Systems of Gibbons-Tsarev type and integrable 3-dimensional models

Systems of Gibb ons-Tsarev t yp e and in tegrable 3-dimension al mo dels A.V. Odesskii 1 , 2 , V.V. Sok olo v 1 1 L.D. Landau Institute for Theore tical Ph ys ics (Russia) 2 Bro ck Univ ers it y (Canada) Abstract W e review the role of Gibb ons -Tsarev-t yp e sys tems in classific a- tion of in tegra ble m ulti-dimensional h ydro dyn amic-t yp e systems. Our main observ ation is an univ ersalit y of Gibb ons-Tsarev-t yp e systems. W e also con- stract explicitly a wide c lass of 3-dimensional h ydro dynamic-type systems corresp onding to the simplest po ssible Gibb ons-Tsarev-t yp e sys tem. MSC nu m b ers : 17B80, 17B63 , 32L81, 1 4H70 Address : L.D. Landau Institute for Theoretical Ph ysics o f Russian Academy of Sciences, Kosygina 2 , 1193 34, Mosco w, Russia E-mail : ao desski@bro c ku.ca, sok olov@itp.ac.ru 1 1 In tro duction In tegrable equations play import a n t role in b ot h Mathematics and Phy sics. Unfortunately , rigorous and univ ersal definition of in tegrabilit y applicable in all situations do es not exist. Differen t viewpoints on the integrabilit y can b e found in [1, 2]. It is well know n that in tegrable 2-dimensional PDEs u t = F ( u, u x , ..., u nx ) (1.1) lik e the KdV- equation, for any N p ossess families of exact solutio ns dep ending o n arbitrar y constan ts c 1 , ..., c N . All these finite-gap and solitonic-type solutions can b e constructed by so called OD E-reductions. A pa ir of compatible N -comp onen t systems of ODEs r i x = f i ( r 1 , ..., r N ) , r i t = g i ( r 1 , ..., r N ) , i = 1 , ..., N (1.2) is called an O D E-reduction of (1.1) if there exists a function U ( r 1 , ..., r N ) suc h that u = U ( r 1 ( x, t ) , ..., r N ( x, t )) satisfies (1.1) for an y solution r 1 ( x, t ) , ..., r N ( x, t ) of (1.2). It is clear that the solution u dep ends on N arbitrary para meters b eing initial v alues for (1.2) at generic p oin t. The existence of special ODE-reductions for a rbitrary N can b e c hosen as a criterion of in tegrability for equation (1.1). F or example, one can assume that (1.1) admits a series of differen tial ODE-constrain ts u m,x = G m ( u, u x , ..., u m − 1 ,x ) , where m is arbitra ry . Clearly , equation (1.1) and t he ODE-constraint can b e rewritten as a pair of compatible dynamical systems with respect to x and t . Anot her example of ODE-reductions is provide d b y Dubrovin’s equations [3]. Ho wev er, in the 2-dimensional case there exis t more efficien t and constructiv e in tegrability criteria, lik e the exis tence of higher lo cal symme tries or conserv a tion la ws (see [4] and references therein). If the n umber d of independent v ariables is greater then 2, then higher lo cal symmetries f o r in tegra ble mo dels do not exist (for some generalization of the symmetry approac h to the case of non-lo cal symmetries see [5]). In suc h a situation the existence of N -comp onen t reductions can b e regarded as o ne of the most p ow erful metho ds of searchin g for new in t egrable mo dels . Notice that one has to consider for the reductions some compatible systems o f PDEs of dimension ≤ d − 1 instead of ODEs (1 .2). In [6] this appro ac h has b een systematically applied t o some classes o f 3-dimensional systems of the form n X j =1 a ij ( u ) u j,t + n X j =1 b ij ( u ) u j,y + n X j =1 c ij ( u ) u j,x = 0 , i = 1 , ..., n + k , (1.3) where u = ( u 1 , . . . , u n ) , and k ≥ 0 . P air s of compatible diag onal semi-Hamiltonian (see formula (2.12)) hy dro dynamic-ty p e systems of the form r i t = v i ( r 1 , ..., r N ) r i x r i y = w i ( r 1 , ..., r N ) r i x i = 1 , 2 , ..., N , (1.4) 2 ha ve been tak en for reductions. According to definition of reductions, t he corresp onding solu- tions of (1.3) are determined by some functions U i ( r 1 , ..., r N ) , i = 1 , ..., n conv erting an y solution of (1.4) to a solution of (1.3). In h ydro dy namics such solutions describe nonlinear in teraction of N planar w a v es. Sometimes they are called N -phase solutions. Clearly , the general solution of (1 .4) contains N arbitrary functions of one v ariable. It turns out that functions v i , w i in the reduction (1.4) ma y con tain additional functions of one v ariables as functional parameters and the nu m b er of these functions is not greater then N . In [6] the existence of hydrodynamic reductions (1.4) lo cally parameterized b y N functions of one v ariables, where N is arbitrary , was prop osed as a criterion of in tegrability fo r systems (1.3). The corresp onding N -phase solutions dep end o n 2 N arbitrary functions of o ne v ariables. Usually the inte grability of systems ( 1 .3) is asso ciated with a represen t a tion of (1.3) as com- m utativity conditions for a pair of vector fields [7]. F o r system s t ha t admit the pseudop otential represen tation [8, 9, 10, 11] these v ector fields a re Hamiltonian whereas for some integrable mo d- els the v ector fields hav e more complicated structure. Moreo ve r, fo r some systems the vec tor fields depend o n a sp ectral para meter. Th us it is v ery difficult to c ho ose any constructiv e class of the v ector fields co v ering all know n examples and to prop ose an unive rsal definition of in te- grabilit y based on the comm uta tivit y of vec tor fields . The same problems arise with definitions of integrabilit y giv en in terms of disp ersionless Lax or zero-curv ature represen tations. Quite the con trary , the hyd ro dynamic reduction approach is univ ersal. This means that all in tegrable mo de ls kno wn b y no w admit the h ydro dy namic r eductions. All notions of t his approac h can b e rigoro usly defined (see Section 2). It w as demonstrated in [6 ] that the existence of h ydro dy namic reductions can b e algo rithmically v erified for a giv en system (1.3) and what is more can b e efficien tly used for classification of inte grable cases. F amilies of system s (1.4) parameterized by N functions of one v ariables can b e desc rib ed in terms of the so-called systems of Gibb ons-Tsarev t yp e (GT-type sys tems). The GT-t yp e sys- tems pla y a crucial r o le in the approac h to in tegrabilit y based on the hydrodynamic reductions. Definition. A compatible system of PDEs of the f o rm ∂ i p j = f ( p i , p j , u 1 , ..., u n ) ∂ i u 1 , i 6 = j, i, j = 1 , ..., N , ∂ i u m = g m ( p i , u 1 , ..., u n ) ∂ i u 1 , m = 2 , ..., n, i = 1 , ..., N , (1.5) ∂ i ∂ j u 1 = h ( p i , p j , u 1 , ..., u n ) ∂ i u 1 ∂ j u 1 , i 6 = j, i, j = 1 , ..., N is called n - fields GT-typ e system . Here p 1 , ..., p N , u 1 , ..., u n are functions of r 1 , ..., r N , N ≥ 3 and ∂ i = ∂ ∂ r i . Notice that the compatibilit y conditions giv e rise to a system of functional equations for the functions f , g k , h and these equations don’t depend on N . Example 1 [10 ]. The system ∂ i p j = p j ( p j − 1) p i − p j ∂ i u 1 , ∂ i u m = u m ( u m − 1 ) p i − u m ∂ i u 1 , m = 2 , ..., n, (1.6) 3 ∂ i ∂ j u 1 = 2 p i p j − p i − p j ( p i − p j ) 2 ∂ i u 1 ∂ j u 1 , i, j = 1 , ..., N , i 6 = j (1.7) is an n -field GT-type system f o r any n, N .  The or ig inal G ibb ons-Tsarev system [1 2] is a degeneration o f (1 .6), (1.7). A wide class of in tegra ble systems ( 1.3) related to (1.6), (1.7) is desc rib ed in [10]. An elliptic vers ion of this GT-t yp e system and the corresp onding in tegrable 3- dimensional systems w ere pro p osed in [11]. Definition. Tw o GT-t yp e systems are called e quivalent if they are related b y a transfor- mation of the form p i → λ ( p i , u 1 , ..., u n ) , i = 1 , ..., N , (1.8) u m → µ m ( u 1 , ..., u n ) , m = 1 , ..., n. (1.9) Remark 1. Our Definitions and form ulas (1 .5), (1.8), (1.8) admit a co o rdinates-free in ter- pretation. Let M b e a bundle with o ne-dimensional fib er E and n -dimensional ba se F . Then eac h of p i is a co ordinate on E and u 1 , ..., u n are some co ordinates on F . Th us we obta in a notion of a GT-type structure on M . It is lik ely that t here exists a canonical GT-type structure on the natural bundle ov er the mo duli space M g of genus g algebraic curv es. Here E is a curv e corresp onding to a p oin t in M g . W e will not use co ordinates-free language in this pap er. F or generic GT- type systems the functions f , h hav e a p ole at p i = p j . Ho we v er, there exist GT-t yp e sy stems holomorphic at p i = p j . Example 2. The system ∂ i p j = 0 , ∂ i u m = g m ( p i ) ∂ i u 1 , ∂ i ∂ j u 1 = 0 (1.10) is an n -field G T-t yp e system for an y n, N and any functions g m ( x ). Notice that only special c hoice of the functions g m ( x ) giv es rise to pairs of compatible semi-Hamiltonian sys tems (1.4).  In this paper w e study systems ( 1.3) related to GT-t yp e systems of the f o rm (1.10). The main motiv ation is the follo wing observ a tion. W e examined the 3-dimensional trav el w av e reductions for known examples of in tegrable d -dimensional systems with d > 3 and found that the GT-type system s corresp onding to these reductions a r e equiv alen t to (1.10) with rational functions g k . W e b e liev e that t his observ a tion gives us an algorithm for constructing of new in teresting examples of in tegra ble m ulti-dimensional systems. The pap er is organized as follows. In Section 2.1 follow ing [6], w e describ e the hydrodynamic reduction method and sho w that any inte grable system (1 .3) is related to a GT-t yp e system. Section 2.2 is devoted to the return w ay from GT-ty p e systems to in tegrable 3-dimensional systems . Mor eov er, w e presen t all kno wn to us GT-type systems and giv e a new in terpretation of results obtained in [10, 11]. In Section 3 we consider GT-t yp e syste ms (1 .10) with rationa l functions g m ( x ) . Using an algorithm describ ed in Section 2 .2, we construct the corresp onding families of compatible pairs 4 of h ydro dynamic-t yp e 2-dimensional systems, and finally 3-dimensional systems of the form (1.3) with arbitrary n, k , whose h ydro dynamic reductions are giv en by our 2- dimensional sys - tems. It turns out that all these 3- dimensional systems p oss ess pseudop otential represen tations with a spectral parameter. In the generic case, the co efficien ts of the 3-dimensional systems are exp ressed in terms of exp onen ts of u i . D egenerations considered in subsection 3.2 in v olves p olynomials in addition to the exponents . In the case of small n and k some of our systems are equiv a len t to known disp ersionless equations o f second order. In particular, the generic system corresp onding to n = 3 , k = 1 is equiv alen t to the disp ersionless Hirota equation a 1 Z x Z y t + a 2 Z y Z xt + a 3 Z t Z xy = 0 , a 1 + a 2 + a 3 = 0 . Ac kno wledgmen t s. Authors thank M.V. P av lo v for fruitful discuss ions. V.S. is gr a teful to IHES for hospitalit y and financial supp o r t. He w as partially supp orted by t he RFBR grants 08-01- 4 61 and NS 3472.2008.2. 2 The G T-typ e systems and i n teg r abi lit y 2.1 The metho d of h ydro dynamic reductions Recall the definitions of the h ydro dynamic reduction metho d and the corresp onding criteria o f in tegra bility fo r 3-dimensional hydrodynamic-type systems [6]. Definition. An ( 1 +1)-dimensional h ydro dynamic-ty p e system of the form r i t = λ i ( r 1 , ..., r N ) r i x , i = 1 , ..., N , (2.11) is called semi-Hamiltonian if the follo wing relation ho lds ∂ j ∂ i λ m λ i − λ m = ∂ i ∂ j λ m λ j − λ m , i 6 = j 6 = m, (2.12) Semi-Hamiltonian systems hav e infinitely man y symmetries and conserv ation la ws of h y- dro dynamic t yp e [14]. Definition. A h ydro dynamic reduction of a system (1.3) is defined by a pair of compatible semi-Hamiltonian hydrodynamic-type systems r i t = λ i ( r 1 , ..., r N ) r i x , r i y = µ i ( r 1 , ..., r N ) r i x , i = 1 , ..., N , (2.13) and b y functions u 1 ( r 1 , ..., r N ) , ..., u n ( r 1 , ..., r N ) suc h that fo r eac h solution of (2.13) the func- tions u 1 = u 1 ( r 1 , ..., r N ) , ..., u n = u n ( r 1 , ..., r N ) (2.14) satisfy (1.3) . 5 According to [6] a system ( 1.3) is called inte gr able if it p oss esses as man y hydrodynamic reductions as p ossible. Namely , substituting (2.14) in to (1.3), eliminating t - and y -deriv a tiv es via (2.13), and equating co efficien ts at r l x to zero, w e obtain n X j =1 a ij ( u ) λ l ∂ l u j + n X j =1 b ij ( u ) µ l ∂ l u j + n X j =1 c ij ( u ) ∂ l u j = 0 , i = 1 , ..., n + k , l = 1 , ..., N . (2.15) F or eac h fixed l this is a linear ov erdetermined system for n unkno wns ∂ l u 1 , ..., ∂ l u n , whose co efficien ts do no t dep end o n l . This linear system m ust hav e non-zero solution so all its n × n minors m ust b e equal to zero. These minors are p olynomials in λ l , µ l indep enden t on l . W e assume that this system of p olynomial equations is equiv alen t to one equation P ( λ l , µ l ) = 0 (2.16) (otherwise λ l , µ l are fixed and we hav e no t sufficien tly man y reductions). Equation (2.16) defines the so-called disp ersion algebr aic curve . Let p b e a co ordinate on this curv e. Then (2.16) is equiv alen t to equations λ l = F ( p l , u 1 , ..., u n ) , µ l = G ( p l , u 1 , ..., u n ) for some functions F , G . Assume that f or generic p l the linear system (2.15) ha s one solution up to prop ortionality . Solving this sys tem, w e obtain ∂ i u m = g m ( p i , u 1 , ..., u n ) ∂ i u 1 , m = 2 , ..., n, i = 1 , ..., N (2.17) for some functions g m . Rewrite (2.13) in the form r i t = F ( p i , u 1 , ..., u n ) r i x , r i y = G ( p i , u 1 , ..., u n ) r i x , i = 1 , ..., N . (2.18) It is easy to see that the compatibility conditions for (2.1 8) ha ve the form ∂ i F ( p j ) F ( p i ) − F ( p j ) = ∂ i G ( p j ) G ( p i ) − G ( p j ) . (2.19) Here w e omit argumen ts u 1 , ..., u n in F , G . F rom (2.19) w e can find ∂ i p j in the form ∂ i p j = f ( p i , p j , u 1 , ..., u n ) ∂ i u 1 , i 6 = j, i, j = 1 , ..., N . (2.20) Finally , the compatibilit y conditions ∂ i ∂ j u m = ∂ j ∂ i u m giv e rise to ∂ i ∂ j u 1 = h ( p i , p j , u 1 , ..., u n ) ∂ i u 1 ∂ j u 1 , i 6 = j, i, j = 1 , ..., N . (2.21) Collecting equations (2.15), (2.2 0 ), (2.2 1 ) together, w e obtain a system of the form (1.5). Since w e wan t to ha ve as ma ny reductions as p ossible, w e assume that this system is in in v olution (i.e. fully compatible). In this case the family of h ydro dynamic reductions (2.18) lo cally dep en ds on N functions in o ne v ariable. 6 2.2 F rom GT-t yp e sy stems to in tegrable mo dels In the classification w o r ks [6] the authors start from a class of systems (1.3) with fixed small n and k , calculate the corresp onding G T-t yp e system and derive in tegrability conditions for (1 .3) from the compatibilit y conditions for the GT-t yp e system. In this section w e trace the return w ay and sho w how to construct wide classes o f in tegrable systems (1.3) with arbitr a ry n and k starting from a g iven GT-t yp e system. W e also describ e our previous results [1 0 , 11] from this p oin t of view. A list of known one-field GT-t yp e systems is giv en b y t he follow ing example s. Example 3. Let P ( x ) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 . Then ∂ ij u = K 2 ( p i , p j ) u 2 + K 1 ( p i , p j ) u + K 0 ( p i , p j ) P ( u )( p i − p j ) 2 ∂ i u∂ j u, ∂ i p j = P ( p j )( u − p i ) P ( u )( p i − p j ) ∂ i u, i, j = 1 , ..., N , i 6 = j, where K 2 ( p i , p j ) = 2 a 3 ( p i − p j ) 2 , K 1 ( p i , p j ) = − a 3 ( p 2 i p j + p i p 2 j ) + a 2 ( p 2 i + p 2 j − 4 p i p j ) − a 1 ( p i + p j ) − 2 a 0 , K 0 ( p i , p j ) = 2 a 3 p 2 i p 2 j + a 2 ( p 2 i p j + p i p 2 j ) + a 1 ( p 2 i + p 2 j ) + a 0 ( p i + p j ) is an one-field GT-type system. Using tra nsfor ma t io ns of the form u → au + b cu + d , p i → ap i + b cp i + d , one can put the p o lynomial P to one of the canonical forms: P ( x ) = x ( x − 1), P ( x ) = x , or P ( x ) = 1 .  Note t ha t in the case P ( x ) = x ( x − 1) we return to the Example 1 with n = 1. Example 4. Let θ ( z , τ ) = X α ∈ Z ( − 1) α e 2 π i ( αz + α ( α − 1) 2 τ ) , ρ ( z , τ ) = θ z ( z , τ ) θ ( z , τ ) . Then ∂ α p β = 1 2 π i  ρ ( p α − p β ) − ρ ( p α )  ∂ α τ , ∂ α ∂ β τ = − 1 π i ρ ′ ( p α − p β ) ∂ α τ ∂ β τ , where α, β = 1 , ..., N , α 6 = β , is an one-field GT-ty p e syste m.  7 It t urns out that if we add t he follow ing equations : ∂ i u m = f ( p i , u m , u ) ∂ i u 1 , m = 2 , ..., n to any one-field G T-t yp e system ∂ i p j = f ( p i , p j , u 1 ) ∂ i u 1 , ∂ i ∂ j u 1 = h ( p i , p j , u 1 ) ∂ i u 1 ∂ j u 1 , then the system o f PDEs thus obtained is in in v o lution. One can obtain n -fields GT-type system for any n in this w a y . W e call this pro cedure r e gular extension . F or example, in the case of Example 4 the regular extension is giv en by ∂ α u β = 1 2 π i  ρ ( p α − u β ) − ρ ( p α )  ∂ α τ , β = 1 , ..., n − 1 . The Example 1 is a regular extension of Example 3 with P ( x ) = x ( x − 1). As f ar as w e kno w, the regular extensions of Examples 3, 4 are the only G T-t yp e system s app eared in the literature. In this pap er w e in v estigate the simplest p ossible GT-t yp e system from Example 2 and obtain the corr esp o nding systems of the t yp e (1.3) whic h are pro bably new. A basic ob ject asso ciated with a g iv en GT-type system is a pair o f compatible (1+1)- dimensional hydrodynamic-ty p e systems o f the form (2.18). One should solve the functional equation (2 .1 9) in order to find a ll p ossible functions F , G . Notice that the deriv ativ es in (2.19) are supp osed to be calculated by virtue of the GT-t yp e system. The existence of non-constant solutions F, G f o r the functional equation (2.19) is a n a ddi- tional condition, which w e imp ose on the GT-t yp e system. F or instance, in the case of Example 2 the solutions exist not f or any functions g m . The following statemen t can b e prov ed straigh tf o rw ardly . Prop osition 1. Any one-field GT-type system having a non-constant solution of the f o rm F ( p, u ) , G ( p, u ) for the functional equation (2.19) is equiv alent to o ne describ ed in Example 3.  In [10] w e found the following solutions F, G for the n -field GT-t yp e system in Example 1: Prop osition 2. Fix s 1 , ..., s n +2 ∈ C . Consider the following compatible ov erdetermined system of linear PDEs: ∂ 2 h ∂ u j ∂ u k = s j u j − u k · ∂ h ∂ u k + s k u k − u j · ∂ h ∂ u j , i, j = 1 , ..., n, j 6 = k , and ∂ 2 h ∂ u j ∂ u j = − 1 + n +2 X k =1 s k ! s j u j ( u j − 1 ) · h + s j u j ( u j − 1) n X k 6 = j u k ( u k − 1) u k − u j · ∂ h ∂ u k + n X k 6 = j s k u j − u k + s j + s n +1 u j + s j + s n +2 u j − 1 ! · ∂ h ∂ u j 8 It is easy to show that the v ector space H of all solutions is n + 1-dimensional. F or an y h ∈ H w e put S ( h, p ) = X 1 ≤ i ≤ n u i ( u i − 1)( p − u 1 ) ... ˆ i... ( p − u n ) h u i + (1 + X 1 ≤ i ≤ n +2 s i )( p − u 1 ) ... ( p − u n ) h. Clearly , S is a p o lynomial of degree n in p . Let h 1 , h 2 , h 3 b e linearly indep enden t elemen ts of H . Then F = S ( h 1 , p ) S ( h 3 , p ) , G = S ( h 2 , p ) S ( h 3 , p ) (2.22) satisfy the functional equation (2.19) for reductions. In the case when the degree o f S is less then n the solutions ar e given by mo r e complicated determinan t formulas (see [10]).  F or t he regular extensions of the Example 4 solutions of the functional equation (2.19) are giv en b y the same form ula (2.22), where S ( h, p ) = X 1 ≤ α ≤ n θ ( u α ) θ ( p − u α − η ) θ ( u α + η ) θ ( p − u α ) h u α − ( s 1 + ... + s n ) θ ′ (0) θ ( p − η ) θ ( η ) θ ( p ) h.. Here η = s 1 u 1 + ... + s n u n + r τ + η 0 , where s 1 , ..., s n , r , η 0 are arbitrary constants and h ( u 1 , ..., u n , τ ) is a solution of the following elliptic h yp ergeometric system: h u α u β = s β  ρ ( u β − u α ) + ρ ( u α + η ) − ρ ( u β ) − ρ ( η )  h u α + s α  ρ ( u α − u β ) + ρ ( u β + η ) − ρ ( u α ) − ρ ( η )  h u β , h u α u α = s α X β 6 = α  ρ ( u α ) + ρ ( η ) − ρ ( u α − u β ) − ρ ( u β + η )  h u β +  X β 6 = α s β ρ ( u α − u β ) + ( s α + 1) ρ ( u α + η )+ s α ρ ( − η ) + ( s 0 − s α − 1 ) ρ ( u α ) + 2 π ir  h u α − s 0 s α ( ρ ′ ( u α ) − ρ ′ ( η )) h, h τ = 1 2 π i X β  ρ ( u β + η ) − ρ ( η )  h u β − s 0 2 π i ρ ′ ( η ) h.  Giv en a GT-type system and a solution F , G of the functional equation (2.1 9) for reduction, one can easily construct an integrable system of the fo rm (1.3). In teger k is called the defe ct of the system. 9 Lemma 1. Consider the linear space V of functions in p spanned b y { F ( p, u 1 , ..., u n ) g j ( p, u 1 , ..., u n ) , G ( p, u 1 , ..., u n ) g j ( p, u 1 , ..., u n ) , g j ( p, u 1 , ..., u n ); j = 1 , ..., n } . Here by definition g 1 = 1. Then the system of the form (1.3) with reductions (2 .18) consists of l equations iff V is (3 n − l )-dimensional. Moreov er, the co efficien ts of (1.3) are defined by relations: n X j =1  a ij ( u ) F ( p, u 1 , ..., u n )+ b ij ( u ) G ( p, u 1 , ..., u n )+ c ij ( u )  g j ( p, u 1 , ..., u n ) = 0 , i = 1 , ..., n + k . An explicit form of in tegrable systems ( 1 .3) corresp onding to Examples 3, 4 can b e found in [10, 11]. 3 W eakly no nlinear 3 - dimension al systems F or generic G T-t yp e systems the functions f , h hav e p oles at p i = p j . How ev er, there exist GT-t yp e sy stems holomorphic at p i = p j . W e call in tegrable system ( 1.3) we akly nonline ar if the corresp onding GT-type system is holomorphic at p i = p j . It is p ossible to c hec k that if k = 0, then an y 2-dimensional system describing trav el w a v e solutions u = u ( c 1 x + c 2 y + c 3 t, c 4 x + c 5 y + c 6 t ) for w eakly nonlinear 3-dimensional sy stem (1.3) is a w eakly nonlinear 2-dimensional system in the sens e of [13]. Example 5. Consider the follow ing 3-dimensional system (see [6]): v t + av x + pv y + q w y = 0 , w t + bw x + r v y + sw y = 0 , (3.23) where a = w , b = v , r = P ( w ) w − v , q = P ( v ) v − w , s = P ( v ) w − v + 1 3 P ′ ( v ) , p = P ( w ) v − w + 1 3 P ′ ( w ) . Here P is an arbitrary p olynomial of degree three. The corresp onding GT-type system is giv en b y ∂ i p j = P ( w ) ( w − v ) P ( v ) p 2 j p i +  1 w − v + P ′ ( v ) P ( v )  p j p i −  1 v − w + P ′ ( w ) P ( w )  p j − P ( v ) ( v − w ) P ( w ) , ∂ i v = p i ∂ i w , ∂ i ∂ j w =  P ( w ) ( v − w ) P ( v ) p i p j + 1 v − w + P ′ ( w ) P ( w )  ∂ i w ∂ j w . 10 This G T-t yp e system is p olynomial in p i , p j and therefore the correspo nding 3-dimensional system is w eakly nonlinear. It is p oss ible to v erify that this GT-type sys tem is equiv alen t to ∂ i p j = 0 , ∂ i u 2 = p i ∂ i u 1 , ∂ i ∂ j u 1 = 0 .  It w as mentioned in [6] that the system (3.2 3) p ossesses a h ydro dynamic-t yp e Lax repre- sen tation dep ending on a sp ectral parameter. It turns o ut that t his is a general prop erty of 3-dimensional sy stems correspo nding t o GT-type syste ms of the form (1.1 0). Prop osition 3. Let F ( p, u 1 , ..., u n ) , G ( p, u 1 , ..., u n ) b e a solution of the functiona l equation (2.19) for a GT-type system (1.10). Then the corresponding 3-dimensional sys tem admits the Lax represen tatio n ψ t = F ( ξ , u 1 , ..., u n ) ψ x , ψ y = G ( ξ , u 1 , ..., u n ) ψ x , where ξ is a sp ectral parameter.  3.1 Generic case Using our observ ation that the GT-t yp e system from Example 5 is equiv alen t to (1.1 0) with rational f unctions g m , w e generalize Example 5 to the case of arbitra r y n and k . Consider the ( n + 1)- field GT-type system (1.10) with g m = M m / M , where M , M 1 , ..., M n +1 are generic p olynomials of degree n . Supp ose that M has pairwise distinct ro ots λ 0 , λ 1 , ..., λ n . Then up to equiv alence the GT-type syste m can be written as ∂ i p j = 0 , ∂ i u m = λ m − λ 0 p i − λ m ∂ i w , ∂ i ∂ j w = 0 (3.24) with fields denoted by u 1 , ..., u n , w . Let H n b e the linear space of functions in u 1 , ..., u n spanned b y 1 , e u 1 , ..., e u n . F or an y function g = a 0 + a 1 e u 1 + ... + a n e u n ∈ H n w e put S n ( g , p ) = a 0 p − λ 0 + n X i =1 a i e u i p − λ i . F or k ∈ N suc h that 0 < k < n − 1 w e fix functions h 1 , ..., h k ∈ H n , where h i = b i, 0 + b i, 1 e u 1 + ... + b i,n e u n , and define S n,k ( g , p ) = det       S n ( g , p ) S n ( h 1 , p ) ... S n ( h k , p ) g h 1 ... h k a n − k +2 b 1 ,n − k +2 ... b k ,n − k +2 ......... ... ... ......... a n b 1 ,n ... b k ,n       . (3.25) 11 By definition, S n, 0 ( g , p ) = S n ( g , p ). Prop osition 4. Let g 1 , g 2 , g 3 b e linearly indep enden t elemen ts of H n . Then for an y 0 ≤ k < n − 1 the functions F = S n,k ( g 1 , p i ) S n,k ( g 3 , p i ) , G = S n,k ( g 2 , p i ) S n,k ( g 3 , p i ) (3.26) satisfy the functional equation (2.19) for h ydro dynamic reductions.  T o find an explicit form of the corresponding 3-dimensional systems we note tha t n X i =1 ( A i u i,t 1 + B i u i,t 2 + C i u i,x ) = 0 is an equation from the 3 - dimensional system iff n X i =1 λ i − λ 0 p − λ i  A i S n,k ( g 1 , p ) + B i S n,k ( g 2 , p ) + C i S n,k ( g 3 , p )  = 0 as function in p . Let g i = a i, 0 + a i, 1 e u 1 + ... + a i,n e u n , i = 1 , 2 , 3. If k = 0, then the correspo nding 3- dimensional system reads a s follows: X 1 ≤ j ≤ n,j 6 = i ( a 2 ,i a 3 ,j − a 2 ,j a 3 ,i ) e u j u i,t 1 − u j,t 1 λ i − λ j + ( a 2 ,i a 3 , 0 − a 3 ,i a 2 , 0 ) u i,t 1 λ i − λ 0 + X 1 ≤ j ≤ n,j 6 = i ( a 3 ,i a 1 ,j − a 3 ,j a 1 ,i ) e u j u i,t 2 − u j,t 2 λ i − λ j + ( a 3 ,i a 1 , 0 − a 1 ,i a 3 , 0 ) u i,t 2 λ i − λ 0 + (3.27) X 1 ≤ j ≤ n,j 6 = i ( a 1 ,i a 2 ,j − a 1 ,j a 2 ,i ) e u j u i,x − u j,x λ i − λ j + ( a 1 ,i a 2 , 0 − a 2 ,i a 1 , 0 ) u i,x λ i − λ 0 = 0 , where i = 1 , ..., n . If k > 0, then the correspo nding 3- dimensional system reads a s follows: X 1 ≤ j ≤ n − k +1 ,j 6 = i  ∆ i ( g 2 )∆ j ( g 3 ) − ∆ j ( g 2 )∆ i ( g 3 )  e u j u i,t 1 − u j,t 1 λ i − λ j +  ∆ i ( g 2 )∆ 0 ( g 3 ) − ∆ 0 ( g 2 )∆ i ( g 3 )  u i,t 1 λ i − λ 0 + X 1 ≤ j ≤ n − k +1 ,j 6 = i  ∆ i ( g 3 )∆ j ( g 1 ) − ∆ j ( g 3 )∆ i ( g 1 )  e u j u i,t 2 − u j,t 2 λ i − λ j +  ∆ i ( g 3 )∆ 0 ( g 1 ) − ∆ 0 ( g 3 )∆ i ( g 1 )  u i,t 2 λ i − λ 0 + X 1 ≤ j ≤ n − k +1 ,j 6 = i  ∆ i ( g 1 )∆ j ( g 2 ) − ∆ j ( g 1 )∆ i ( g 2 )  e u j u i,x − u j,x λ i − λ j +  ∆ i ( g 1 )∆ 0 ( g 2 ) − ∆ 0 ( g 1 )∆ i ( g 2 )  u i,x λ i − λ 0 = 0 , 12 where i = 1 , ..., n − k + 1 and n − k +1 X j =1 e u j ∆ j ( g r ) u j,t s = n − k +1 X j =1 e u j ∆ j ( g s ) u j,t r , n − k +1 X j =1 ∆ j ( g r ) e u j u i,t s − u j,t s λ i − λ j + ∆ 0 ( g r ) u i,t s λ i − λ 0 = n − k +1 X j =1 ∆ j ( g s ) e u j u i,t r − u j,t r λ i − λ j + ∆ 0 ( g s ) u i,t r λ i − λ 0 , where i = n − k + 2 , ..., n . Here r , s = 1 , 2 , 3, t 3 = x a nd ∆ j ( g ) = det       g h 1 ... h k a j b 1 ,j ... b k ,j a n − k +2 b 1 ,n − k +2 ... b k ,n − k +2 ......... ... ... ......... a n b 1 ,n ... b k ,n       for j = 1 , ..., n , g = a 0 + a 1 e u 1 + ... + a n e u n , h 1 = b 1 , 0 + b 1 , 1 e u 1 + ... + b 1 ,n e u n ,..., h k = b k , 0 + b k , 1 e u 1 + ... + b k ,n e u n . Note that this equations are linearly dep enden t and the system is equiv alen t to a system with n + k linearly indep e nden t equations. Prop osition 5. If k = 0, then the corresp onding system (3.27) p ossesses the fo llo wing n + 1 conserv ation laws of hy dro dynamic ty p e:     S r eg n ( g 2 , λ i ) S r eg n ( g 3 , λ i ) a 2 ,i e − u i a 3 ,i e − u i     t 1 +     S r eg n ( g 3 , λ i ) S r eg n ( g 1 , λ i ) a 3 ,i e − u i a 1 ,i e − u i     t 2 +     S r eg n ( g 1 , λ i ) S r eg n ( g 2 , λ i ) a 1 ,i e − u i a 2 ,i e − u i     x = 0 , where i = 1 , ..., n and     S r eg n ( g 2 , λ 0 ) S r eg n ( g 3 , λ 0 ) a 2 , 0 a 3 , 0     t 1 +     S r eg n ( g 3 , λ 0 ) S r eg n ( g 1 , λ 0 ) a 3 , 0 a 1 , 0     t 2 +     S r eg n ( g 1 , λ 0 ) S r eg n ( g 2 , λ 0 ) a 1 , 0 a 2 , 0     x = 0 . Here S r eg n ( g , λ i ) =  S n ( g , p ) − a i e u i p − λ i     p = λ i and S r eg n ( g , λ 0 ) =  S n ( g , p ) − a 0 p − λ 0     p = λ 0 . Prop osition 6. If k > 0 , then the corresp onding system p o ssess es the follow ing 3 k conser- v ation law s of h ydro dynamic t yp e: ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t s = ∆( g s , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) ! t r , (3.28) 13 where i = 1 , ..., k , r , s = 1 , 2 , 3, t 3 = x and ∆( f 1 , ..., f k ) = det     f 1 ... f k f 1 ,u n − k +2 ... f k ,u n − k +2 ......... ... ......... f 1 ,u n ... f k ,u n     . Remark 2. It is lik ely that for k > 0 the corresp o nding 3-dimensional system p o ssess es additional n + 1 conse rv ation la ws of hy dro dynamic ty p e. Remark 3. Prop osition 6 allows us to define functions z 1 , ..., z k suc h that ∆( g r , h 1 , ... ˆ i...h k ) ∆( h 1 , ..., h k ) = z i,t r (3.29) for all i = 1 , ..., k and r = 0 , 1 , 3. See [10] or [11] f or further discussion. 3.2 Degenerations Our constructions of GT-ty p e systems and functions S n,k are v a lid in the case of pa ir wise distinct ro ots λ 0 , ..., λ n . In this section w e study degenerations of the GT-t yp e systems described in Section 3.1 . Define p olynomials P i ( u 1 , u 2 , ... ) as coefficien ts of the follo wing T aylor expansion exp ( ε u 1 + ε 2 u 2 + ... ) = 1 + P 1 ε + P 2 ε 2 + · · · . In part icular, P 1 = u 1 , P 2 = u 2 + 1 2 u 2 1 , P 3 = u 3 + u 1 u 2 + 1 6 u 3 1 . Denote the partial sums 1 + P k i =1 P i ε i b y Q k ( ε ). By definition, P 0 = Q 0 ( ε ) = 1 . Degeneration 1 . This degeneration corresp onds to the case λ 0 6 = λ 1 = ... = λ n . Consider the fo llo wing ( n + 1) -field GT-type system with fields u 1 , ..., u n , w : ∂ i p j = 0 , ∂ i u m =  λ 1 − λ 0 ( p i − λ 1 ) m + 1 ( p i − λ 1 ) m +1  ∂ i w , ∂ i ∂ j w = 0 . (3.30) F or a n y v ector ( a 0 , a 1 , ..., a n ) define g = a 0 + e u 1 n X i =1 a i P i − 1 and S n ( g , p ) = a 0 p − λ 0 + e u 1 n X i =1 a i Q i − 1 ( p − λ 1 ) ( p − λ 1 ) i . 14 Degeneration 2 . This degeneration corresp onds to the case λ 0 = λ 1 = ... = λ n . Consider the fo llo wing ( n + 1) -field GT-type system : ∂ i p j = 0 , ∂ i u m = 1 ( p i − λ 0 ) m ∂ i w , ∂ i ∂ j w = 0 . (3.31) F or a n y v ector ( a 0 , a 1 , ..., a n ) define g = n X i =0 a i P i . and S n ( g , p ) = n X i =0 a i Q i ( p − λ 0 ) ( p − λ 0 ) i +1 . Com bining these degenerations, one obta ins the general case. Let λ 0 , ..., λ l b e pairwise dis- tinct ro o ts of m ultiplicities n 0 + 1 , n 1 ..., n l corresp ondingly . Note that n 0 + ... + n l = n . Consider the fo llo wing ( n + 1) -field system with fields u 0 , 1 , ..., u 0 ,n 0 , u 1 , 1 , ..., u 1 ,n 1 , ..., u l, 1 , ..., u l,n l w : ∂ i p j = 0 , ∂ i u 0 ,m = 1 ( p i − λ 0 ) m ∂ i w , ∂ i u s,m =  λ s − λ 0 ( p i − λ s ) m + 1 ( p i − λ s ) m +1  ∂ i w , ∂ i ∂ j w = 0 . (3.32) F or a n y v ector ( a 0 , 0 , a 0 , 1 , ..., a 0 ,n 0 , a 1 , 1 , ..., a 1 ,n 1 , ..., a l,n l ) define g = n 0 X i =0 a 0 ,i P i + l X s =1 n s X i =1 a s,i e u s,i P i − 1 and S n ( g , p ) = n 0 X i =0 a 0 ,i Q i ( p − λ 0 ) ( p − λ 0 ) i +1 + l X s =1 n s X i =1 a s,i e u s,i Q i − 1 ( p − λ s ) ( p − λ s ) i . Let k > 0. Fix v ectors ( b 0 , 0 ,j , b 0 , 1 ,j , ..., b 0 ,n 0 ,j , b 1 , 1 ,j , ..., b 1 ,n 1 ,j , ..., b l,n l ,j ) , j = 1 , ..., k . Let g = n 0 X i =0 a 0 ,i P i + l X s =1 n s X i =1 a s,i e u s,i P i − 1 and similarly h j = n 0 X i =0 b 0 ,i,j P i + l X s =1 n s X i =1 b s,i,j e u s,i P i − 1 , j = 1 , ..., k . Define S n,k ( g , p ) by S n,k ( g , p ) = det       S n ( g , p ) S n ( h 1 , p ) ... S n ( h k , p ) g h 1 ... h k g v n − k +2 h 1 , v n − k +2 ... h k , v n − k +2 ......... ... ... ......... g v n h 1 , v n ... h k , v n       (3.33) 15 where ( v 1 , ..., v n ) = ( u 0 , 1 , ..., u 0 ,n 0 , u 1 , 1 , ..., u 1 ,n 1 , ..., u l, 1 , ..., u l,n l ). Then Prop osition 4 ho lds. The explicit form of the corresponding 3 - dimensional systems can b e calculated using the general recip e. 4 Summary and di scussi o n: to w ard a classificatio n of in teg rable 3-dimens ional h ydro dynamic-t yp e s ystems W e summarize our previous remarks as the fo llo wing pro ject of classification of in tegr a ble 3- dimensional h ydro dynamic-ty p e systems: 1. Classify (up to equiv alence (1.8), (1.9)) all GT-type systems (1.5) p oss essing a non-trivial solution o f the functional equation (2.19). 2. F or each suc h GT- type system classify all solutions of the functional equation (2.1 9). 3. F o r each GT-ty p e system a nd solutio n of (2 .1 9) construct the corresp onding 3-dimensional system (see Lemma 1). This a pproac h has the follow ing adv a n ta g es: a. It turns out that GT-ty p e systems are unive rsal: for eac h GT-t yp e system there exist sev eral families of t he corresp onding 3-dimensional syste ms depending on essen tial para meters. Moreo ver, it is lik ely that f or each gen us g = 0 , 1 , ... there exists an unique GT-type sys tem (with one field for g = 0 , 1 and with 3 g − 3 fields for g > 1) suc h that a n y generic 3-dimensional system with the disp ersion curv e of gen us g corresp onds to this GT-type system or its regular extension. b. In tegrable systems (1.3) are defined up to arbitrary p oin t transformat io ns. Since the equiv a lence problem for systems (1 .3) is highly no n- trivial, it is not easy to find the simplest co ordinates in whic h a given 3- dimensional system has the simplest form. As the rule, the simplest co ordinates for the GT-t yp e system are at the same time the most natural co ordinates for the corresp onding system (1.3). F or example , eac h of the v ariables p i , u from the Example 3 admits natural interpre tation as a co ordinate on C P 1 . Similarly , eac h of the v ariables p i from the Example 4 can b e naturally interpreted as a co ordinate o n an elliptic curv e and τ as a mo dular parameter of this curve. W e exp ect that for g > 1 there exists a canonical G T-t yp e system with 3 g − 3 fields b eing co or dina t es on the mo duli space of gen us g curves . This G T- t yp e syste m should ha v e (an analog of ) regular extensions with additional fields b eing p oin ts on this curv e. One could find this GT-type system b y study of h ydro dynamic reductions for 3-dimensional systems from [9]. On the other hand, taking into accoun t t he results related to g = 0 , 1 we exp ect that there ar e m uch more 3- dimensional systems corresp o nding to the canonical G T-t yp e system, than it w a s fo und in [9]. The main disadv an tage of the approac h outlined ab ov e is that the realization o f the item 1 is v ery hard for n > 1. W e hop e to address this classification problem later. In this pap er w e 16 consider the simplest p oss ible GT-t yp e system of the form (1.10) and demonstrate that ev en in this case there exists a r ic h family of in teresting integrable 3-dimensional systems asso ci- ated with (1.10). The co e fficien ts of these systems are quasi-p o lynomials. Note that sligh tly more complicated GT-t yp e systems from Examples 3 and 4 r equire generalized hypergeometric functions for description o f the correspo nding 3- dimensional systems. References [1] What is In tegrability? edt. V.E. 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