New Semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

NEW SEMI–HAMIL TONIAN HIERARCHY RELA TE D TO INTEGRABLE M A GNETIC F LO WS O N SURF A CES MISHA BIAL Y AND A NDREY MIRONO V Abstract. W e consider magnetic geo desic flows on the 2-tor us . W e prov e that the question of existence of po lynomial in momen ta first in tegrals on one ener gy le vel leads to a Semi-Hamiltonian sys- tem of quasi-linea r equa tions, i.e. in the hyperb olic regions the system has Riemann in v ar iants and can be written in conserv ation laws form. 1. Intr oduction In t his pap er w e in troduce a new hierarc h y of Semi-Hamiltonian sys- tem which is naturally related to inte grable magnetic flows on surfaces. More precisely w e c onsider ma g netic g eo desic flo ws o n t w o - torus. Consider one energy lev el and assume it admits a p olynomial in mo- men ta in tegral of motion. Then w e pro v e that t he system of quasi-linear equations on the co efficien ts is in fact Semi-Hamiltonian sys tem. T hese systems w ere introduced b y S.Tsarev and later on studied extensiv ely . It is prov ed in [1] that these systems are in tegrable b y the generalized ho dograph metho d. A remark able theorem by Sev ennec [2] states that Semi-Hamiltonian prop erty is equiv alen t to existence of t wo sp ecial co- ordinate sys tems in the space of field v ariables: Riemann inv arian t s and Conserv ation la ws. It is remark able fa ct that b oth forms natura lly app ear for many systems . F or example for Benney chains [3] and for geo desic flo ws [4] the Riemann in v ariants corresp ond to critical v a l- ues of the in tegral, and Conserv ation la ws are related to t he inv arian t torii of the flo w. In this pap er the Semi-Hamiltonian prop ert y app ears naturally in the same manner a s w e shall prov e b elo w. The problem of existence of in tegra ble systems in the presence of gyroscopic forces ( which is equiv alen t to magnetic field) w as studied b y V.V.Kozlov and his studen ts [5],[6],[7]. T op ological obstructions to the in tegrabilit y of mec hanical systems on surfaces with magnetic fields w ere obtained in [7]. Date : 13 No vem b er 2011. 2000 Mathematics Subje ct Classific ation. 3 5L65,35 L67,70 H0 6 . Key wor ds and phr ases. Integral of motion, magnetic geo desic flows, Riemann inv ariants, Systems of Hydro dynamic t y pe. M.B. w as suppo rted in part b y Israel Science foundation grant 128 /10 a nd A.M. was supp orted by RFBR grant 11-0 1-121 06-ofi-m- 2011. 1 2 MISHA BIA L Y AND A NDREY MIRONO V First of all let us recall some facts a b out g eo desic flows on 2- torus (without magnetic field). If the geo desic flow is integrable then on the torus there a re global semi-geo desic co ordinates ds 2 = g 2 ( t, x ) dt 2 + dx 2 . This co ordinates can b e constructed using in v aria n t Liouville torus ha ving the diffeomorphic pro jection on the base T 2 . The existence of suc h a torus is prov en in [8]. In the semi-geo desic co ordinat es the p olynomial in momen ta in tegral has the form F n = n X k =0 a k ( t, x ) p n − k 1 g n − k p k 2 , where a n − 1 ≡ g and a n ≡ 1 . The co efficien ts satisfies the system U t + A ( U ) U x = 0 , (1) where U = ( a 0 , . . . , a n − 2 , a n − 1 ) T , and matrix A has the form A =        0 0 . . . 0 0 a 1 a n − 1 0 . . . 0 0 2 a 2 − na 0 0 a n − 1 . . . 0 0 3 a 3 − ( n − 1) a 1 . . . . . . . . . . . . . . . . . . 0 0 . . . a n − 1 0 ( n − 1) a n − 1 − 3 a n − 3 0 0 . . . 0 a n − 1 na n − 2 a n − 2        . In [9] it is pro ven tha t in the case of in tegrals of degree 3 or 4 in t he elliptic regions (where matr ix A has all eigen v a lues differen t and t w o of them are complex-conjugate) the inte gral can b e reduced to in tegra ls of the first or second degree or t he metric is flat. W e prov ed in [4 ] that the system (1) is Semi-Hamiltonian. In t his pap er w e generalize this result to the case of non trivial magnetic field. 2. The main t heorem The geo desic flow on the tor us T 2 with the Riemannian metric ds 2 = g ij dx i dx j giv en b y the Hamiltonian equations o n T ∗ T 2 ˙ x j = ∂ H ∂ p j , ˙ p j = − ∂ H ∂ x j , j = 1 , 2 , where H = 1 2 g ij p i p j . The function F : T ∗ T 2 → R is called the first in tegra l of the geo desic flow if ˙ F = { F , H } g = 0 , where { F , H } g is P oisson brac ke t { F , H } g = 2 X j =1  ∂ F ∂ x j ∂ H ∂ p j − ∂ H ∂ x j ∂ F ∂ p j  . NEW S EMI–HAMIL TONI AN HI ERARCHY 3 In the case of magnetic geo desic flo w the P oisson brack et gets the form { F , H } mg = 2 X j =1  ∂ F ∂ x j ∂ H ∂ p j − ∂ H ∂ x j ∂ F ∂ p j  + Ω  ∂ F ∂ p 1 ∂ H ∂ p 2 − ∂ F ∂ p 2 ∂ H ∂ p 1  , where Ω is the magnetic field. The magnetic g eo desic flo w giv en b y the Hamiltonian equations with resp ect to ma g netic geo desic brack et { ., . } mg ˙ x i = { x i , H } mg , ˙ p i = { p i , H } mg . (2) Let us choose conformal co o rdinates ( x, y ) on the t o rus. The metric b ecomes the form d s 2 = Λ( x, y ) ( dx 2 + dy 2 ) . W e shall consider the problem of existence of po lynomial in momenta in tegra l o f motion F of degree N on one energy lev el H = p 2 1 + p 2 2 2Λ = 1 2 . W e can parameterize the fibres of the energy lev el o ve r T 2 as fo llo ws. W rite p 1 = √ Λ cos ϕ, p 2 = √ Λ sin ϕ. The equations (2) b ecomes of the fo r m ˙ x = 1 √ Λ cos ϕ, ˙ y = 1 √ Λ sin ϕ, ˙ ϕ = Λ y 2Λ √ Λ cos ϕ − Λ x 2Λ √ Λ sin ϕ − Ω . The in tegral F on the energy lev el has the f o rm F = k = N X k = − N a k e ik ϕ , (3) where a k = u k ( x, y ) + iv k ( x, y ) , a − k = ¯ a k . The condition ˙ F = 0 is equiv alent to ( F ) x cos ϕ + ( F ) y sin ϕ + F ϕ  Λ y 2Λ cos ϕ − Λ x 2Λ sin ϕ − Ω √ Λ  = 0 . W e substitute (3) in the last equation and collect terms with resp ect to e ik ϕ . W e get ( a k − 1 ) x + ( a k +1 ) x 2 + ( a k − 1 ) y − ( a k +1 ) y 2 i + Λ y 2Λ i ( k − 1) a k − 1 + i ( k + 1) a k +1 2 − − Λ x 2Λ i ( k − 1) a k − 1 − i ( k + 1) a k +1 2 i − ik Ω √ Λ a k = 0 , (4) where k = 0 , . . . , N + 1 (w e assume a s = 0 at s > N ). Let k b e N + 1 in (4): ( a N ) x − N Λ x 2Λ a N + 1 i  ( a N ) y − N Λ y 2Λ a N  = 0 . Multiply the last iden tit y b y Λ − N/ 2 4 MISHA BIA L Y AND A NDREY MIRONO V ( a N Λ − N/ 2 ) x − i ( a N Λ − N/ 2 ) y = 0 . Th us a N Λ − N/ 2 is a holomorphic function, consequen tly a N = Λ N/ 2 ( α + iβ ) for some constants α, β . T aking appropriate rotation in the plane ( x, y ) and dividing F by appropriate constant we can assume that α = 1 , β = 0. Thus w e hav e a N = Λ N/ 2 . Notice that for k = 0 the equation (4) do es no t contain magnetic field and has the form ( a − 1 ) x + ( a 1 ) x 2 + ( a − 1 ) y − ( a 1 ) y 2 i + Λ y 2Λ ia 1 − ia − 1 2 + Λ x 2Λ a − 1 + a 1 2 = 0 . ( 5 ) Let us intro duce the notation Q k = ( a k − 1 ) x + ( a k +1 ) x 2 + ( a k − 1 ) y − ( a k +1 ) y 2 i + + Λ y 2Λ i ( k − 1) a k − 1 + i ( k + 1) a k +1 2 − Λ x 2Λ ( k − 1) a k − 1 − ( k + 1) a k +1 2 . F rom equation (4 ) for k = N w e can find the mag netic field: Ω = Q N iN √ Λ a N . Let us substitute this expression of Ω in to (4) for ev ery k = 1 , . . . , N − 1 . W e get N Q k Λ N/ 2 = k Q N a k . (6) The equations (5 ) and (6) form a system of quasi-linear equations on 2 N unkno wn functions Λ , u 0 , . . . , u N − 1 , v 1 , . . . , v N − 1 . This is a quasi- linear system of the form A ( U ) U x + B ( U ) U y = 0 , where U = (Λ , u 0 , . . . , u N − 1 , v 1 , . . . , v N − 1 ) ⊤ . W e shall write in the last section this system explicitly for N = 2. Our main result is Theorem 1. F or a ny N , the quasi-lin e a r system (5),(6) on c o efficients of the inte gr al F is Se m i-Hamiltonian system. 3. Riemann inv ariants Consider F as a function on the unite circle S 1 ⊂ C . It is a remark- able fact that the critical v alues of F = Λ N/ 2 z N + a N − 1 z N − 1 + · · · + Λ N/ 2 z − N are Riemann inv ariants of the system (5), (6). Indeed, let x 1 , . . . , x 2 N b e the set of distinct critical p o ints (hyperb olic region) x i ∈ S 1 ⊂ C . In tro duce r k = F ( x k ) , k = 1 , . . . , 2 N . F ro m the identit y ˙ F = F x ˙ x + F y ˙ y + F ϕ ˙ ϕ = 0 NEW S EMI–HAMIL TONI AN HI ERARCHY 5 it follows that r k are Riemann inv arian ts, b ecause hav ing F ϕ = 0 w e are left with F x cosϕ + F y sinϕ = 0 and th us the equation on r k follo ws: ( r k ) x + λ k ( r k ) y = 0 , λ k = tan ϕ k , x k = e iϕ k . Let us c hec k that r 1 , ..., r 2 N form a regular co ordinate system, that is Riemann inv arian ts are functionally indep enden t. W rite z F ′ = N Λ N/ 2 z N + ( n − 1) a n − 1 z N − 1 + · · · + a 1 z − a − 1 z − 1 − · · · − N Λ N/ 2 z − N . Notice tha t the critical p oints x 1 , . . . , x 2 N are ro ots of z F ′ and so by Vieta fo r m ula x 1 . . . x 2 N = − 1 . (7) F or con venie nce w e denote field v ariables as ( µ 1 , . . . , µ 2 N ) = (Λ N/ 2 , a N − 1 , . . . , a 0 , . . . , a 1 − N ) . Then ∂ r k ∂ µ s = ∂ F ∂ µ s ( x k ) + F ′ ( x k ) ∂ x k ∂ µ s = ∂ F ∂ µ s ( x k ) . Using (7) w e hav e det  ∂ r k ∂ µ s  = ( − 1) N det   x 2 N 1 + 1 x 2 N − 1 1 . . . x 1 . . . . . . . . . . . . x 2 N 2 N + 1 x 2 N − 1 2 N . . . x 2 N   . Splitting t he first column and again using (7) w e get det  ∂ r k ∂ µ s  = ( − 1) N ( x 1 . . . x 2 N − 1) W = ( − 1) N +1 2 W . Where W = Q i>j ( x i − x j ) is the V andermonde determinant. So we ha ve that ( µ ) ↔ ( r ) is a regular ch ange o f v ariable near ev ery p oint A in the strictly h yp erb olic region. Remark. The field v ariables Λ , a 0 are real, a s and a − s , s > 0 are complex conjugate. Therefore computation of  ∂ r k ∂ µ s  is equiv alent to the computation of ∂ ( r 1 ,...,r 2 N ) ∂ (Λ N/ 2 ,a 0 ,u 1 ,...,u N − 1 ,v 1 ,...,v N − 1 ) . 4. Conser v a tion la ws The aim of this section is to prov e that the system (5),(6) can b e written in the form of conserv ation law s. This system has man y ex- plicit conserv atio n laws . F o r example, the real part of (4) for k = N m ultiplied by Λ 1 − N 2 has the form of conserv ation low  u N − 1 Λ 1 − N 2  x +  v N − 1 Λ 1 − N 2  y = 0 . Another series of conserv a t ion la ws can b e obtained in t he following w ay . The identit y (4) a t k = 0 giv es us a conserv ation la w ( √ Λ u 1 ) x − ( √ Λ v 1 ) y = 0 . 6 MISHA BIA L Y AND A NDREY MIRONO V Similarly w e can get this conserv ation la w for the p ow ers of the in tegral. Namely F m generates t he conserv ations law ( √ Λ u ( m ) 1 ) x − ( √ Λ v ( m ) 1 ) y = 0 . where a ( m ) 1 = u ( m ) 1 + iv ( m ) 1 are corresp onding F ourier co efficien ts of F m . So, we hav e infinitely many explicit conserv a t io n lo ws. Remark ably they are in fact p olynomial in the field v ariables. Ho we v er w e do no t kno w if it is p ossible to get b y this metho d functionally indep enden t conserv ation lows. F or this reason w e pro ceed in a differen t w ay . W e sho w that is p ossible to generate functionally indep endent conserv a tion la ws b y in v ariant tori of the magnetic flow , in a similar w a y w e did it in [4] for geo desic flows . Imaginary par t of (4) for k = N m ultiplied b y Λ 1 − N 2 giv es us ΩΛ = 1 2 N  v N − 1 Λ (1 − N ) / 2  x − 1 2 N  u N − 1 Λ (1 − N ) / 2  y . (8) Let ϕ = f ( x, y ) b e a surface inv arian t under the flow. The in v ariance condition reads as f o llo ws: f x cos f √ Λ + f y sin f √ Λ + Λ x sin f 2Λ √ Λ − Λ y cos f 2Λ √ Λ + Ω = 0 or equiv a lently multiplying by Λ ( √ Λ sin f ) x − ( √ Λ cos f ) y + ΩΛ = 0 . (9) Substituting (8) into (9) w e get a conserv ation la w correspo nding to in v ariant surface ϕ = f ( x, y ):  √ Λ sin f + 1 2 N v N − 1 Λ (1 − N ) / 2  x −  √ Λ cos f + 1 2 N u N − 1 Λ (1 − N ) / 2  y = 0 . Let us show now that b y this metho d w e can get, in the strictly h y- p erb olic regio n, 2 N conserv at io n law s with functionally indep enden t G k , G k = Im[ √ Λ z k + 1 2 N a N − 1 Λ (1 − N ) / 2 ] , z k = e iϕ k . Let Γ b e t he domain of strict h yp erb olicit y of F ( F ′ has 2 N distinct ro ots on the unite circle). Prop osition 2. Denote by ˆ Γ the op en dense subset of Γ define d by the c o ndition that ± i is no t among the r o ots of F . L et A ∗ = ( µ ∗ 1 , . . . , µ ∗ 2 N ) b e any p oin t of ˆ Γ . Then in a neighb orh o o d of A ∗ ther e exist 2 N func- tional ly inde p e ndent c on servation laws. Corollary 1. Quasi-line ar system (5),(6) is Se m i-Hamiltonian at any p oint of Γ . NEW S EMI–HAMIL TONI AN HI ERARCHY 7 Pro of of Corollary . By Sev ennec theorem and the Prop osition 1 the system is Semi-Hamiltonia n at any p oin t of ˆ Γ. But it is dense, so the condition of Semi-Hamiltonicit y extends to the whole Γ. Applying Sev ennec theorem again w e hav e that also for A ∗ ∈ Γ \ ˆ Γ there are 2 N independen t conserv ations laws , but w e were not able to construct them explicitly . This prov es the Corollary .  Pro of of Prop osition. Define a neighborho o d of A ∗ with the help of Riemann inv arian ts r 1 , . . . , r 2 N . Recall r k are lo cal co ordinates. Let r ∗ k , k = 1 , . . . , 2 N are Riemann in v ariants of A ∗ . That is x ∗ k are critical po in ts of F ∗ = (Λ ∗ ) N/ 2 z N + a ∗ N − 1 z N − 1 + · · · + (Λ ∗ ) N/ 2 z − N . Notice that all critical p oin ts x ∗ k are non-degenerate since w e are in the strictly Hyp erb olic region. W e shall assume that o dd and ev en v a lues of the index k correspo nd to maxima and minima resp ective ly . L et ε > 0 b e a small n umber. Define a neigh b orho o d o f F ∗ R ε = { F : | r k − r ∗ k | < ε 10 , k = 1 , . . . , 2 N } . Here ε should b e ch osen smaller then 1 4 min | r ∗ k − r ∗ k +1 | . So for an y p olynomial from R ε it has critical p oin ts x k close to x ∗ k on the unite circle and critical v alues close to r ∗ k . Define now z k ( ε, µ ) as solutions of the equations on the unite circle: F ( z k ( ε, µ ) , µ ) = r k + ( − 1) k ε, z k lies in a neighborho o d of x k . Since w e assume that x 1 , x 3 , . . . are p oin ts of maxima and x 2 , x 4 , . . . are p oints o f minima then it follo ws that r k + ( − 1) k ε are regular v alues for an y p o lynomial F with co ef- ficien ts taken from R ε ( F should b e restricted to a neigh b orho o ds of x ∗ k ). Imp ortant for us is that z k ( ε, µ ) dep end smo o t hly on ε and µ . When w e shrink ε to zero then z k ( µ, ε ) → x k b y the construction. In the follo wing w e shall decrease ε , but ke eping a w ay from zero, in o rder to get the needed neigh b orho o d R ε . Define the follow ing functions on R ε : G k = Im[ √ Λ z k + 1 2 N a N − 1 Λ (1 − N ) / 2 ] = 1 2 i Im  √ Λ  z k − 1 z k  + 1 2 N Λ (1 − N ) / 2 ( a N − 1 − a 1 − N )  (where z k dep end on µ s implicitly). W e claim that one can c ho ose ε > 0 sufficien tly small in order to hav e det ∂ ( G 1 , . . . , G 2 N ) ∂ ( µ 1 , . . . , µ 2 N ) ( A ∗ ) 6 = 0 . This would imply the claim. W e ha v e ∂ G k ∂ µ l | µ = µ ∗ = − √ Λ ∗ 2 i  1 + 1 ( z ∗ k ) 2  ∂ F ∂ µ l | µ = µ ∗ F ′ ( µ ∗ , z k ) + R ∗ k l . 8 MISHA BIA L Y AND A NDREY MIRONO V Where R ∗ k l con tains all terms of explicit deriv ation of ∂ G k ∂ µ l . W e ha v e det  ∂ G ∂ µ  | µ = µ ∗ = √ Λ ∗ 2 i ! 2 N 2 N Y k =1  1 + 1 ( z ∗ k ) 2  2 N Y k =1 1 F ′ ( µ ∗ , z k ) × × det "  ∂ F ( z ∗ k , µ ) ∂ µ l  | µ = µ ∗ + R ∗ k l 1 1 + 1 ( z ∗ k ) 2 ! F ′ ( µ ∗ , z ∗ k ) # . Men tion that when ε → 0, 1 1+ 1 ( z ∗ k ) 2 → 1 1+ 1 ( x k ) 2 whic h is finite since x k 6 = i b y assumptions. Also z ∗ k ( ε ) → x ∗ k and so F ′ ( µ ∗ , z ∗ k ) → 0. Therefore the determinan t in brack ets when ε → 0 t ends to the determinant of the matrix  ∂ F ( x ∗ k , µ ) ∂ µ l  . This is exactly the determinant of the matrix considered in t he section 3. So it is equal − 2 W ( x ∗ 1 , . . . , x ∗ 2 N ) and do es not v a nish. But then it follo ws that for small ε > 0 det  ∂ G ∂ µ ( A )  6 = 0 in a neigh b or ho o d of A ∗ . Prop osition 2 and Theorem 1 are prov ed. 5. Discussion In this section w e discus some op en problems. In the case of n = 2 the equations (5),(6) o n functions Λ , u 0 , u 1 , v 1 ha ve the form ( √ Λ u 1 ) x − ( √ Λ v 1 ) y = 0 ,  u 1 √ Λ  x +  v 1 √ Λ  y = 0 , ( u 0 ) x + 2Λ x − v 1 2 √ Λ  u 1 √ Λ  y −  v 1 √ Λ  x ! = 0 , − ( u 0 ) y + 2Λ y − u 1 2 √ Λ  u 1 √ Λ  y −  v 1 √ Λ  x ! = 0 . In tro duce the new functions f , g : f = u 1 √ Λ , g = v 1 √ Λ . W e ha v e f x + g y = 0 , ( f Λ) x − ( g Λ) y = 0 , ( u 0 ) x + 2Λ x − 1 2 g ( f y − g x ) = 0 , − ( u 0 ) y + 2Λ y + 1 2 f ( f y − g x ) = 0 . NEW S EMI–HAMIL TONI AN HI ERARCHY 9 This system can b e written in the form A ( U ) U x + B ( U ) U y = 0 , where U = (Λ , u 0 , f , g ), A =     0 0 1 0 f 0 Λ 0 2 1 0 1 2 g 0 0 0 − 1 2 f     , B =     0 0 0 1 − g 0 0 − Λ 0 0 − 1 2 g 0 2 − 1 1 2 f 0     . The problem of existence of p erio dic solution is v ery interes ting. The systems of suc h form (no n-ev olution form) w ere considered in [10] from the p oin t of view of blo w- up analysis along c haracteristic curv es. It w ould b e v ery in teresting to apply these ideas to our system. Notice g f y = ( f g ) y − f g y = ( f g ) y + f f x = 0 , f g x = ( f g ) x − g f x = ( f g ) x + g g y . So w e hav e ( u 0 ) x + 2Λ x + 1 2 ( g g x − f f x ) − 1 2 ( f g ) y = 0 , − ( u 0 ) y + 2Λ y + 1 2 ( f f y − g g y ) − 1 2 ( f g ) x = 0 . Th us w e hav e explicit conserv ation la ws form for our system f x + g y = 0 , ( f Λ) x − ( g Λ) y = 0 , ( u 0 + 2Λ + 1 4 ( g 2 − f 2 )) x − ( 1 2 f g ) y = 0 , (10) ( − u 0 + 2Λ − 1 4 ( g 2 − f 2 )) y − ( 1 2 f g ) x = 0 . (11) Lust t w o conserv ation lows are v ery in teresting. Let us recall that a h yp erb olic diagonal system ( r i ) x + λ i ( r 1 , . . . , r n )( r i ) y = 0 , i = 1 , .., n is Semi-Hamiltonian if ∂ j Γ k k i = ∂ i Γ k k j , i 6 = j 6 = k , where Γ k k i = ∂ i λ k λ i − λ k . It can b e pro ved that a diagonal system is Semi-Hamilto nia n if and only if it can be written in some co ordinates as a system of conserv ation laws (see [2]). W e hav e the diagonal metric (see [1]) g ii = H 2 i , and Lame co efficien ts can b e found from t he o v er-determined system ∂ k ln H i = Γ i ik . 10 MISHA BIA L Y AND A NDREY MIRONO V By P avlo v–Tsarev t heorem [11] if t he Semi-Hamiltonian system has t wo conserv ation lows of the form F x + G y = 0 , F y + H x = 0 , then the corresp onding metric g ii is Egorov metric i.e. β ij = β j i , β ij = ∂ i H j H i , i 6 = j. In suc h a case the metric is po t ential: g ii = ∂ i a ( r ) for a function a . So it follo ws that our system for N = 2 is Ego r o v Semi-Hamiltonian system, since we ha v e tw o conserv ation low s (10),(11). By similar calculations one can chec k that for N = 3 our system is also Egorov system. It w ould b e in teresting to pro v e this fact for a rbitrary N . Another in teresting problem is to find P oisson brack et of h ydro dy- namical t ype for the system in the form of Dubrovin–No vik ov [1 2] or F erap on tov–Mokho v [13]. Reference s [1] S.P . Tsarev. 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Non-lo cal Hamiltonian op erator s of hydrody - namic type related to metrics of consta nt curv ature // Russian Math. Surv eys. 1990. V. 45 . N. 3 . P . 218-2 19. NEW S EMI–HAMIL TONI AN HI ERARCHY 11 M. Bial y, Scho ol of Ma thema tical Sciences, Ra ymond and Beverl y Sackler F acul ty of Exact Sciences, Tel A viv University, Israel E-mail addr ess : bial y@post .tau.a c.il A.E. Mir onov, Sobolev Institute of Ma thema tics and Labora tor y of Geometric Methods in Ma thema tical Physics, Mo scow St a te Uni- versity E-mail addr ess : miro nov@ma th.nsc .ru