 c 3 1 S 2 ( p ) , q ′ 3 = 1 c 3 2 S 2 ( q ) , r ′ 3 = 1 c 3 3 S 2 ( r ) . (25) W e p oint out that the ansatz (22) p ossesses the ob vious S L (2 , R )-symmetry , p → αp + β γ p + δ , q → αq + β γ q + δ , r → αr + β γ r + δ , P → P γ p + δ , Q → Q γ q + δ , R → R γ r + δ , whic h can b e used to b ring the p olynomial S to a canonical f orm. In th e case of three distinct ro ots one can redu ce S to a qu adratic, S ( x ) = x 2 + g 3 , so that the ODEs (25) imply p = ℘ ′ ( a 1 u 1 ) , q = ℘ ′ ( a 2 u 2 ) , r = ℘ ′ ( a 3 u 3 ) where 27 a 3 i = 2 /c 3 i , and ℘ is the W eierstrass ℘ -function: ( ℘ ′ ) 2 = 4 ℘ 3 − g 3 (notice th at w e are d ealing with an equianharmonic case: g 2 = 0). This leads to the inte grable equation ℘ ′ ( a 1 u x ) − ℘ ′ ( a 2 u y ) ℘ ( a 1 u x ) ℘ ( a 2 u y ) u xy + ℘ ′ ( a 3 u t ) − ℘ ′ ( a 1 u x ) ℘ ( a 1 u x ) ℘ ( a 3 u t ) u xt + ℘ ′ ( a 2 u y ) − ℘ ′ ( a 3 u t ) ℘ ( a 2 u y ) ℘ ( a 3 u t ) u y t = 0 . Up to an appropr iate rescaling, this equation is equiv alen t to α ℘ ′ ( u x ) − ℘ ′ ( u y ) ℘ ( u x ) ℘ ( u y ) u xy + β ℘ ′ ( u t ) − ℘ ′ ( u x ) ℘ ( u x ) ℘ ( u t ) u xt + γ ℘ ′ ( u y ) − ℘ ′ ( u t ) ℘ ( u y ) ℘ ( u t ) u y t = 0 . It p ossesses a d egeneration g 3 → 0 , ℘ ( x ) → 1 /x 2 , resulting in αu t ( u 3 x − u 3 y ) u xy + β u y ( u 3 t − u 3 x ) u xt + γ u x ( u 3 y − u 3 t ) u y t = 0 , compare w ith (20). 4.5 In t egrable equations in terms of mo dular forms and theta functions This section cont ains a num b er of m ore ‘exotic’ in tegrable examples whic h are n ot expressible in elementa ry functions. Let u s b egin with equations of the form ( u y p ( u t )) x + ( u x q ( u t )) y + ( u x u y r ( u t )) t = 0 . The in tegrabilit y cond itions yield a complicated system of three third order ODEs for the fun c- tions p, q and r . Let us analyze sp ecial cases. Case 1. The choice q = p , r = p ′ corresp onds to Lagrangian equations with the Lagrangian densit y u x u y p ( u t ). In this case the int egrabilit y conditions redu ce to a single four th order ODE for p , p ′′′′ ( p 2 p ′′ − 2 pp ′ 2 ) − p 2 p ′′′ 2 + 2 pp ′ p ′′ p ′′′ + 8 p ′ 3 p ′′′ − 9 p ′ 2 p ′′ 2 = 0 . It w as sho wn in [22] that the generic solution of this equation is a mo d ular form of lev el thr ee, kno wn as the Eisens tein series E 1 , 3 ( z ): p ( 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 + ... ; 14 (the Eisenstein series E 1 , 3 ( z ) results up on the substitution u t → 2 π iz ). It can also b e w ritten in the form p ( u t ) = 1 − 6 ∞ X k =1 e (3 k − 1) u t 1 − e (3 k − 1) u t − e (3 k − 2) u t 1 − e (3 k − 2) u t ! . Notice that a similar c hoice q = p, r = − p ′ corresp onds to equations of the f orm u xy − p ′′ 2 p u x u y u tt = 0. Here the in tegrabilit y cond itions r esult in a fourth order ODE p 2 p ′′ p ′′′′ − p 2 p ′′′ 2 − 2 pp ′′ 3 + p ′ 2 p ′′ 2 = 0 , whose prop erties are quite diﬀeren t from those of the abov e equation: ﬁ r st of al l, one can reduce the ord er by setting p ′′ = 2 ph . This r esu lts in the second ord er ODE hh ′′ − h ′ 2 − 2 h 3 = 0, wh ich implies h ( u t ) = 4 s 2 e 2 su t ( e 2 su t − 1) 2 , s = const. This is the case of the singular manifold dT o d a equation (19), see also Ex. 2 of Sect. 5. Case 2. Another interesting choice is p = 1 , q = 0 w h ic h corresp onds to equations of the f orm u xy + ( u x u y r ( u t )) t = 0. The in tegrabilit y conditions r esult in a single third order ODE for r , r ′′′ ( r ′ − r 2 ) − r ′′ 2 + 4 r 3 r ′′ + 2 r ′ 3 − 6 r 2 r ′ 2 = 0 , whic h app eared recently in a diﬀerent cont ext in the th eory of mo dular form s of lev el tw o: set r = y / 2 to obtain E q. (4.7) fr om [1]. This equ ation p ossesses a r emark able S L (2 , R )-in v ariance, ˜ z = αz + β γ z + δ , ˜ r = ( γ z + δ ) 2 r + γ ( γ z + δ ) , αδ − γ β = 1; here z = u t . Mo du lo this S L (2 , R )-action, th e generic solution is giv en by the series r ( u t ) = 1 + 8 e 4 u t − 8 e 8 u t + 32 e 12 u t − 40 e 16 u t + 48 e 20 u t − 32 e 24 u t + ..., whic h, u p on setting e 4 u t = q , coincides with th e Eisenstein series, E ( q ) = 1 − 8 ∞ X n =1 ( − 1) n nq n 1 − q n , asso ciated with the congruence su bgroup Γ 0 (2) of the mo dular group [1]. Case 3. As a generalization of Case 2, let us consider equations of the form u xy + ( u x r ( u y , u t )) t = 0. The inte grabilit y cond itions tak e the form r bbb r bb = r r bb − r bc + r 2 b r r b − r c , r bbc r bb = r r bc − r cc + r b r c r r b − r c , r bcc = 1 r r b − r c (2 r r 2 bc − rr bb r cc − r cc r bc + r 2 b r cc − 2 r b r c r bc + 2 r 2 c r bb ) , r ccc = 1 r r b − r c (2 r 2 r 2 bc − 2 r 2 r bb r cc + r r cc r bc + 4 r r 2 b r cc − 8 rr b r c r bc + 4 r r 2 c r bb − r 2 cc − r b r c r cc + 2 r 2 c r bc ) . 15 Here the ﬁr st t w o equations imply r bb = α ( rr b − r c ), which is the w ell-kno w n Burgers equation. Without an y loss of generalit y w e will assu me α = 1. Under the sub stitution r = 2 v b /v , the Burgers equation linearizes to the h eat equation, v c = v bb . Mo d ulo this equation, the in tegrabilit y conditions redu ce to a single s ixth order ODE f or the fu nction v , v 4 v ′′′ v (6) − v 4 v (4) v (5) + 3 v 3 v ′′ 2 v (5) − 5 v 3 v ′′ v ′′′ v (4) + 2 v 3 v ′′′ 3 − 3 v 3 v ′ v ′′ v (6) − 2 v 3 v ′ v ′′′ v (5) + 5 v 3 v ′ v (4) 2 + 2 v 2 v ′ v ′′ v ′′′ 2 + 6 v 2 v ′ 2 v ′′ v (5) − 10 v 2 v ′ 2 v ′′′ v (4) + 2 v 2 v ′ 3 v (6) − 6 v v ′ 2 v ′′ 2 v ′′′ + 12 v v ′ 3 v ′′′ 2 − 6 v v ′ 4 v (5) + 6 v ′ 3 v ′′ 3 − 12 v ′ 4 v ′′ v ′′′ + 6 v ′ 5 v (4) = 0 , here pr ime denotes diﬀeren tiation with resp ect to b . On e can show that the generic solution of th e heat equation constrained b y this sixth order ODE is give n b y the formula v ( u y , u t ) = θ  u y 2 π , − u t π i  , wh er e θ is the Jacobi theta-function: θ ( z , τ ) = 1 + 2 ∞ X n =1 e π in 2 τ cos(2 π nz ) . Remark. F urth er generalizati on, u xy + f ( u x , u y , u t ) t = 0, w as discussed in [19] in the disp er- sionless Hirota f orm Ω xy + f (Ω xt , Ω y t , Ω tt ) = 0 (these tw o represen tations are relat ed via u = Ω t ). It was demonstrated that the generic solution is giv en b y a ratio of tw o Jacobi theta functions: f ( u x , u y , u t ) = − 1 4 ln θ 1 ( u t , u y − u x ) θ 1 ( u t , u y + u x ) . 5 Conserv ation la ws Let u s b egin with some general r emarks wh ic h clarify the geometric meaning of conserv ation la w s of Eq . (14). Consider a ﬁ r st ord er conserv ation la w in the form ∂ x 1 g 1 + ... + ∂ x n g n = 0 , (26) where g i ( p ) are functions of the ﬁ rst order deriv ativ es p 1 = u x 1 , ..., p n = u x n . Th e requirement that the equalit y (26 ) is satisﬁed iden tically mo d ulo Eq. (14) is equiv alen t to the set of relations ∂ p i g i = k f ii , ∂ p i g j + ∂ p j g i = 2 kf ij , h ere k is a prop ortionalit y factor. These r elations can b e rewritten in the form dg 1 dp 1 + ... + dg n dp n = kf ij dp i dp j , (27) whic h has a clear geometric inte rpr etation. Let us intro d uce a ﬁb er bundle B with th e base P n supplied w ith a ﬂ at conform al stru cture dg i dp i (here p i are co ordinates on the base P n , g i are co ordinates along th e ﬁb er). Th e iden tit y (27) provides an emb edding of the conformal stru cture f ij dp i dp j , deﬁn ed on the b ase, int o the ﬁ b er bun dle B with the ﬂat conformal structure dg i dp i . T o un co ver the geometry of B let us return to the equiv alence group S L ( n + 1 , R ) whic h acts via (15) on the space of indep end en t v ariables x i . The indu ced actio n on conserv ation la ws g = ( g 1 , ..., g n ) t is ˜ g = κ ( C + b p ) g , recall that κ = 1 det ( C + b p ) , see Sect. 3. Combining this with the transformation la w d ˜ p = κd p ( C + b p ) − 1 w e obtain d ˜ p ˜ g = κ 2 d p g . (28) 16 In tro du cing K = dp 1 ∧ ... ∧ dp n w e can see that ˜ K = κ n +1 K so th at (28 ) tak es th e f orm d ˜ p ˜ g ˜ K − 2 n +1 = d p g K − 2 n +1 . (29) This sho ws that g has a natural interpretatio n as a 1-form with v alues in the canonical bund le K − 2 n +1 , n amely , g ∈ H 0 ( P n , Ω 1 ⊗ K − 2 n +1 ). No tice that for n = 3 (whic h is our main case of in terest) th is bundle ap p eared in [27] in the geomet ric theory of solutio ns of Einstein’s equations. Let us return to th e 3-dimensional case. Our main resu lt is that any integ rable equation of the form (1) p ossesses exactly four ﬁrst order conserv ation la ws of the form (2), g 1 ( u x , u y , u t ) x + g 2 ( u x , u y , u t ) y + g 3 ( u x , u y , u t ) t = 0 . According to the discussion ab o v e, this means that there exists a 4-parameter linear family of sections g ∈ H 0 ( P 3 , Ω 1 ⊗ K − 1 2 ) p ro viding conformal em b eddings of the conformal stru cture f ij dp i dp j . Let us b egin with illustrating examples. Example 1. The equation a 1 u x u y t + a 2 u y u xt + a 3 u t u xy = 0 , whic h w as found in S ect. 4.4, p ossesses four conserv ation la ws of the form ( a 2 + a 3 − a 1 )( u y u t ) x + ( a 1 + a 3 − a 2 )( u x u t ) y + ( a 1 + a 2 − a 3 )( u x u y ) t = 0 , a 2  u t a 1 + a 2 a 3 u y  x + a 1  u t a 1 + a 2 a 3 u x  y = 0 , a 3  u y a 1 + a 3 a 2 u t  x + a 1  u y a 1 + a 3 a 2 u x  t = 0 , a 3  u x a 2 + a 3 a 1 u t  y + a 2  u x a 2 + a 3 a 1 u y  t = 0 . Example 2. The dT o da sin gu lar manifold equation [8], u xy − u x u y ( e u t − 1)(1 − e − u t ) u tt = 0 , p ossesses t w o conserv ation la ws of the form  log u y e u t − 1  x +  u x e u t e u t − 1  t = 0 ,  log u x e u t − 1  y +  u y e u t e u t − 1  t = 0 , as well as ( p ( u t ) u y ) x + ( p ( u t ) u x ) y −  p ′ ( u t ) u x u y  t = 0 where p satisﬁes the second order linear ODE p ′′ 2 p = e u t ( e u t − 1) 2 ; as p oin ted out by M. P a vlo v, it p ossesses tw o linearly indep en den t solutions, p = e u t + 1 e u t − 1 and p = e u t + 1 e u t − 1 u t − 2 , 17 whic h pro vide tw o extra conserv ation la ws. Example 3. The equation u t ( u 3 x − u 3 y ) u xy + u y ( u 3 t − u 3 x ) u xt + u x ( u 3 y − u 3 t ) u y t = 0 , whic h app eared in Sect. 4.4, p ossesses four conserv ation laws, ( u x u y ( u 3 x − u 3 y )) t + ( u x u t ( u 3 t − u 3 x )) y + ( u y u t ( u 3 y − u 3 t )) x = 0 , as well as  G  u t u y  x −  G  u x u y  t = 0 ,  G  u y u x  t −  G  u t u x  y = 0 ,  G  u x u t  y −  G  u y u t  x = 0 , here the fun ction G ( s ) is deﬁned as G ′ = 1 s 3 − 1 . Exp licitly , one has G ( s ) = 1 3  ln( s − 1) + ǫ ln( s − ǫ ) + ǫ 2 ln( s − ǫ 2 )  , here ǫ = e 2 π i/ 3 , ǫ 3 = 1; notice that G is real-v alued. Example 4. The equation ℘ ′ ( u x ) − ℘ ′ ( u y ) ℘ ( u x ) ℘ ( u y ) u xy + ℘ ′ ( u t ) − ℘ ′ ( u x ) ℘ ( u x ) ℘ ( u t ) u xt + ℘ ′ ( u y ) − ℘ ′ ( u t ) ℘ ( u y ) ℘ ( u t ) u y t = 0 , whic h app eared in Sect. 4.4, p ossesses four conserv ation laws, the ﬁrst one b eing H ( u y , u t ) x + H ( u t , u x ) y + H ( u x , u y ) t = 0 , where th e fun ction H ( r , s ) is deﬁned by the equations H r = − ζ 2 ( s ) − ℘ ( s ) − 4 ℘ ( s ) ζ ( s ) ℘ ( s ) − ζ ( r ) ℘ ( r ) ℘ ′ ( s ) − ℘ ′ ( r ) , H s = ζ 2 ( r ) + ℘ ( r ) + 4 ℘ ( r ) ζ ( s ) ℘ ( s ) − ζ ( r ) ℘ ( r ) ℘ ′ ( s ) − ℘ ′ ( r ) , here ζ is the W eierstrass zeta-function: ζ ′ = − ℘ (w e p oin t out that the equations for H are automatica lly consisten t: H r s ≡ H sr ). Th r ee extra conserv ation la ws are of the form F ( u x , u y ) t − F ( u x , u t ) y = 0 , F ( u y , u x ) t − F ( u y , u t ) x = 0 , F ( u t , u x ) y − F ( u t , u y ) x = 0 , where th e fun ction F ( r , s ) is deﬁn ed by the equations F r = ℘ ( r ) ℘ ( s ) ℘ ′ ( r ) − ℘ ′ ( s ) , F s = ℘ 2 ( r ) ℘ ′ ( s ) − ℘ ′ ( r ) − 1 2 ζ ( r ) . Explicitly , one has (see [23], w here th e fun ction F app eared in a diﬀerent con text): F ( r, s ) = 1 6  ln σ ( r − s ) + ǫ ln σ ( ǫr − s ) + ǫ 2 ln σ ( ǫ 2 r − s )  ; here ǫ 3 = 1 and σ is the W eierstrass sigma-fun ction: σ ′ /σ = ζ . Notice that F is r eal-v alued. The m ain resu lt of this section is the follo wing Theorem 2 Any inte gr able quasiline ar PDE of the form (1) p ossesses four ﬁrst or der c onser- vation laws. 18 Remark 1. W e wo uld lik e to stress that the con verse statement is n ot tru e: the existence of four conserv ation la w s do es not necessarily imp ly the integ rabilit y . F or instance, any Euler- Lagrange equation of the form ( g u x ) x + ( g u y ) y + ( g u t ) t = 0, corresp onding to the Lagrangian densit y g ( u x , u y , u t ), is manifestly conserv ativ e and, m oreo ver, p ossesses three extra ﬁrst order conserv ation la ws ( u x g u x − g ) x + ( u x g u y ) y + ( u x g u t ) t = 0 , ( u y g u x ) x + ( u y g u y − g ) y + ( u y g u t ) t = 0 , ( u t g u x ) x + ( u t g u y ) y + ( u t g u t − g ) t = 0 , whic h constitute comp onents of the energy-momen tum tensor. On the other hand, as sh o wn in [20], the int egrabilit y conditions are v ery r estrictive and reduce to a complicated system of fourth order PDEs for the Lagrangian d ensit y g , resulting in the ﬁnite-dimensionalit y of the mo duli sp ace of integ rable L agrangians. Remark 2. In tegrable multi- dimensional equations n ormally p ossess inﬁnite hierarc hies of higher ord er conserv ation la ws, w hic h are generically non-lo cal. The discussion of higher con- serv ation la ws is b eyo nd the scop e of our p ap er. Pro of of T he orem 2: Diﬀeren tiating Eq. (2) we obtain ( g 1 ) a u xx + ( g 2 ) b u y y + ( g 3 ) c u z z + (( g 1 ) b + ( g 2 ) a ) u xy + (( g 1 ) c + ( g 3 ) a ) u xz + (( g 2 ) c + ( g 3 ) b ) u y z = 0 . Let u s introdu ce the new v ariables s i , r i via r 1 = ( g 1 ) a , r 2 = ( g 2 ) b , r 3 = ( g 3 ) c , 2 s 1 = ( g 2 ) c + ( g 3 ) b , 2 s 2 = ( g 1 ) c + ( g 3 ) a , 2 s 3 = ( g 1 ) b + ( g 2 ) a ; one can ve rify that these v ariables automatically satisfy the compatibilit y conditions s 1 aa = s 2 ab + s 3 ac − r 1 bc , 2 s 3 ab = r 1 bb + r 2 aa , s 2 bb = s 1 ab + s 3 bc − r 2 ac , 2 s 1 bc = r 2 cc + r 3 bb , (30) s 3 cc = s 1 ac + s 2 bc − r 3 ab , 2 s 2 ac = r 1 cc + r 3 aa . F or a relatio n of the from (2) to b e a conserv ation la w of Eq. (1) one h as to requ ire the existence of a factor k ( a, b, c ) s u c h th at r 1 = kf 11 , r 2 = kf 22 , r 3 = kf 33 , s 1 = kf 23 , s 2 = kf 13 , s 3 = kf 12 . Substituting this in to the compatibilit y conditions (30) we arrive at six linear equ ations f or the ‘inte grating factor’ k . Solvi ng this system for the second order partial deriv ativ es k ij (the v ariables a, b, c are lab elled by in d ices 1 , 2 , 3, resp ectiv ely), one obtains k ij = 1 2 ( f is f r j + f j s f r i − f ij f r s )( k f pq ,tl + 2 k p f tl,q ) ǫ ptr ǫ q ls . (31) Here ǫ ij k is the tota lly an tisymmetric tensor dual to th e v olume form of the metric f ij corre- sp ond in g to the equation (1), that is, ǫ 123 = 1 / √ F , ǫ 213 = − 1 / √ F , etc, F = det f ij . The sys tem (31) app ears to b e in inv olution mo dulo th e int egrabilit y conditions (13 ). Since the v ariable k and its ﬁrst order deriv ativ es k a , k b , k c are n ot r estricted by an y additional constraint s, th ere is a 4-parameter fr eedom for the int egrating factor. This ﬁnishes the pr o of. 19 Example 5. Let u s write out Eqs. (31) f or the disp ersionless Hirota equation d iscussed in Ex. 1. W e ha v e f 11 = f 22 = f 33 = 0 , f 12 = a 3 c, f 13 = a 2 b, f 23 = a 1 a , and the sub stitution in to (31) implies k ab = k ac = k bc = 0 , k aa = − 3 a k a , k bb = − 3 b k b , k cc = − 3 c k c . This leads to the f our v alues for the integrat ing factor: k = const , k = a − 2 , k = b − 2 , k = c − 2 , whic h corresp ond to th e f our conserv ation la w s from Ex. 1. 6 Disp ersionless Lax pairs In this section we pro v e that any in tegrable equ ation of the form (1) p ossesses a disp ersionless Lax pair, S t = F ( S x , u x , u y , u t ) , S y = G ( S x , u x , u y , u t ) . Let u s b egin with illustrating examples. Example 1. The disp ersionless Hirota equation, a 1 u x u y t + a 2 u y u xt + a 3 u t u xy = 0 , a 1 + a 2 + a 3 = 0 , p ossesses the Lax p air S t u t = µ S x u x , S y u y = λ S x u x ; here the constants λ and µ s atisfy a single quadr atic constraint a 1 λµ + a 2 µ + a 3 λ = 0. Notice that this Lax pair is linear in S . It w as discussed in [46 ] in the cont ext of V eronese w ebs, and was used to solv e the disp ersionless Hirota equation via a n on-linear Riemann problem. W e will see b elo w that the case a 1 + a 2 + a 3 6 = 0 is far more complicated, leading to Lax pairs parametrized b y hyp ergeometric fu nctions. Example 2. The dT o da sin gu lar manifold equation, u xy − u x u y ( e u t − 1)(1 − e − u t ) u tt = 0 , p ossesses the Lax p air S t = F ( S x /u x , u t ) , S y = F 2 ( S x /u x , u t ) u y ; here the fun ction F ( x 1 , x 2 ) is deﬁ ned as F ( x 1 , x 2 ) = − 1 λ ln  1 − λx 1 (1 − e − x 2 )  , λ = const , F 2 = ∂ F /∂ x 2 . Although th e constan t λ can b e eliminated b y a rescaling S → λS , one can consider the limit as λ → 0. This r esu lts in a linear Lax pair for the same equation: S t = (1 − e − u t ) S x u x , S y = e − u t S x u x u y . P arametric Lax pairs . In many cases it turns out to b e more con v enien t to work w ith parametric L ax pairs, S x = f ( p, u x , u y , u t ) , S y = g ( p, u x , u y , u t ) , S t = h ( p, u x , u y , u t ) , (32) here p is a parameter. Exp r essing p from the ﬁrst equation and sub stituting into the last t w o one gets a Lax pair in the form (3). Similar parametric Lax pairs app eared previously in the co nt ext 20 of the unive rsal Whitham hierarc h y [32], see also [39]. The condition for (32) to b e a Lax p air for Eq. (1) can b e d eriv ed as follo ws. Thinking of p as a fu nction of x, y , t , and calculating the compatibilit y cond itions S xy = S y x , S xt = S tx , S y t = S ty , one obtains thr ee relations wh ic h are linear in p x , p y and p t . It is easy to s ee that the rank of the matrix at the d eriv ativ es p x , p y , p t equals tw o, so th at one can obtain a single relation whic h do es not cont ain the deriv ativ es of p : ( h a g p − g a h p ) u xx + ( f b h p − h b f p ) u y y + ( g c f p − f c g p ) u tt + (( f a − g b ) h p + h b g p − h a f p ) u xy + (( h c − f a ) g p + g a f p − g c h p ) u xt + (( g b − h c ) f p + f c h p − f b g p ) u y t = 0; (33) this r elation m ust b e satisﬁed identica lly m o dulo Eq. (1). This requir emen t leads to the r elations h a g p − g a h p = kf 11 , f b h p − h b f p = kf 22 , g c f p − f c g p = kf 33 , ( f a − g b ) h p + h b g p − h a f p = 2 kf 12 , ( h c − f a ) g p + g a f p − g c h p = 2 kf 13 , ( g b − h c ) f p + f c h p − f b g p = 2 kf 23 , (34) where k = k ( p, a, b, c ) is the co eﬃcien t of p rop ortionalit y . Th e set of r elations (34) can b e represent ed in a compact form det   f p g p h p d f dg dh da db dc   = kf ij dp i dp j , recall that ( u x , u y , u t ) = ( a, b, c ) = ( p 1 , p 2 , p 3 ). W e p oint out th at, b y virtue of (34), the triple ( f p , g p , h p ) satisﬁes the disp ersion r elation: f 11 f 2 p + f 22 g 2 p + f 33 h 2 p + 2 f 12 f p g p + 2 f 13 f p h p + 2 f 23 g p h p = 0 . P arametric L ax pairs are particularly u seful wh en the equation under stu d y is symmetric under the interc hange of x , y , t : Example 3. Let us consider the equation a 1 u x u y t + a 2 u y u xt + a 3 u t u xy = 0 , where, in con trast to Ex. 1, the constan ts a 1 , a 2 , a 3 are arbitrary . Without any loss of generalit y w e will norm alize th em so that a 1 + a 2 + a 3 = 1. W e seek a Lax p air in parametric f orm S x = f ( p ) u x , S y = g ( p ) u y , S t = h ( p ) u t . The corresp ondin g E q. (33) is ( g − h ) f ′ u x u y t + ( h − f ) g ′ u y u xt + ( f − g ) h ′ u t u xy = 0 , here ′ = d/dp , so that one can set f ′ = a 1 k ( f − g )( f − h ) , g ′ = a 2 k ( g − f )( g − h ) , h ′ = a 3 k ( h − f )( h − g ) , (35) 21 where k = k ( p ) is a co eﬃcient of pr op ortionalit y . W e p oint out that the system (35) p ossesses a conserv ation la w ( g − h ) a 1 ( h − f ) a 2 ( f − g ) a 3 = const . In tro du cing the new indep end en t v ariable q = ( f − g ) / ( f − h ) (notice that we hav e a reparametriza- tion freedom p → ϕ ( p )), and u sing the identit y q ′ = k ( f − g )( g − h ) / ( f − h ), one can linearize the system (35), d f dq = a 1 1 − q ( f − h ) , dg dq = a 2 ( h − f ) , dh dq = a 3 q ( f − h ) , Noticing that f − h = (1 − q ) − a 1 q − a 3 , one can reduce these equatio ns to h yp ergeometric int egrals, d f dq = a 1 (1 − q ) − a 1 − 1 q − a 3 , dg dq = − a 2 (1 − q ) − a 1 q − a 3 , dh dq = a 3 (1 − q ) − a 1 q − a 3 − 1 . Explicitly , one arriv es at th e ‘reparametrized’ Lax pair S x u x = f ( q ) , S y u y = g ( q ) , S t u t = h ( q ) , (36) where f = F ( q ) , g = F ( q ) − (1 − q ) − a 1 q 1 − a 3 , h = F ( q ) − (1 − q ) − a 1 q − a 3 , and F ( q ) is a the hyp ergeometric function: F ( q ) = a 1 1 − a 3 q 1 − a 3 2 F 1 ( a 1 + 1 , 1 − a 3 ; 2 − a 3 ; q ) . In the symm etric case a 1 = a 2 = a 3 = 1 3 w e obtain a Lagrangian equation u x u y t + u y u xt + u t u xy = 0 , whic h corresp onds to the Lagrangian den sit y u x u y u t . Its parametric Lax pair was calculated in [20] in the form (here the parameter is d enoted b y z ), S x u x = ζ ( z ) , S y u y = ζ ( z ) + ℘ ′ ( z ) + λ 2 ℘ ( z ) , S t u t = ζ ( z ) + ℘ ′ ( z ) − λ 2 ℘ ( z ) , (37) where ℘ ( z ) is the W eierstrass ℘ -fun ction, ( ℘ ′ ) 2 = 4 ℘ 3 + λ 2 (notice th at g 2 = 0 , g 3 = − λ 2 ), and ζ ( z ) is the corresp ond ing zeta-function: ζ ′ = − ℘ . T he Lax pair (37) transform s into (36 ) via a substitution q = ( ℘ ′ ( z ) + λ ) / ( ℘ ′ ( z ) − λ ). Example 4. The equation u t ( u 3 x − u 3 y ) u xy + u y ( u 3 t − u 3 x ) u xt + u x ( u 3 y − u 3 t ) u y t = 0 p ossesses the p arametric Lax p air S x = G  u x p  , S y = G  u y p  , S t = G  u t p  , where th e fun ction G ( s ) satisﬁes the equations G ′ = 1 s 3 − 1 . Exp licitly , one has G ( s ) = 1 3  ln( s − 1) + ǫ ln( s − ǫ ) + ǫ 2 ln( s − ǫ 2 )  . 22 Example 5. The equation ℘ ′ ( u x ) − ℘ ′ ( u y ) ℘ ( u x ) ℘ ( u y ) u xy + ℘ ′ ( u t ) − ℘ ′ ( u x ) ℘ ( u x ) ℘ ( u t ) u xt + ℘ ′ ( u y ) − ℘ ′ ( u t ) ℘ ( u y ) ℘ ( u t ) u y t = 0 p ossesses the p arametric Lax p air S x = F ( p, u x ) , S y = F ( p, u y ) , S t = F ( p, u t ) , where th e fun ction F ( r , s ) is deﬁn ed by the equations F r = ℘ ( r ) ℘ ( s ) ℘ ′ ( r ) − ℘ ′ ( s ) , F s = ℘ 2 ( r ) ℘ ′ ( s ) − ℘ ′ ( r ) − 1 2 ζ ( r ) . Explicitly , one has F ( r, s ) = 1 6  ln σ ( r − s ) + ǫ ln σ ( ǫr − s ) + ǫ 2 ln σ ( ǫ 2 r − s )  , compare with Ex. 4 of Sect. 5. Similar p arametric Lax pairs ca n b e constructed for all in tegrable examples obtained in Sect. 4.4. The m ain resu lt of this Section is the f ollo wing Theorem 3 Any inte gr able e q u ation of the form (1) p ossesses a disp ersionless L ax p air. F ur- thermor e, the existenc e of a disp ersionless L ax p air is e quivalent to the existenc e of an inﬁnity of hydr o dynamic r e ductions and, thus, is ne c essary and suﬃcient for the i nte gr ability. Pro of: Setting S x = ξ , u x = a, u y = b, u t = c , and calculating the consistency condition for the Lax pair (3), one obtains S ty − S y t = ( F ξ G a − G ξ F a ) u xx + F b u y y − G c u tt + ( F a + F ξ G b − G ξ F b ) u xy − ( G a + G ξ F c − F ξ G c ) u xt + ( F c − G b ) u y t . Since this expression has to v anish m o dulo Eq. (1), on e arr iv es at the r elations F ξ G a − G ξ F a = kf 11 , F b = kf 22 , G c = − k f 33 , F a + F ξ G b − G ξ F b = 2 kf 12 , G a + G ξ F c − F ξ G c = − 2 kf 13 , F c − G b = 2 kf 23 , where k = k ( ξ , a, b, c ) is the co eﬃcien t of prop ortionalit y . The last ﬁ v e relations imply the expressions for th e deriv ativ es of F and G in the f orm F a = 2 kf 12 + G ξ k f 22 + F ξ ( k f 23 − p ) , G a = − 2 kf 13 − F ξ k f 33 − G ξ ( k f 23 + p ) , F b = kf 22 , G b = − k f 23 + p, (38) F c = kf 23 + p, G c = − k f 33 , where p = p ( ξ , a, b, c ) is y et another auxiliary fu nction. Su bstituting these expr essions into the ﬁrst relation, one can see that F ξ and G ξ ha v e to satisfy the d isp ersion r elation, f 11 + f 22 G 2 ξ + f 33 F 2 ξ + 2 f 12 G ξ + 2 f 13 F ξ + 2 f 23 F ξ G ξ = 0 . (39) T o close the s ystem (38) – (39) one p ro ceeds as follo ws. Calculating th e consistency conditions for Eqs. (38 ), F ab = F ba , G ab = G ba , etc, six conditions altogether, and d iﬀeren tiating the disp ers ion relation (39) by a, b, c and ξ , one obtai ns ten relations whic h can b e solve d for F ξ ξ , G ξ ξ and the ﬁrs t order deriv ative s of k and p . The resulting s ystem is in in v olution if and only if the in tegrabilit y conditions of Sect. 2 are satisﬁed. Th is ﬁnishes the pro of of T heorem 3. 23 7 Diﬀeren tial-geometric asp ects of the in tegrabilit y conditions As explained in Sect. 3, the diﬀerentia l-geometric p icture b ehin d equations of th e form (14) is the pro jectiv e space P n with co ordinates p 1 , ..., p n supplied w ith the conformal structur e f ij dp i dp j . The equiv alence group S L ( n + 1 , R ) acts b y p ro jectiv e transf ormations of P n . Two equations are equiv alent if and only if conformal classes of the corresp on d ing metrics are pro jectiv ely equiv alen t. Let us consider Lagrangian equations of the form (14) whic h arise as the Euler-Lagrange equations from the fun ctionals R g ( p ) d x w here the densit y g ( p 1 , ..., p n ) dep ends on the ﬁrst order deriv ative s p i = u x i only . In this case the coeﬃcient matrix f ij is the Hessian matrix of g . Our ﬁrst remark, which is true in any d imension, is that the class of Lagrangian systems is in v arian t under the action of the equiv alence group deﬁn ed by Eqs. (16), (18). One can sho w that the extension of the p ro jectiv e action (16) to the Lagrangian d ensit y g is given by the form ula ˜ g = g 1 + p C − 1 b . (40) The pro jectiv e inv ariance of the class of Lagrangian systems can b e seen as follo ws. L et us consider th e Lagrangian densit y g ( p ) as the equ ation of a h yp ersurface in P n +1 deﬁned as p n +1 = g ( p ). The s econd fun damen tal form of this hyp ersurf ace coincides with the conformal class of the second diﬀerenti al d 2 g . T he transformation (16 ), (40 ) is a p r o jectiv e tran s formation in P n +1 . Thus, the fact th at d 2 g transforms into d 2 ˜ g (up to a conformal factor) is nothin g b ut the well- kno wn p ro jectiv e inv ariance of the second fu ndamenta l form . A geometric c haracterizati on of linearizable equations (14) is provided in S ect. 7.1 (Th eo- rem 4). T o b e p recise, we will b e interested in those equations whic h can b e linearized b y a transformation fr om the equiv alence group . A simple tensorial c haracterizatio n of linearizable and Lagrangian equations is giv en in Sect. 7.2. The answe r is form ulated in terms of the tens or a ij k and the p ro jectiv ely ﬂ at connection ∇ with C h ristoﬀel’s symbols Γ i j k = s j δ i k + s k δ i j whic h are naturally asso cated with the conformal structure f ij dp i dp j . In v ariant diﬀerenti al-geometric formulat ion of the int egrabilit y conditions (13) derive d in Sect. 2 is p ro vided in S ect. 7.3. T h is inv olve s th e tensor a ij k and its co v arian t deriv ativ e with resp ect to ∇ . Finally , a simp le diﬀeren tial-geo metric charac terization of conformal stru ctur es corresp ond- ing to inte grable equations is p r op osed in Sect. 7.4. 7.1 Linearizab le equations and quadratic line complexes Before formulati ng the main result, let us summarize the pr op erties of linear (linearizable) equations. Example 1. Consider the 3-dimensional linear w a v e equation, u tt = u xx + u y y ; notice ﬁrst that it is Lagrangian with the qu adratic Lagrangian density g = u 2 x + u 2 y − u 2 t . Th e asso ciated conformal s tr ucture in P 3 corresp onds to the standard Loren tzian metric ( dp 1 ) 2 + ( dp 2 ) 2 − ( dp 3 ) 2 ; here p 1 = u x , p 2 = u y , p 3 = u t . This conform al structure p ossesses a 3- parameter family of null lines deﬁn ed by th e equ ations p 1 = αp 3 + β , p 2 = γ p 3 + δ , (41) 24 where the constan ts α, β , γ , δ satisfy a single quadratic constrain t α 2 + γ 2 = 1. Th e last prop ert y can b e reform ulated in a p ro jectiv ely in v ariant w a y as follo ws. Recall that a 3-parameter family of lines in P 3 is called a line c omplex . A complex is said to b e quadr atic if it is deﬁ n ed b y a single quadr atic relation among the Pluck er co ordinates in the space of lines (whic h is identiﬁed with th e Pluck er quadric in P 5 ). In the parametrization (41), the Pluc k er co ordinates are (1 : α : β : γ : δ : αδ − γ β ) . Fixing a p oint in P 3 with co ordinates p 1 0 , p 2 0 , p 3 0 , the lines of th e complex p assing throu gh this p oint generate a qu ad r atic cone with the equation ( p 1 − p 1 0 ) 2 + ( p 2 − p 2 0 ) 2 = ( p 3 − p 3 0 ) 2 ; th ese cones are nothing b ut the null cones of the corresp on d ing conformal structur e. Introd ucing in P 3 homogeneous co ord inates q 0 : q 1 : q 2 : q 3 via p i = q i /q 0 , one can s ee that the inte rsection of null cones with the plane at inﬁnit y (deﬁned by th e equation q 0 = 0) is the conic ( q 1 ) 2 + ( q 2 ) 2 = ( q 3 ) 2 . Th us, all q u adratic cones of our complex p ass through one and the same plane conic (complexes of this t yp e h a v e the S egre symb ol [(222)], see [28, 4]). Summarizing, w e see that (a) the linear wa v e equation is Lagrangian; (b) the corresp onding conformal structure p ossesses a three-parameter family of n u ll lines whic h form a quadr atic complex with the S egre symbol [(222)] (equiv alen tly , quadratic cones of the complex p ass thr ough one and the s ame p lane conic). Reform ulated in these terms, b oth prop erties are manifestly pro jectiv ely inv arian t, and h old for arbitrary equ ations related to the linear wa ve equation via the action of the equiv alence group. Example 2. Let us consider the equation (1 − u 2 x − u 2 y ) u tt − u 2 t ( u xx + u y y ) + 2 u t ( u x u xt + u y u y t ) = 0 , whic h is L agrangian with the Lagrangian d ensit y g = u 2 x + u 2 y − 1 u t . T he corresp onding conformal structure, (1 − ( p 1 ) 2 − ( p 2 ) 2 )( dp 3 ) 3 − ( p 3 ) 2 (( dp 1 ) 2 + ( dp 2 ) 2 ) + 2 p 3 ( p 1 dp 1 dp 3 + p 2 dp 2 dp 3 ) , p ossesses a 3-parameter family of n ull lines (41) sp eciﬁed b y a sin gle qu adratic r elation β 2 + δ 2 = 1. This deﬁ n es a quadratic complex whose null cones pass thr ou gh one and the same planar conic deﬁned by the equation ( q 1 ) 2 + ( q 2 ) 2 = ( q 0 ) 2 in the plane q 3 = 0 (recall that q i are homogeneous co ordinates in P 3 ). T h e equation linearizes (to the w a ve equation from Ex. 1) under the transf orm ation ˜ u = − t, ˜ t = − u, ˜ y = y , ˜ x = x, whic h generates a pr o jectiv e transformation of the deriv ative s, ˜ u ˜ t = 1 u t , ˜ u ˜ y = u y u t , ˜ u ˜ x = u x u t . This extend s to the transformation of the Lagrangian densities as ˜ g = g /u t . One can v erify that the quadratic Lagrangian densit y g of the linear wa ve equation transf orms to the densit y ˜ g of the linearizable equation from Ex. 2. Geometrically , this transformation is nothing b ut a pro jectiv e transform ation which sends the p lane q 3 = 0 to the plane at in ﬁnit y . Our observ ations are summarized in the follo wing theorem w hic h, in f act, holds in any dimension. Theorem 4 The fol lowing c onditions ar e e quivalent: (1) Eq . (14) is line arizable by a tr ansformation fr om the e quivalenc e gr oup. 25 (2) Eq. (14) is L agr angian with the L agr angian density g = Q ( p ) l ( p ) wher e Q and l ar e arbitr ary quadr atic and line ar forms in p 1 , ..., p n , r esp e ctively (not ne c essarily homo gene ous). (3) The c onformal structur e f ij dp i dp j p ossesses a c omplex (that is, a (2 n − 3) -p ar ameter family) of nul l lines whose qu adr atic c ones p ass thr ough a stationary hyp erplane quadric. F or n = 3 these c onditions ar e e quiavalent to the r e quir ement that the c omplex has Se gr e symb ol [(222)]. Pro of: The equiv alence of (1) and (2) can b e s een as follo w s. S upp ose that Eq. (14) is linearizable. Then the corresp on d ing conform al structure f ij dp i dp j is transformable to a constan t co eﬃcien t form. Sin ce an y constan t co eﬃcient linear system is Lagrangia n with a quadratic Lagrangian densit y g , and the class of Lagrangian systems is p ro jectiv ely in v arian t, an y linearizable equation is necessarily Lagrangian. Applying th e pro jectiv e transformation (16), (40) to a quadratic Lagrangian densit y , one obtains a density of the form g = Q ( p ) l ( p ) where Q and l are quadratic and linear expr essions in p 1 , ..., p n , r esp ectiv ely . C on v ersely , giv en a Lagrangian d ensit y of th e form g = Q ( p ) l ( p ) , and app lying an y pro jectiv e tr ansformation whic h has the linear form l ( p ) in the d enominator, one obtains a pur ely quadr atic Lagrangian densit y wh ich giv es rise to a lin ear equation. This establishes the equiv alence of (1) and (2). The implication (1) = ⇒ (3) is straight forwa rd: an y constant co eﬃcient conformal s tr uc- ture p ossesses a complex of null lines w hose quadratic cones pass through a s tationary quadr ic b elonging to the h yp erplane at inﬁnit y . Conv ersely , consider an equat ion whose conformal struc- ture p ossesses a quadratic complex of null lin es such that all null cones pass throu gh a stationary quadric b elonging to a stationary hyp erplane H . Applying a pro jectiv e transform ation whic h sends H to a hyperp lane at in ﬁ nit y , w e obtain a linear equation with constan t coeﬃcients. In the case n = 3 one can also r efer to the Prop osition 4.3. of [4] which implies that, up to pro jectiv e equiv alence, there exists a u nique quadratic complex with the Serge symb ol [(222)]. The constraints on the complex are crucial for the linearizabilit y . Example 3. Let us consider the d isp ersionless Hirota equation a 1 u x u y t + a 2 u y u xt + a 3 u t u xy = 0 , a 1 + a 2 + a 3 = 0; w e p oin t out that the equatio n is integ rable for an y v alues of constan ts, not necessarily satisfying this r elation. The corresp on d ing conformal structure a 1 p 1 dp 2 dp 3 + a 2 p 2 dp 1 dp 3 + a 3 p 3 dp 1 dp 2 p ossesses a three-parameter family of null lines which form a qu adratic complex d eﬁned by the equation a 1 β γ + a 2 αδ = 0. This complex is not of the Segre t yp e [(222)], ther efore, the equation is not linearizable. In fact, it is easy to see that it is not Lagrangian. Example 4. Let us consider the equation u xx + ( u 2 y − 1) u tt − 2 u y u t u y t + u 2 t u y y = 0; one can show th at it is not integ rable, and n ot Lagrangian. The corresp onding conformal structure ( dp 1 ) 2 + (( p 2 ) 2 − 1)( dp 3 ) 2 − 2 p 2 p 3 dp 2 dp 3 + ( p 3 ) 2 ( dp 2 ) 2 p ossesses a three-parameter family of null lines which form a qu adratic complex d eﬁned by the equation α 2 + δ 2 = 1, which is again n ot of the Segre type [(222)]. Remark. The condition for a conformal structur e f ij dp i dp j in P n to p ossess a complex of n ull lines is equiv alen t to a simp le d iﬀeren tial-geo metric constraint ∂ ( k f ij ) = ϕ ( k f ij ) ; (42) 26 here ∂ k = ∂ p k , ϕ k is a co v ector, and br ac kets denote a complete symmetrization in i, j, k . Con tracting (42 ) with f ij w e obtain ϕ k = 1 n + 2 f pq ( ∂ k f pq + 2 ∂ p f q k ) . The condition (42) charact erizes conformal structures coming from quadratic line complexes, see e.g. [3, 42] and references therein. Thus, the identi t y (42) is n ecessary (although not suﬃcien t) for the linearizabilit y . An inv arian t c haracterization of linearizable equ ations is p r o vided in Sect. 7.2. b elo w. 7.2 T ensorial c haracterization of linearizable and Lagrangian equations The r esults of this section are v alid in any dimension, and pro vid e a simple diﬀerential -geometric criterion of the linearizabilit y for equations of the form (14). Lemma . An e quation of the f orm (14) is line arizable by a tr ansformation fr om the e q u ivalenc e gr oup S L ( n + 1 , R ) iﬀ ther e exists a ﬂat c onne ction ∇ with Christoﬀel symb ols Γ i j k = s j δ i k + s k δ i j such that ∇ k f ij = c k f ij ; (43) (c onne ctions satisfying Eq. (43 ) ar e known as Weyl c onne ctions). Pro of: The necessit y is straigh tforw ard: given a linear equation w ith constant co eﬃcients f ij , we im- mediately arriv e at (43) where ∇ is a ﬂat connection w ith zero Christoﬀel symb ols, ∇ k = ∂ k = ∂ /∂ p k , and c k = 0. App lying a transformation from the equiv alence group (whic h acts pr o jec- tiv ely on the sp ace P n with co ord inates p 1 , ..., p n ) to a ﬂat connection ∇ , w e will obtain a ﬂat connection with nonzero Christoﬀel sym b ols of the form Γ i j k = s j δ i k + s k δ i j , whic h s till satisﬁes Eqs. (43). Moreo ver, the condition (43) is manifestly in v arian t under rescalings f ij → τ f ij (under su c h r escalings, the co v ector c k transforms to c k + ∇ k ln τ ). Con v ersely , su p p ose a connection ∇ has C h ristoﬀel symb ols of the f orm Γ i j k = s j δ i k + s k δ i j (connections of this form are known as pro jectiv ely ﬂat: their geo desics are str aight lines). If, in addition, ∇ is ﬂat (has zero cur v ature tensor), there exists a pr oje ctive trans formation br in ging Christoﬀel symb ols to zero. In the new co ordinates Eq. (43) w ill tak e the form ∂ k f ij = c k f ij , whic h implies that the co eﬃcien t m atrix f ij is prop ortional to a constan t matrix. Th is ﬁ nishes the pr o of. Relations (43) lead to explicit tensorial constraint s f or f ij as follo ws. T aking in to accoun t that Γ i j k = s j δ i k + s k δ i j one can rewr ite (43) as ∂ k f ij = ( c k + 2 s k ) f ij + s i f k j + s j f k i . (44) Con tacting Eqs. (44) with the inv erse matrix f ij one arriv es at th e relations f ij ∂ k f ij = nc k + 2( n + 1) s k , f ij ∂ j f ik = c k + ( n + 3) s k , with a d ouble summation o v er i and j . This implies s k = f ij ( n +2)(1 − n ) ( ∂ k f ij − n∂ j f ik ) , c k = f ij ( n +2)( n − 1) (( n + 3) ∂ k f ij − 2( n + 1) ∂ j f ik ) . (45) 27 Giv en an arbitrary conformal stru cture f ij dp i dp j in P n , let u s introd uce the tensor a ij k = ∂ k f ij − ( c k + 2 s k ) f ij − s i f k j − s j f k i , here s k , c k are the same as in Eqs. (45). O n e can r eadily verify the ap olarit y r elations f ij a ij k = 0 , f ij a ik j = 0. Thus, w e can formulate the follo wing Prop osition 1. E q uation (14) is line arizable by a tr ansformation fr om the e quivalenc e gr oup S L ( n + 1 , R ) i ﬀ the c orr esp onding c onforma l structur e satisﬁes the fol lowing two pr op erties: (1) the tensor a ij k vanishes; (2) the c onne ction Γ i j k = s j δ i k + s k δ i j is ﬂat; a simple c alculation shows that this c ondition is e quivalent to ∂ j s i − s i s j = 0 . Remark 1. In a somewhat diﬀeren t form, the tensor a ij k app eared previously in [42] in the study of manifolds of quadratic cones in P n . I t w as pro v ed that the v anishing of a ij k alone implies the existence of a stationary hyp erquadric Q n − 1 ⊂ P n suc h that all cones of the family are tangen tial to Q n − 1 . C learly , all suc h conform al stru ctures are pr o jectiv ely equiv alent , and can b e br ough t to a canonical form [( p , p ) − 1]( d p , d p ) − ( p , d p ) 2 = 0 , where p = ( p 1 , ..., p n ) are co ord inates in P n , and ( , ) is the s tand ard scalar pro d uct. Th e null cones of this conformal structure are tangen tial to a unit sphere cen tered at the origin. The corresp ondin g second order equation tak es th e form [( ∇ u ) 2 − 1] △ u − ( ∇ u ) H ( ∇ u ) t = 0 , where ∇ u = ( u x 1 , ..., u x n ) is the gradient of u , △ is the Laplacian, an d H is the Hessian matrix of u . In the th ree-dimensional case w e arrive at the equation ( u 2 y + u 2 t − 1) u xx + ( u 2 x + u 2 t − 1) u y y + ( u 2 x + u 2 y − 1) u tt − 2( u x u y u xy + u x u t u xt + u y u t u y t ) = 0; notice that it is Lagrangian: the corresp onding Lagrangian den s it y q 1 − u 2 x − u 2 y − u 2 t go verns minimal hypersu rfaces z = u ( x, y , t ) in the Lorentzia n space w ith th e metric dx 2 + dy 2 + dt 2 − dz 2 . W e h a v e v eriﬁed that this equation do es not satisfy the in tegrabilit y conditions of Sect. 2. Remark 2. An other result of [42] (also formulate d in diﬀerent terms ) s tates that, imp osed si- m ultaneously , conditions (1) and (2) imply the existence of a stationary hyp erplane H in P n with a s tationary quadric Q n − 2 ⊂ H su c h that all cones of th e f amily p ass through Q n − 2 . Th is p r o- vides an alternativ e pro j ectiv ely-inv ariant c haracterizat ion of conformal sr uctures co rresp onding to linearizable equations: th e linearizing transformation is any pro jectiv e transformation whic h sends H to the hyp erplane at inﬁ nit y (see Sect. 7.2). The tensor a ij k pro vides a s im p le c haracterization of Lagrangian equations: Prop osition 2. Equ ation (14) is L agr angian i ﬀ the c orr esp onding c onformal structur e satisﬁes the fol lowing two pr op erties: (1) the tensor a ij k is total ly symmetr ic; in fact, sinc e a ij k is manifestly symmetric i n the ﬁrst two indic e s, it is suﬃcient to r e q u ir e a ij k = a ik j , (2) the c ove ctor s i is a g r adient: ∂ j s i = ∂ i s j ; these co nditions are obta ined b y weak ening the corresp onding co nditions of Prop osition 1 (recall that any linearizable equation is automatically Lagrangian). 28 Pro of: T o sh o w that a giv en equation is Lagrangian, one has to ﬁnd an in tegrating factor τ su c h that the matrix τ f ij is the Hessian matrix of a function, equiv alentl y , ∂ k ( τ f ij ) = ∂ j ( τ f ik ), whic h giv es ∂ k τ τ f ij + ∂ k f ij = ∂ j τ τ f ik + ∂ j f ik . Con tracting this expression with f ij one gets ∂ k τ τ = f ij n − 1 ( ∂ j f ik − ∂ k f ij ) . (46) Substituting this bac k in to th e previous equation on e obtains the relation ∂ k f ij + f ij f pq n − 1 ( ∂ q f pk − ∂ k f pq ) = ∂ j f ik + f ik f pq n − 1 ( ∂ q f pj − ∂ j f pq ) , whic h is iden tical with a ij k = a ik j . Finally , the right hand side of (46) m ust b e a gradient. S ince the expression f ij ∂ k f ij is automaticall y a gradien t by virtue of the id en tit y tr F − 1 ∂ k F = ∂ k ln det F , one has to r equire that f ij ∂ j f ik is a gradient. Th is, how eve r, is equiv alen t to the requ iremen t that th e co v ector s i m ust b e a gradient. This ﬁnish es the p ro of. Finally , we ha v e the follo w ing Prop osition 3. Conformal structur e f ij dp i dp j in P n p ossesses a q u adr atic c omplex of nu l l lines iﬀ the symmetrize d tensor a ij k vanishes: a ( ij k ) = 0 . Indeed, the condition a ( ij k ) = 0 is ident ical to (42). 7.3 T ensorial form ulation of the integrabilit y conditions Let us b egin with a general diﬀeren tial-geometric digression. Given a metric f ij and a connection ˆ ∇ with C h ristoﬀel symbols ˆ Γ i j k on a n -dimensional man if old, let us in tro duce th e follo wing ob jects: — co v ectors s k and c k : s k = f ij ( n +2)(1 − n )  ˆ ∇ k f ij − n ˆ ∇ j f ik  , c k = f ij ( n +2)( n − 1)  ( n + 3) ˆ ∇ k f ij − 2( n + 1) ˆ ∇ j f ik  ; — tensor a ij k : a ij k = ˆ ∇ k f ij − ( c k + 2 s k ) f ij − s i f k j − s j f k i ; — tensor a ij k l : a ij k l = ˆ ∇ l a ij k − ( c l + 3 s l ) a ij k − s i a lj k − s j a ilk − s k a ij l . The imp ortance of these ob j ects is explained by th eir tr ansformation pr op erties: supp ose th at the metric f ij and th e conn ection ˆ ∇ are allo w ed to v ary within their conformal and pro j ectiv e classes, resp ectiv ely , that is, f ij → ϕf ij , ˆ Γ i j k → ˆ Γ i j k + ψ k δ i j + ψ j δ i k . 29 One can readily ve rify the transformation pr op erties s k → s k − ψ k , c k → c k + ∂ k ϕ ϕ , a ij k → ϕa ij k , a ij k l → ϕa ij k l , whic h, in p articular, imp ly that the n ew Ch ristoﬀel symbols, Γ i j k = ˆ Γ i j k + s k δ i j + s j δ i k , giv e rise to a w ell-deﬁned aﬃne connection ∇ wh ic h d ep ends neither on the c hoice of ˆ Γ i j k in its pro jectiv e class, nor on th e conformal f actor ϕ . The expressions for a ij k nd a ij k l compactify to a ij k = ∇ k f ij − c k f ij , a ij k l = ∇ l a ij k − c l a ij k . In our con text, n = 3, f ij is a conformal structur e in P 3 , and the pr o jectiv e structure is asso ciated with the p ro jectiv e class of a ﬂat connection: ˆ Γ i j k = 0 (indeed, only pr o jectiv e transformations preserve the pr o jectiv e class of a ﬂat connection). Thus, ˆ ∇ k = ∂ /∂ p k , and th e connection ∇ is giv en by Γ i j k = s k δ i j + s j δ i k ; notice that, although ∇ is manifestly pr o jectiv ely ﬂat, is do es n ot n eed to b e ﬂat (that is, hav e zero curv atute tensor) in general. The tensors a ij k , a ij k l and the aﬃne connection ∇ constitute a complete set of p ro jectiv e in v ariant s of the conformal stru cture f ij dp i dp j . F o r n = 3 the in tegrabilit y conditions (13 ) can b e formulated as follo ws (w e b egin with the Lagrangian case, whic h is computationally simp ler): In tegrability conditions in the Lagrangian case In the Lagrangia n case the tensor a ij k is totally symmetric, and the in tegrabilit y conditions tak e the form ∂ j s i − s i s j = − 1 20 a tmµ a r sν f iq f j p f µν ε ptr ε q ms − 3 20 a ipr a j q s f pq f r s , (47) and a ij k l = 9 10 S y m a ij p a k lq f pq − S y m  9 20 f sl a ipq a j tm f r k + 3 2 a ltm a k pq f r i f sj − 3 20 a tmµ a r sν f iq f j p f k l f µν  ε ptr ε q ms , (48) resp ectiv ely . Here S y m d enotes a complete symmetrization in i, j, k , S y m T ij k = 1 3! X σ ∈ S 3 T σ ( i ) σ ( j ) σ ( k ) , and ǫ ij k is the totally an tisymmetric tensor du al to the vol ume form of the metric f ij , that is, ǫ 123 = 1 / √ F , ǫ 213 = − 1 / √ F , etc, F = det f ij . T his pro vides yet another form of the inte grabilit y conditions in the Lagrangian case, compare w ith [20 ]. Reca ll th at linearizable equations are c haracterized by the relations a ij k = 0, ∂ j s i − s i s j = 0 (Prop. 2 of Sect. 7.2), wh ich clearly annihilate b oth of the conditions (47), (48). T h is is in agreement with th e ob vious fact th at any linearizable equation is automatically in tegrable. In tegrability conditions in the general case In the general case the tensor a ij k is no longer symmetric (only in the ﬁrs t t wo ind ices), and th e in tegrabilit y cond itions b ecome considerably more complicated. T h us, the analogue of Eq. (47) tak es the form ∂ j s i − s i s j = − 1 20 (2 a µtm a ν r s + 2 a µmt a ν s r − 3 a tmµ a r sν ) f iq f j p f µν ε ptr ε q ms − 1 20 (6 a ipr a j q s − 5 a pr i a q sj + a ipr a q sj + a pr i a j q s ) f pq f r s ; (49) 30 one can sho w that the right h and side of (49) is symmetric w ith resp ect to i and j , so that ∂ j s i = ∂ i s j . This means that, for in tegrable equations, the co v ector s i m ust b e a gradien t. In this case the left hand sid e of Eq. (49) can b e represen ted in the form ∂ j s i − s i s j = 1 n − 1 R ij where R ij is the Ricci tensor on ∇ . The analogue of Eq. (48) tak es the form a ij k l = − 1 20 S y m ((8 a k lq + 44 a q k l − 70 a q lk ) a ij p + (64 a k lq − 82 a q k l + 30 a q lk ) a pij − 12 a pk j a q il ) f pq − 1 20 S y m  (8 a iq p f r k − 42 a pq k f r i ) f sj a mlt + (3 a tmµ a r sν − 4 a µmt a ν r s ) f ip f j q f k l f µν + (64 a k q p f sj − 12 a k pq f sj + 10 a pq j f sk − 20 a j q p f sk ) a ltm f r i + (102 a iq p a tmj − 51 a pq i a tmj − 48 a iq p a j mt ) f r k f sl + (50) (28 a k q p a imt − 32 a k pq a imt + 10 a pq k a tmi ) f r j f sl + (42 a pq i f r k f sj − 40 a ipq f r k f sj + 20 a j q p f r i f sk ) a tml + (32 a µtm a ν r s − 34 a µtm a r sν ) f k p f j q f il f µν  ε ptr ε q ms , here S y m denotes symmetrization with resp ect to i and j . Both conditions (49 ), (50) simp lify to (47), (48) u nder the Lagrangian assum p tion. These conditions pr o vide a straight forward computer test of the int egrabilit y for equations from the class (1). 7.4 In t egrable equations and conformal structures p ossessing conjugate n ull co ordinate systems Our ﬁ rst result is the follo wing Theorem 5 The c onformal structur e f ij dp i dp j c orr esp onding to any inte gr able thr e e- dimensional e quation (1) i s c onformal ly ﬂat. The pro of is a str aigh tforw ard calculation of the corresp onding Cotton tensor, b ased on th e in tegrabilit y conditions deriv ed in Sect. 2. Recall that for th r ee-dimensional Lagrangian systems this result was established earlier in [20]. W e emp hasize that the transformation wh ic h brings the metric to a constant co eﬃcien t form is not necessarily pro j ectiv e (it do es b ecome pro j ectiv e for linearizable sy s tems only). Thus, the theory of in tegrable equations of the f orm (1) has t w o ‘ﬂat’ counterparts: the ﬁrst one is a ﬂ at p ro jectiv e structure pr o vided by the pro jectiv e space P n with coordin ates p 1 , ..., p n , and the standard pr o jectiv e action of S L ( n + 1 , R ). Th e sec ond is the ﬂat conformal stru cture f ij dp i dp j . Although, view ed separately , b oth structures are trivial, this is n o longer true when they are imp osed simultaneously: their ‘ﬂ at co ordinate systems’ do not coincide in general. Our next goal is to provide a d iﬀerential- geometric c haracterizati on of h ydro dynamic reduc- tions. Although our discussion will b e restricted to the dim en sion three, all conclusions hold in an y d im en sion. W e will follo w the notatio n of Sect. 2. Let p = ( p 1 , p 2 , p 3 ) = ( a, b, c ) b e functions of the Riemann in v arian ts R i (for our p urp oses it will b e su ﬃcien t to consider one-, t w o- and three-comp onent reductions only). By v ir tue of (53), the deriv ativ e of p with resp ect to R i is giv en b y ∂ i p = (1 , µ i , λ i ) ∂ i a. The disp ersion relation (54) implies that ∂ i p a null v ector of the conformal structure f ij dp i dp j . Th us, the Riemann in v ariant s R i pro vide a net of null curves on the corresp onding submanifold p ( R ). F u rthermore, th e r elation ∂ i ∂ j p ∈ span { ∂ i p , ∂ j p } , 31 whic h r eadily follo w fr om (7), implies that th is net is conjugate. Th us, we ha v e the follo w ing geometric p icture: One-comp onen t reductions corresp ond to null curv es of the asso ciated conform al str ucture. Tw o-comp onen t reductions are in one-to-one corresp ondence with su rfaces w hic h carry a conjugate net of n ull curves. Notice th at w e are in the realm of t w o diﬀerent geometries, namely , conformal geometry (resp onsible f or the pr op ert y of b eing null), and pro jectiv e ge ometry (resp onsible f or the pr op ert y of b eing conju gate). Three-comp onen t reductions corresp ond to th ree-conjugate null co ord inate systems in P 3 . Since th e exist ence of ‘su ﬃcien tly man y’ three-comp onent r eductions is a necessary and su ﬃcien t condition f or the integ rabilit y , the problem of the classiﬁcation of three-dimensional integ rable equations of the form (1) can b e reformulate d geometrically as follo w s: classify c onformal struc- tur es in the pr oje ctive sp ac e P 3 which p ossess inﬁnitely many thr e e-c onjugate nul l c o or dinate systems p ar ametrize d by thr e e arbitr ary functions of one variable . It is a tru ly remark able fact that the m o duli sp ace of su ch structur es is only 20-dimensional! 8 App endix: pro of of Theorem 1 Here we provide f u rther details of the pro of w hic h wa s only sk etc h ed in Sect. 2. O ur starting p oint is the quasilinear repr esen tation of Eq. (1), a y = b x , a t = c x , b t = c y , f 11 a x + f 22 b y + f 33 c t + 2 f 12 a y + 2 f 13 a t + 2 f 23 b t = 0 . (51) F ollo win g the metho d of h ydro dynamic r eductions, we seek multi- phase solutions in the form a = a ( R 1 , . . . , R N ) , b = b ( R 1 , . . . , R N ) , c = c ( R 1 , . . . , R N ) , (52) where the phases R 1 ( x, y , t ), . . . , R N ( x, y , t ) are arbitr ary solutions of Eqs. (6). S ubstituting the ansatz (52) int o (51) one obtains the equations ∂ i b = µ i ∂ i a, ∂ i c = λ i ∂ i a, (53) along with the disp ersion r elation D ( λ i , µ i ) = f 11 + f 22 ( µ i ) 2 + f 33 ( λ i ) 2 + 2 f 12 µ i + 2 f 13 λ i + 2 f 23 µ i λ i = 0 . (54) Hereafter, w e assume the conic (54) to b e irreducible. This condition is equ iv alen t to the non - v anishing of th e d eterminan t of the co eﬃcient matrix F =   f 11 f 12 f 13 f 12 f 22 f 23 f 13 f 23 f 33   , det F 6 = 0 . The consistency cond itions f or Eqs. (53) imply ∂ i ∂ j a = ∂ j λ i λ j − λ i ∂ i a + ∂ i λ j λ i − λ j ∂ j a. (55) Diﬀeren tiating the disp ersion relation (54) with resp ect to R j , j 6 = i, and k eeping in mind Eqs. (53) and (7), one obtains explicit expressions for ∂ j λ i and ∂ j µ i in the f orm ∂ j λ i = ( λ i − λ j ) B ij ∂ j a, ∂ j µ i = ( µ i − µ j ) B ij ∂ j a, (56) 32 where B ij are r ational expressions in λ i , λ j , µ i , µ j whose co eﬃcien ts dep end on f ij ( a, b, c ) and ﬁrst order deriv ativ es thereof. Exp licitly , one has B ij = N ij D ij = 1 2 N ij f 11 + f 22 µ i µ j + f 33 λ i λ j + f 12 ( µ i + µ j ) + f 13 ( λ i + λ j ) + f 23 ( µ i λ j + µ j λ i ) ; notice that, modu lo the disp ersion r elation (54), the denominator D ij equals 4 D  λ i + λ j 2 , µ i + µ j 2  . The numerator N ij is a p olynomial exp ression of the form N ij = f 33 , 3 λ i 2 λ j + f 33 , 2 µ j λ i 2 + f 33 , 1 λ i 2 + 2 f 23 , 3 µ i λ i λ j + 2 f 23 , 2 µ i µ j λ i + 2 f 23 , 1 µ i λ i + f 22 , 3 µ i 2 λ j + f 22 , 2 µ i 2 µ j + f 22 , 1 µ i 2 + 2 f 13 , 3 λ i λ j + 2 f 13 , 2 µ j λ i + 2 f 13 , 1 λ i +2 f 12 , 3 µ i λ j + 2 f 12 , 2 µ i µ j + 2 f 12 , 1 µ i + f 11 , 3 λ j + f 11 , 2 µ j + f 11 , 1 ; w e adopt the con v en tion that v ariables a, b, c corresp ond to indices 1 , 2 , 3: th us, f 11 , 1 = f 11 ,a , etc. T aking into accoun t Eqs. (56), E qs. (55) assum e the form ∂ i ∂ j a = − ( B ij + B j i ) ∂ i a∂ j a. (57) The compatibilit y conditions ∂ k ∂ j λ i = ∂ j ∂ k λ i , ∂ k ∂ j µ i = ∂ j ∂ k µ i and ∂ k ∂ j ∂ i a = ∂ j ∂ k ∂ i a are equiv alen t to the equations ∂ k B ij = ( B ij B k j + B ij B ik − B k j B ik ) ∂ k a, (58) whic h m ust b e satisﬁed iden tically b y virtue of Eqs. (53), (54), (56) and (57). In ord er to obtain equations with ‘simplest p ossible’ coeﬃcient s at th e second ord er deriv ative s of f ij ( a, b, c ) we rewrite Eqs. (58) as ∂ k N ij = N ij  1 D ij ∂ k D ij + B k j ∂ k a + B ik ∂ k a  − D ij B k j B ik ∂ k a. (59) The second order d eriv ativ es of f ij ( a, b, c ) are presen t only in th e l.h.s. term ∂ k N ij . F ur th er reduction of the complexit y of th e expr ession in th e r.h.s. is ac hiev ed b y representing 1 /D ij in the form 1 D ij = U ij = [2( λ i f 23 + f 12 )( λ j f 23 + f 12 ) − f 22 ( λ i λ j f 33 + ( λ i + λ j ) f 13 + f 11 ) + f 22 ( λ j f 23 + f 12 ) µ i + f 22 ( λ i f 23 + f 12 ) µ j + f 2 22 µ i µ j ] / (2( λ i − λ j ) 2 det F ) , whic h holds identi cally mo d ulo the disp ersion relation (54), and a subsequent substitution B st = N st /D st = N st U st . T he denominators of the r .h.s. terms in Eqs. (59) cancel ou t, p ro ducing a p olynomial in λ i , λ j , λ k , µ i , µ j , µ k with co eﬃcients dep end ing on th e fun ctions f ij ( a, b, c ) and their d eriv ativ es up to second order. This wa s the most essent ial tec hnical p art of the calculation: the original expression (59) has more than 1.000.0 00 terms with diﬀeren t denominators; after prop erly organized cancella tions it redu ces to a p olynomial expression with less than 4500 terms. Using the disp ers ion r elation (54) and assum in g, say , f 22 6 = 0 (this can alw a y s b e ac h iev ed b y a linear change of the indep endent v ariables x, y , t ), we s implify this p olynomial b y excluding the p o w ers ( µ i ) s , ( µ j ) s , ( µ k ) s , s ≥ 2, arr ivin g at a p olynomial of degree one in eac h of µ i , µ j , µ k , and degree t w o in λ ’s. Equating similar co eﬃcient s of these p olynomials in b oth sides of Eqs. (59), w e arrive at a set of 45 equations for the deriv ativ es of the co eﬃcients f ij ( a, b, c ), whic h are linear in the second ord er deriv ative s. One can ve rify that only 30 of these equations 33 are linearly indep end en t. Solving them, we get closed form expressions for the second ord er partial deriv ativ es of the co eﬃcient s f 11 , f 12 , f 13 , f 23 , f 33 in terms of the ﬁrst order deriv ativ es thereof, 30 equations altogether (withou t any loss of generalit y one can set f 22 = 1), wh ic h can b e repr esen ted in s ym b olic form (13 ), d 2 f ij = 1 F R ( f k l , d f k l ); here R is quadratic in b oth f k l and d f k l . A straigtforwa rd computation shows that this system is in in v olution: all compatibilit y conditions are satisﬁed ident ically . Since the v alues of the ﬁv e functions f 11 , f 12 , f 13 , f 23 , f 33 , and ﬁrst ord er deriv ativ es thereof, are not restricted by any additional constraint s, w e obtain a 5+ 3 · 5 = 20- dimensional mo d uli space of integrable equations. This ﬁn ishes the pr o of of Theorem 1. Ac kno wledgemen ts W e thank B Dubro vin, K Khusn u tdino v a, M P a vlo v and A V eselo v for their interest, con- stan t supp ort an d v aluable remarks. EVF thanks A Odesskii for clarifying discussions on mo dular forms. The researc h of EVF and P AB w as partially su pp orted by the EPSRC gran t EP/D0361 78/1, the Europ ean Union through the FP6 Marie C urie R TN pro ject ENIGMA (Con tract num b er MR TN-CT-2004-56 52), and the ES F programme MISGAM. SPT is grateful to the Institute of Mathematic s in T aip ei (T aiw an) wher e a part of this w ork has b een com- pleted, and esp ecially to Jen-Hsu Chang for the hospitalit y at the National Defense Universit y . 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Second order quasilinear PDEs and conformal structures in projective space

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