nlin.SI 2009-12-31 0

Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions

A complete classification of integrable conservative hydrodynamic chains is presented. These hydrodynamic chains are written via special coordinates -- moments, such that right hand sides of these infinite component systems depend linearly on a discr…

Authors: Maxim V. Pavlov, Sergej A. Zykov

Classification of conserv ativ e h ydro dynamic c hains. Vlaso v t yp e kinetic equation, Riemann mapping and the metho d of symmetric h ydro dynamic reduction s . Maxim V. P avlo v 1 , Sergei Zyko v 2 , 3 1 Department of Mathemat ical Ph ysics, P .N. Leb edev Physical Institut e of Russian Academ y of Sciences, Mosco w , Leninskij Prosp ekt, 53 2 Department of Mathemat ical Ph ysics, SISSA, T rieste, Ita ly 3 Institut e of Metal P h ysics, Ur al bra nc h of RAS, Ek ateri n b u r g, Russia Abstract A complete classification of in tegrable conserv ativ e hydro dynamic c hains is pr e- sen ted. These h ydr odyn amic c hains are wr itten via sp ecial co ordinates – momen ts, suc h that righ t hand sides of these infinite comp onen t sy s tems dep end linearly on a d iscrete indep end ent v ariable k . All v ariable co efficie nts of these hydrod ynamic c hains can b e expressed via mo dular form s with resp ect to moment A 0 , via hy- p ergeometric fu nctions w ith resp ect to moment A 1 ; they dep end p olynomia lly on momen t A 2 and linearly on all other higher momen ts A k . A disp ersionless Lax rep- resen tation is found. Corresp onding collisionless Boltzmann (Vlaso v lik e kinetic) equation is deriv ed. A Riemann mapping is constru cte d . A generating fu nctio n of conserv ation la ws and commuting flows is p resen ted. Con tents 1 In tro duction 3 2 Zakharo v’s momen t decomp osition 4 3 Canonical v ariables 13 4 General solution in the “triangular” c ase 14 5 Conserv ativ e h ydro dynam ic c hains 18 6 Egoro v’s case 20 1 7 Conclusion 21 8 App endix 21 References 23 h ydro dynamic chains , Riemann in v ariants, symmetric h ydro dynamic t yp e systems . MSC: 35L40, 35L65, 37K10; P A CS: 02.30.J, 11.10.E. 2 1 In tro du ction In past years (2003 up to now) significan t results w ere obtained in the theory of integrable h ydro dynamic c hains (se e [ 13 ], [ 14 ], [ 6 ], [ 16 ], [ 19 ]). A firs t in tegrable hy dro dynamic c hain A k t = A k +1 x + k A k − 1 A 0 x , k = 0 , 1 , 2 , ... (1) w as deriv ed by D. Benney (see [ 2 ]) in 19 7 3. An integrabilit y of the Benney h ydro dynamic c hain can be illustrated b y an existence of a generating function of conserv a tion la ws (see [ 2 ], [ 11 ]) p t =  p 2 2 + A 0  x , (2) where a generating function of conserv ation law dens ities is giv en b y p = λ − H 0 λ − H 1 λ 2 − H 2 λ 2 − ..., (3) whose all conserv ation law densities are p olynomial functions with resp ect to moments A k , i.e. H 0 = A 0 , H 1 = A 1 , H 2 = A 2 + ( A 0 ) 2 , H 3 = A 3 + A 0 A 1 , ... It means that h ydro dynamic c hain ( 1 ) also can be written in the conserv ative form (see, for instance, [ 17 ]) ∂ t H 0 = ∂ x H 1 , ∂ t H k = H k +1 − 1 2 k − 1 X m =0 H m H k − 1 − m ! x , k = 1 , 2 , ... (4) W e are in terested in a de scription of integrable hydrodynamic chains w ritt en in the form (cf. ( 1 )) A k t = f 1 A k +1 x + f 0 A k x + A k +1 ( s 0 A 0 x + s 1 A 1 x ) + A k ( r 0 A 0 x + r 1 A 1 x ) (5) + k [ A k +1 ( w 0 A 0 x + w 1 A 1 x + w 2 A 2 x ) + A k ( v 0 A 0 x + v 1 A 1 x + v 2 A 2 x ) + A k − 1 ( u 0 A 0 x + u 1 A 1 x + u 2 A 2 x )] , where co effic ients f i , s j , r k dep end on first t w o momen ts A 0 and A 1 only , while all other co efficien ts w m , v n , u p dep end just on first three momen ts A 0 , A 1 and A 2 . Recen tly , tw o p a r ticular cases of hy dro dynamic c hains ( 5 ) w ere completely in v esti- gated. The Hamiltonian h ydro dynamic c hains (here H 1 ,k ≡ ∂ H 1 /∂ A k , k = 0 , 1) A k t = ( α + β ) H 1 , 1 A k +1 x + β H 1 , 0 A k x + [ α ( k + 1) + 2 β ] A k +1 ( H 1 , 1 ) x + ( αk + 2 β ) A k ( H 1 , 0 ) x are asso ciated with the Kup ershmidt P o isson brac kets (see [ 5 ] and [ 10 ]); while the Hamil- tonian h ydro dynamic c ha ins (here H 2 ,k ≡ ∂ H 2 /∂ A k , k = 0 , 1 , 2 ) A k t = 2 H 2 , 2 A k +1 x + H 2 , 1 A k x + ( k + 2) A k +1 ( H 2 , 2 ) x + ( k + 1) A k ( H 2 , 1 ) x + k A k − 1 ( H 2 , 0 ) x are asso ciated w ith the Kup ershmidt–Manin Poiss o n brack et (see the second part in [ 6 ], [ 8 ] and [ 10 ]). These hydrodynamic c hains are inte gr able if and only if all comp onen ts of corresponding Haan tjes tensors v anish. It means that the corresp onding Hamiltonian densities H 1 ( A 0 , A 1 ) a nd H 2 ( A 0 , A 1 , A 2 ) cannot b e arbitra ry . A full list of admissible expressions is giv en in [ 5 ] and [ 6 ], resp ectiv ely . In a general case, the co efficie nt f 1 in ( 5 ) is reducible to the unit y , the co efficien t f 0 can b e eliminate b y an appropriate change of momen ts A k ; while all other co effic ients can 3 b e simplified in the in tegrable case only . F ollowin g the approach based on an existence of first three conserv atio n laws and v a nishing of the Haan tjes tensor (see the first part in [ 6 ]), one can extract the inte gr able case A k t = A k +1 x − k [( A k +1 + u 0 A k + u − 1 A k − 1 )[ln( A 2 + σ )] x − A k ( u 0 ) x − A k − 1 ( u − 1 ) x ] , (6) where f unctions u 0 , u − 1 , σ satisfy to an ov erdetermined system in an in v olution (s ee ( 54 )). The same result can b e obtained b y the metho d of hy dro dynamic reductions es tablished b y J. Gibb ons and S.P . Tsarev in [ 9 ] and dev elop ed b y E.V. F erap on tov a nd K.R. Khus - n utdinov a in [ 4 ]. In this pa p er, w e utilize the concept of the so-called symmetric h y- dro dynamic r eductions (see [ 16 ], [ 18 ]). In this case, an existence of a Riemann mapping λ ( q , A 0 , A 1 , A 2 , ... ) connecting t he Vlaso v t yp e kinetic equation (see [ 7 ], [ 12 ], [ 21 ]) with h ydro dynamic c hain ( 6 ) leads to an o ve rdetermined system in in volution ( 3 6 ), ( 37 ), ( 38 ), ( 39 ), ( 4 0 ), ( 41 ), whose general solutio n can b e parameterized b y hypergeometric func- tions. Then a generating function of conserv atio n law s can b e found in quadratures. Th us, an infinite s eries of conserv a tion la ws densities H k ( A 0 , A 1 , ..., A k ) allo ws to rewrite h ydro dynamic chain ( 6 ) in the conserv a t iv e form 1 (cf. ( 4 )) ∂ t H k = ∂ x F k ( H 0 , H 1 , ..., H k +1 ) , k = 0 , 1 , 2 , ... (7) W e pro v e that its first t w o conserv ation laws coincide with first tw o conserv ation laws found in [ 6 ]. In the general case, E.V. F erap on tov and D.G. Marshall f ound that F 2 ( H 0 , H 1 , H 2 ) = ln H 2 + G ( H 0 , H 1 ) and functions F 0 ( H 0 , H 1 ) , G ( H 0 , H 1 ) satisfy to another ov erdetermined system in in volution (see [ 6 ]). Moreo v er, we prov e that this system in inv olutio n ( 55 ) is equiv alen t to system in in v olution ( 36 ), ( 37 ), ( 38 ), ( 39 ), ( 4 0 ), ( 41 ). Th us, a complete classification of inte gr able c onservative hydr o dynamic chains ( 7 ) is giv en in this pap er. This pap er is organized in the fo llowing w ay . In Section 2, symmetric 2 N comp onen t h ydro dynamic reductions are extracted b y virtue of Zakharov’s momen t decomp osition (see [ 18 ], [ 2 1 ]). Suc h 2 N comp onen t h ydro dynamic ty p e systems con tain N comp onen t symmetric sub-systems, whic h are still h ydro dynamic reductions. These N component h ydro dynamic t yp e systems imply to the Vlasov t yp e kinetic equation. W e show that an existance o f the Riemann mapping conne cting this Vlaso v t yp e kinetic equation with h ydro dynamic c hain ( 6 ) allo ws to select a ll inte g rable hydrodynamic chains. In Section 3, canonical co ordinates in a moment space are in tro duced. Then a further in ves tig a tion sim- plifies. In Section 4, a sp ecial “triangular” case is completely in tegrated. Three v a riable co efficien ts in ( 6 ) can b e parameterized by solutions of the so-called Halphen–Da rboux system (see [ 1 ]). In Section 5, a generating function of conserv atio n law s is f o und. In Conclusion, a generalization of the approach pre sente d in this pap er is discus sed. 2 Zakharo v’s momen t decomp os ition The momen t decompo sition approac h de veloped in [ 18 ] (see also [ 16 ]) is based on a conce pt of an existence of symmetric h ydro dynamic ty p e systems a i t = ∂ x F ( a ; p ) | p = a i , i = 1 , 2 , ..., N , 1 A problem of a description of in tegrable h ydr odynamic c ha ins ( 7 ) was formulated in [ 1 3 ]. A particular and impor tan t E gorov’s case F 0 ( H 0 , H 1 ) ≡ H 1 was in vestigated in [ 14 ]. 4 whic h are nothing but hydrodynamic reductions of t he hydrodynamic c hains, where in all kno wn cases b efore (se e, for instance , [ 5 ], [ 16 ], [ 17 ], [ 18 ], [ 19 ], [ 15 ]) eac h corresp onding momen t A k dep ends on N functions of a single v ariable 2 , i.e. A k = N X m =0 f mk ( a m ) , k = 1 , 2 , 3 , ... Another momen t decomp osition (intro duced by V.E. Zakharo v, see [ 21 ]) A k = N X m =0 ( a m ) k b m (8) also is applicable in all these know n cases (see [ 18 ]). Benney hyd ro dynamic c hain ( 1 ) under this momen t decomp osition reduces to the 2 N comp onen t h ydro dynamic t yp e system a i t =  ( a i ) 2 2 + A 0  x , b i t = ( a i b i ) x , whic h p ossesses a formal reduction to the N comp onen t case (cf. ( 2 )) a i t =  ( a i ) 2 2 + A 0  x , if all field v ariables b k v anish, and A 0 b ecomes a function of all rest field v a r ia bles a n only . Let us replace a i b y q ( x, t, λ ), where λ is a parameter. It means a i = q ( x, t, ξ i ), where ξ i are arbitrary constan ts. Then ( 2 ) q t = q q x + A 0 x b y a semi-ho dograph transformatio n q ( x, t, λ ) → λ ( x, t, q ) reduces to the linear equation λ t = q λ x − λ q A 0 x , whic h is kno wn as the Vlaso v kinetic equation (see [ 21 ]; or the collisionless Boltzmann equation, see [ 7 ]). Supp ose λ ( x, t, q ) is a function λ ( q , A 0 , A 1 , ... ), where a ll momen t s A k ( x, t ) satisfy Benney hyd ro dynamic c hain ( 1 ). Since w e supp ose a ll momen ts A k are indep ende nt, one can obtain an infinite series of equations ∂ k λ = q − k ∂ 0 λ, k = 0 , 1 , 2 , ..., (9) where ∂ k ≡ ∂ /∂ A k , and ( ∂ q ≡ ∂ /∂ q ) ∂ 0 λ = q − ∞ X m =0 mA m − 1 q m ! − 1 ∂ q λ. (10) 2 let us emphasize that N is an a rbitrary natura l num ber . 5 A solution of ( 9 ) is giv en by 3 λ = B 1 ( q ) ∞ X m =0 A m q m +1 + B 2 ( q ) , (11) where B 1 ( q ) and B 2 ( q ) ar e not determined y et functions. Ho w eve r, a substitution ( 11 ) in to ( 10 ) yie lds B 1 ( q ) = 1 and B 2 ( q ) = q . Then ( 11 ) b ecomes nothing else but an in ve rse series to ( 3 ). Th us, we conclude that a symptom of an inte gr abilit y of hydrodynamic c hains is an existence of a Riemann mapping λ ( q , A 0 , A 1 , ... ) connecting with Vlaso v t yp e kinetic equation (see b elo w). In this pap er, w e utilize this pro perty for a class ification of in tegrable h ydro dynamic c hains. This momen t decomp osition approac h can b e extended on a wide class of h ydro dy- namic c hains (cf. ( 5 )) A k t = K X n =0 f n A k + n x + M X m =0 K X n =0 A k + n s nm + k K X n = − 1 A k + n w nm ! A m x (12) where K and M are ar bit r a ry natural n umbers, all functions f i , s j k , w lp dep end on first M + 1 moments (if K = 1 , M = 2 , s 0 , 2 = 0 , s 1 , 2 = 0 a nd f 0 , f 1 , s 0 , 0 , s 0 , 1 , s 1 , 0 , s 1 , 1 dep end just on tw o first momen ts A 0 , A 1 , these hy dro dynamic c hains reduce to ( 5 )). Indeed, ( 12 ) reduces to N separate expressions for eac h index i (let remind that N is arbitrary) ( a i ) k b i t + k ( a i ) k − 1 b i a i t = K X n =0 f n [( a i ) k + n b i x + ( k + n )( a i ) k + n − 1 b i a i x ] + M X m =0 K X n =0 ( a i ) k + n b i s nm + k K X n = − 1 ( a i ) k + n b i w nm ! A m x , due to a subs titut io n ( 8 ) in mo m ents e quipp e d by the i n dex k only . Moreov er, the a bov e N expres sions (due to their linear explicit dep enden ce on a discrete v ariable k ) can b e split on t w o parts b i t = K X n =0 f n [( a i ) n b i x + n ( a i ) n − 1 b i a i x ] + M X m =0 K X n =0 ( a i ) n b i s nm A m x , a i t = K X n =0 f n · ( a i ) n a i x + M X m =0 K X n = − 1 ( a i ) n +1 w nm A m x . (13) As in the previous case, this 2 N comp onen t h ydro dynamic t yp e system p osse sses N comp onen t reduction ( 13 ), where all momen ts A m and all v ariable co efficien ts w ik dep end on N field v ar ia bles a n only . 3 It is well known th a t a general solution of the above linear eq uation is parameter iz e d by one arbitra ry function o f a s ingle v a riable ˜ λ ( λ ). How ever, in this approach, an e xistence of any solution is essential. 6 Our main observ ation success fully utilized in this a pproac h is that inte gr able hy- dro dynamic c hain ( 12 ) is asso ciated with the auxiliary equation q t = K X n =0 f n q n q x + M X m =0 K X n = − 1 q n +1 w nm A m x , (14) whic h obtains due to a formal replacemen t a i → q in ( 1 3 ). It me ans, that eq ua t io n ( 14 ) is compatible with hy dro dynamic chain ( 12 ), where f unction q m ust dep end on mome nts A k ( x, t ) and t he parameter λ . The sem i- hodo g raph transformatio n q ( x, t, λ ) ↔ λ ( x, t, q ) reduces ( 14 ) to the linear equation λ t − K X n =0 f n q n λ x + M X m =0 K X n = − 1 q n +1 w nm A m x λ q = 0 , (15) whic h w e call the Vlasov t yp e kinetic equation (cf. [ 12 ]). The f unction λ ( x, t, q ) de p ends o n x, t implicitly via an explicit dep endence on moments A k ( x, t ). W e shall call hydrodynamic c hain ( 12 ) inte gr able if a Riemann mapping λ ( q , A 0 , A 1 , ... ) connecting Vlaso v t yp e kinetic equation ( 15 ) with ( 12 ) exists. Examples : Hamiltonian h ydro dynamic c hains asso ciated with the Kup ershmidt– Manin P oisson brack et (se e [ 8 ]; h n ≡ ∂ h/∂ A n , h nm ≡ ∂ 2 h/∂ A n ∂ A m ) A k t = M − 1 X n =0 ( n + 1) h n +1 A k + n x + M X m =0 M − 1 X n =0 ( n + 1) A k + n h n +1 ,m + k M − 1 X n = − 1 A k + n h n +1 ,m ! A m x are connected with the Vlaso v type kinetic equation λ t − M − 1 X n =0 ( n + 1) h n +1 q n λ x + M X m =0 M − 1 X n = − 1 q n +1 h n +1 ,m A m x λ q = 0 , where the Hamiltonian is giv en by H = R h ( A 0 , A 1 , ..., A M ) dx and (see ( 12 )) f n = ( n + 1 ) h n +1 , K = M − 1 , s nm = ( n + 1 ) h n +1 ,m , w nm = h n +1 ,m . Hamiltonian h ydro dynamic c hains asso ciated with the Kup ersh midt P oisson brack ets (see [ 5 ] and [ 10 ]) A k t = M X n =0 ( αn + β ) h n A k + n x + M X m =0 M X n =0 ( αn + 2 β ) A k + n h nm + αk M X n =0 A k + n h nm ! A m x are connected with the Vlaso v type kinetic equation λ t − M X n =0 ( αn + β ) h n q n λ x + α M X m =0 M X n =0 q n +1 h nm A m x λ q = 0 where the Hamiltonian is giv en by H = R h ( A 0 , A 1 , ..., A M ) dx and (see ( 12 )) f n = ( αn + β ) h n , K = M , s nm = ( αn + 2 β ) h nm , w nm = α h nm , w − 1 ,m = 0 . 7 Without loss of generality and for simplicit y let us consider a h ydro dynamic c hain written in the form A k t = 1 X n =0 f n A k + n x + M X m =0 1 X n =0 A k + n s nm + k 1 X n = − 1 A k + n w nm ! A m x . (16) If this h ydro dynamic chain is in tegrable, then also all its higher commuting flo ws b elong to the general class determined b y ( 12 ) with appro priate choices nat ura l num b ers K a nd M . Lemma : The c o e ffi c ient f 1 c an b e fixe d to the unity by the invertible p oint tr ansfor- mation ˜ A k = ( f 1 ) k A k , then the c o efficient f 0 c an b e elimin ate d by the i n vertible p oint tr ansforma tion 4 ˜ A k = k X m =0  k m  ( f 0 ) k − m A m , wher e  k m  is a binomial c o efficient . I f ∂ M f 1 = 0 and ∂ M f 0 = 0, then h ydr o dynamic chain ( 16 ) r e duc es to the c anonic al form A k t = A k +1 x + M X m =0 1 X n =0 A k + n s nm + k 1 X n = − 1 A k + n w nm ! A m x ; (17) if ∂ M f 1 6 = 0 or ∂ M f 0 6 = 0, then ( 16 ) r e duc es to ( 1 7 ), but M r eplac es by M + 1, c orr e- sp ond ingly . Pro of : Hydro dynamic c hain ( 16 ) is asso ciated with the reduced v ersion of ( 14 ) q t = ( f 1 q + f 0 ) q x + M X m =0 1 X n = − 1 q n +1 w nm A m x . Th us, the transformation ˜ q = f 1 q + f 0 reduces the ab ov e eq uatio n to a more simple case with f 1 = 1 and f 0 = 0. Corresp onding Zakharov’s momen t decomp osition ( 8 ) transforms accordingly ˜ A k = N X m =0 ( f 1 a m + f 0 ) k b m . (18) This is no thing else but a linear combination of af oremen tioned transformations. This p oin t transformation ˜ A 0 = A 0 , ˜ A 1 = f 1 A 0 + ( f 0 ) 2 , ˜ A 2 = ( f 1 ) 2 A 2 + 2 f 0 f 1 A 1 + ( f 0 ) 3 , ..., ˜ A M = ( f 1 ) M A M + M f 0 ( f 1 ) M − 1 A M − 1 + ... + ( f 0 ) M +1 , ... cannot b e inv erted to a sim- ilar fo rm due to complexit y of functions f 0 ( A 0 , A 1 , ..., A M ) and f 1 ( A 0 , A 1 , ..., A M ). Just higher momen ts A M + k ( ˜ A 0 , ˜ A 1 , ..., ˜ A M + k ) b ecame line ar expressions with resp ect to higher momen ts ˜ A M +1 , ˜ A M +2 , ... On the other hand, 2 N comp onen t hydrodynamic t yp e sys tem ( 13 ) b i t = ( f 1 a i + f 0 ) b i x + f 1 b i a i x + b i M X m =0 1 X n =0 ( a i ) n s nm A m x , a i t = ( f 1 a i + f 0 ) a i x + M X m =0 1 X n = − 1 ( a i ) n +1 w nm A m x 4 Similar transfo rmations pr eserving the K upershmidt–Manin Poisson bracket were considered in [ 6 ], but with c onstant co efficien ts f 0 and f 1 . 8 under the aforemen tioned tra nsformation c i = f 1 a i + f 0 reduces to b i t = c i b i x + b i c i x + b i M X m =0 1 X n =0 ( c i ) n ¯ s nm A m x , c i t = c i c i x + M +1 X m =0 1 X n = − 1 ( c i ) n +1 ¯ w nm A m x , (19) where ¯ s 1 m = s 1 m f 1 − ∂ m ln f 1 , ¯ s 0 m = s 0 m − f 0 f 1 s 1 m + f 0 ∂ m ln f 1 − ∂ m f 0 , ¯ w 1 m = w 1 m f 1 − ∂ m ln f 1 , ¯ w 0 ,M +1 = ∂ M f 1 , ¯ w − 1 ,M +1 = f 1 ∂ M f 0 − f 0 ∂ M f 1 , ¯ w 0 m = w 0 m + (1 − δ m, 0 ) ∂ m − 1 f 1 − ∂ m f 0 + 2 f 0 ∂ m ln f 1 − 2 f 0 w 1 m f 1 + M X p =0 1 X n =0 A p + n s nm + 1 X n = − 1 pA p + n w nm ! ∂ p ln f 1 , ¯ w − 1 m = (1 − δ m, 0 )( f 1 ∂ m − 1 f 0 − f 0 ∂ m − 1 f 1 )+ f 0 ∂ m f 0 − ( f 0 ) 2 ∂ m ln f 1 + ( f 0 ) 2 w 1 m f 1 − f 0 w 0 m + f 1 w − 1 m − f 0 M X p =0 1 X n =0 A p + n s nm + 1 X n = − 1 pA p + n w nm ! ∂ p ln f 1 + M X p =0 1 X n =0 A p + n s nm + 1 X n = − 1 pA p + n w nm ! ∂ p f 0 . Due to ( 18 ), ( 19 ) can b e written in the final form b i t = c i b i x + b i c i x + b i M X m =0 1 X n =0 ( c i ) n ˜ s nm ˜ A m x , c i t = c i c i x + M +1 X m =0 1 X n = − 1 ( c i ) n +1 ˜ w nm ˜ A m x , where co efficien ts ˜ s ij and ˜ w k l are expres sed v ia new momen ts ˜ A n . This is nothing else but a h ydro dynamic reduction of hyd ro dynamic chain ( 17 ) ˜ A k t = ˜ A k +1 x + M +1 X m =0 1 X n =0 ˜ A k + n ˜ s nm + k 1 X n = − 1 ˜ A k + n ˜ w nm ! ˜ A m x . Th us, Lemma is prov ed. Example : The remark able Kup ershmidt hy dro dynamic chain (see [ 10 ], [ 15 ]) A k t = A k +1 x + β A 0 A k x + ( k + γ ) A k A 0 x , k = 0 , 1 , ... reduces to canonical form ( 17 ) ˜ A k t = ˜ A k +1 x + [(1 − β ) k + γ − β ] ˜ A k ˜ A 0 x + β k ˜ A k − 1  ˜ A 1 + γ − β − 1 2 ( ˜ A 0 ) 2  x , k = 0 , 1 , ... Th us, w e can in v estigate an in tegra bilit y of h ydro dynamic c hain ( 16 ) written in a more con v enien t form ( 17 ) instead ( 16 ). In suc h a case, ( 14 ) reduc es to q t = q q x + M X m =0 1 X n = − 1 q n +1 w nm A m x . (20) 9 A consistency of ( 20 ) with ( 17 ) leads to an infinite set of equations ∂ M + k q = q − k ∂ M q , k = 0 , 1 , 2 , ... (21) and M + 1 equations ( m = 0 , 1 , 2 , ..., M ) reduces b y virtue of ( 21 ) to (1 − δ m, 0 ) ∂ m − 1 q + M − 1 X k =0 1 X n =0 s nm A k + n + k 1 X n = − 1 w nm A k + n ! ∂ k q +Σ m · ∂ M q = 1 X n = − 1 w nm q n +1 + q ∂ m q , where M + 1 infinite sums are determined b y Σ m = ∞ X k =0 1 X n =0 s nm A M + n + k + ( M + k ) 1 X n = − 1 w nm A M + n + k ! 1 q k . (22) Ho w eve r, all these infinite su ms can b e reduced to a sole sum only (see b elo w). A cons is- tency of the Vlaso v type kinetic equation (cf. ( 15 ), see also ( 20 )) λ t = q λ x − M X m =0 1 X n = − 1 q n +1 w nm A m x λ q with ( 17 ) yields an infinite set of equations (whic h is equiv alen t to ( 21 ) due to the tra ns- formation ∂ k q = − ∂ k λ/∂ q λ ) ∂ M + k λ = q − k ∂ M λ, k = 0 , 1 , 2 , ..., whose solution is giv en by λ = B 1 ( q , A 0 , A 1 , ..., A M − 1 )[Σ + B 2 ( q , A 0 , A 1 , ..., A M − 1 )] , where Σ = ∞ X p =0 A p q p +1 , (23) while other M + 1 equations (1 − δ m, 0 ) ∂ m − 1 λ + M − 1 X k =0 1 X n =0 s nm A k + n + k 1 X n = − 1 w nm A k + n ! ∂ k λ +Σ m · ∂ M λ + 1 X n = − 1 w nm q n +1 ∂ q λ = q ∂ m λ, reduce to a linear system 5 G m ( q , A 0 , A 1 , ..., A M )Σ + Q m ( q , A 0 , A 1 , ..., A M ) = 0 due to ( 22 ) expresse s via ( 23 ) Σ m = 1 X n =0 s nm q M + n +1 − 1 X n = − 1 ( n + 1) w nm q M + n +1 ! Σ − 1 X n = − 1 w nm q M + n +2 ∂ q Σ + 1 X n = − 1 w nm M + n − 1 X p =0 n − p q p − n − M A p − 1 X n =0 s nm M + n − 1 X p =0 A p q p − n − M . 5 This linear system does not contain a par t pro p ortional to ∂ q Σ, be c ause its cor respo nding co efficien t v anis hes authomatically . 10 Since b oth c o efficien ts G m and Q m m ust v anish indep enden tly , a full 2 M + 2 component system can b e split on the t w o M + 1 comp onen t sub-systems of linear eq uatio ns q ∂ m ln B 1 + ( δ m, 0 − 1) ∂ m − 1 ln B 1 − M − 1 X k =0 1 X n =0 s nm A k + n + k 1 X n = − 1 w nm A k + n ! ∂ k ln B 1 − 1 X n = − 1 w nm q n +1 ∂ q ln B 1 = 1 X n =0 [ s nm − ( n + 1) w nm ] q n (24) and q ∂ m B 2 +( δ m, 0 − 1) ∂ m − 1 B 2 − M − 1 X k =0 1 X n =0 s nm A k + n + k 1 X n = − 1 w nm A k + n ! ∂ k B 2 − 1 X n = − 1 w nm q n +1 ∂ q B 2 + 1 X n =0 [ s nm − ( n + 1) w nm ] q n B 2 = A 0 ( s 1 m − w 1 m ) − δ m, 0 . All deriv a tiv es of functions B 1 and B 2 can b e expressed f r o m this linear system with v ariable co efficien t s s ij and w k l . A consistency o f these deriv ativ es leads to an o v erdetermined system in partial deriv ative s on s ij and w k l with r esp ect to momen ts A 0 , A 1 , ..., A M − 1 and q , while a dependence on the hig hes t momen t A M can b e found b y a straigh tforward differen tiatio n of linear system ( 24 ) written in the matrix 6 form       q + ∗ ∗ ... ∗ − w 1 , 1 q 2 − w 0 , 1 q + ∗ ∗ q + ∗ ... ∗ − w 1 , 2 q 2 − w 0 , 2 q + ∗ ... ... ... ... ... ∗ ∗ ... q + ∗ − w 1 ,M − 1 q 2 − w 0 ,M − 1 q + ∗ ∗ ∗ ... ∗ − w 1 ,M q 2 − w 0 ,M q + ∗             ∂ 0 ln B 1 ∂ 1 ln B 1 ... ∂ M − 1 ln B 1 ∂ q ln B 1       =       ˜ w 0 q + ∗ ˜ w 1 q + ∗ ... ˜ w M − 1 q + ∗ ˜ w M q + ∗       , where ˜ w k = s 1 k − 2 w 1 k and the mark “ ∗ ” means elemen ts independen t on q . Indeed, suc h a differen tial consequence is g iv en b y       ∗ ∗ ... ∗ − w ′ 1 , 1 q 2 − w ′ 0 , 1 q + ∗ ∗ ∗ ... ∗ − w ′ 1 , 2 q 2 − w ′ 0 , 2 q + ∗ ... ... ... ... ... ∗ ∗ ... ∗ − w ′ 1 ,M − 1 q 2 − w ′ 0 ,M − 1 q + ∗ ∗ ∗ ... ∗ − w ′ 1 ,M q 2 − w ′ 0 ,M q + ∗             ∂ 0 ln B 1 ∂ 1 ln B 1 ... ∂ M − 1 ln B 1 ∂ q ln B 1       =       ˜ w ′ 0 q + ∗ ˜ w ′ 1 q + ∗ ... ˜ w ′ M − 1 q + ∗ ˜ w ′ M q + ∗       , where the mark “ ′ ” means a partial deriv a tiv e with respect to the m o men t A M . Lemma : A ny r ow of the ab ove line ar system  ∗ , ∗ , ..., ∗ , − w ′ 1 ,m q 2 − w ′ 0 ,m q + ∗ , ˜ w ′ m q + ∗  (25) is pr op ortiona l to the last r ow fr om the pr evio us line ar system  ∗ , ∗ , ..., ∗ , − w 1 ,M q 2 − w 0 ,M q + ∗ , ˜ w M q + ∗  . (26) 6 a determina n t of this ( M + 1) × ( M + 1) matrix is a p olynomial of degree M + 2 with resp ect to q , except some sp ecial ca ses, like w 1 ,M = 0, which should b e considered separ ately . 11 Pro of : Indeed, let us consider the linear system         q + ∗ ∗ ... ∗ − w 1 , 1 q 2 − w 0 , 1 q + ∗ − ˜ w 0 q − ∗ ∗ q + ∗ ... ∗ − w 1 , 2 q 2 − w 0 , 2 q + ∗ − ˜ w 1 q − ∗ ... ... ... ... ... ... ∗ ∗ ... q + ∗ − w 1 ,M − 1 q 2 − w 0 ,M − 1 q + ∗ − ˜ w M − 1 q − ∗ ∗ ∗ ... ∗ − w 1 ,M q 2 − w 0 ,M q + ∗ − ˜ w M q − ∗ ∗ ∗ ... ∗ − w ′ 1 ,m q 2 − w ′ 0 ,m q + ∗ − ˜ w ′ m q − ∗                 ∂ 0 B 1 ∂ 1 B 1 ... ∂ M − 1 B 1 ∂ q B 1 B 1         = 0 , determined b y the ( M + 2) × ( M + 2) mat r ix incorp orating all rows of the orig inal linear system and an y row from its differen tial consequence . A de terminant of this matrix equals zero for non trivial solutions B 1 . Th us, the last ro w (see ( 25 )) m ust b e a linear com binatio n of all other ro ws. Ho we ver, most of them ( m = 0 , 1 , ..., M − 1) contain an eleme nt q + ∗ , whic h do es no t exist in first M entries of this last row. Thus , the last ro w cannot b e expresse d via these highe r flows except the ro w with the n um b er M (see ( 26 )). It means, that all elemen ts of these two ro ws m ust b e pro portio nal to eac h other. Lemma is pro v ed. Th us, the full set o f equations is giv en by ( n = 0 , 1 , ..., M − 2) β ′ M β M = δ ′ M δ M = ( ǫ M − 1 M ) ′ 1 + ǫ M − 1 M = ( ǫ n M ) ′ ǫ n M , (27) β ′ m β M = δ ′ m δ M = ( ǫ M − 1 m ) ′ 1 + ǫ M − 1 M = ( ǫ n m ) ′ ǫ n M , m = 0 , 1 , ..., M − 1 , (28) where β m = s 0 ,m − w 0 ,m + ( s 1 ,m − 2 w 1 ,m ) q , ǫ n m = nw − 1 ,m A n − 1 + ( s 0 ,m + nw 0 ,m ) A n + ( s 1 ,m + nw 1 ,m ) A n +1 , δ m = w − 1 ,m + w 0 ,m q + w 1 ,m q 2 . All these equations can b e subs equen tly in tegrat ed. Indeed, t he first ra t io in ( 27 ) β ′ M β M = δ ′ M δ M is not hing else but a cubic p olynomial with resp ect to q . Since q is arbitrar y , all four co efficien ts m ust v anish indep enden tly . A general solution of corresp o nding four ordinary differen tial equations (with respect to A M only) is giv en by s 0 ,M = ( r 0 − M + 1) w 1 ,M , s 1 ,M = ( r 1 − M + 1) w 1 ,M , w 0 ,M = u 0 w 1 ,M , w − 1 ,M = u − 1 w 1 ,M , (29) where functions r 0 , u 0 , r 1 , u − 1 dep end on first M momen ts A 0 , A 1 , ..., A M − 1 . An in tegration of the second ratio in ( 27 ) δ ′ M δ M = ( ǫ M − 1 M ) ′ 1 + ǫ M − 1 M 12 leads to w 1 ,M = − 1 σ + A M r 1 , (30) where the function σ dep ends on first M momen ts A 0 , A 1 . . . , A M − 1 . An integration of the first ratio in ( 28 ) β ′ m β M = δ ′ m δ M leads to w − 1 ,m = u − 1 w 1 ,m + γ − 1 ,m , w 0 ,m = u 0 w 1 ,m + γ 0 ,m , (31) s 1 ,m = ( r 1 − M + 1) w 1 ,m + ρ 1 ,m , s 0 ,m = ( r 0 − M + 1) w 1 ,m + ρ 0 ,m , where functions γ − 1 ,m , γ 0 ,m , ρ 0 ,m , ρ 1 ,m dep end on first M momen ts A 0 , A 1 , ..., A M − 1 . An in tegration o f t he second ra tio in ( 28 ) δ ′ m δ M = ( ǫ M − 1 m ) ′ 1 + ǫ M − 1 M leads to w 1 ,m = ω m − A M ρ 1 ,m σ + A M r 1 , (32) where functions ω m dep end on first M moments A 0 , A 1 , ..., A M − 1 . It is easy to see, that all other ratios in ( 27 ) and ( 28 ) are fulfilled b y virtue of ( 29 ), ( 30 ), ( 31 ), ( 32 ). In the next Section, a mo r e deep analysis is presen ted for h ydro dynamic c hain ( 5 ) written in form ( 17 ). 3 Canonical v ariable s The function B 1 dep ends o n first M mo ments A 0 , A 1 , A M − 1 only , but co efficie nts of linear system ( 24 ) dep end also on A M explicitly via ( 2 9 ), ( 30 ), ( 31 ) , ( 32 ). Th us, eac h deriv a t ive of ln B 1 can b e expres sed as a ratio of t w o p olynomials with resp ect to q . In a general case (if w 1 ,M 6 = 0), the common denominator is a p olynomial of a degree M + 2 . All n umerators are p olynomials of the same degree, except a numerator of deriv at iv e ln B 1 with respect to q . Its degree is M + 1. Let us in tro duce ro ots q k ( A 0 , A 1 , ..., A M − 1 ) of this p olynomial as basic field v ariables for further computations. In suc h a c a se, ∂ q ln B 1 = − M +2 X m =1 α m q − q m , (33) where α m ( A 0 , A 1 , ..., A M − 1 ) are not y et determined functions. It means, that B 1 = α 0 M +2 Y m =1 ( q − q m ) − α m , (34) where α 0 ( A 0 , A 1 , ..., A M − 1 ) is not yet determined function. A substitution ( 34 ) ba c k to the fir st deriv a tiv e o f ln B 1 with r esp ect to q allo ws to express few (not all) v ariable 13 co efficien ts ( r 0 , u 0 , r 1 , u − 1 , σ , γ − 1 ,m , γ 0 ,m , ρ 0 ,m , ρ 1 ,m , ω m , see the previous Section) via new field v ariables q k . Moreo v er, α 0 and all other α m m ust b e constan t parameters (this is a conseque nce of an absenc e of logarithmic terms in deriv ativ es o f ln B 1 with resp ec t to momen ts A k ); r 1 is constan t due to the constrain t M +2 X m =1 α m = 1 r 1 − M − 1 , (35) follo wing from comparison of r.h.s. in ( 33 ) with a correspo nding expression fr o m linear system ( 24 ). Without loss of generalit y , one can fix α 0 on the unit y . The compatibilit y conditions ∂ k ( ∂ q ln B 1 ) = ∂ q ( ∂ k ln B 1 ) , ∂ k ( ∂ n ln B 1 ) = ∂ n ( ∂ k ln B 1 ) imply t o explicit rela- tionships betw een some co efficien ts as w ell as dep endenc ies ∂ k q n via q m and res t of initial co efficien ts. Finally , the compatibility conditions ∂ k ( ∂ m q n ) = ∂ k ( ∂ m q n ) should lead to a parametrization o f all co efficien ts via q m and their deriv ativ es with resp ect t o momen ts A 0 , A 1 , ..., A M − 1 . Ho w eve r, this is not precisely true. Hydro dynamic c hain ( 16 ) p ossesses a la r g e class of in v ertible transformations, allo wing to significan t ly reduce a n umber of distinguish c o effi- cien ts. F or instance , h ydro dynamic c hain ( 5 ) con ta ins 15 co efficien ts, while its in tegrable v ersion ( 6 ) con tains just 3 co efficien t s. It means, that transformation ( 18 ) is necessary but not sufficien t for a most appropriate c hoice of reduced n umber o f coefficien ts for a satisfactory in v estigation. T o a void comple xity of this problem, in this Section, w e restrict our consideration on the case M = 2 asso ciated with h ydro dynamic c hain ( 5 ). 4 General s o lution i n th e “triangular” case In t his Section, w e restrict o ur consideration o n a most imp ortan t case determined b y the c hoice r 0 = 1 a nd r 1 = 1 ( see ( 5 ) and commen ts to ( 12 ), i.e. the restrictions s 0 , 2 = 0 , s 1 , 2 = 0), i.e. (see ( 35 )) 4 X m =1 α m = − 2 . Moreo v er, w e essen tially can simplify further computations fixing a ll ρ k n = 0, where k , n = 0 , 1. Neve rtheless, this is no t a p articular case. A c omplete description of conserv ative in tegrable h ydro dynamic c hains ( 7 ) is giv en b y ( 6 ). In general case ( 17 ), a n infinite set of conserv ation laws can be written in the form (cf. ( 7 )) ∂ t H k = ∂ x F k ( H 0 , H 1 , ..., H M ) , k = 0 , 1 , 2 , ..., M − 1 , ∂ t H M + k = ∂ x F M + k ( H 0 , H 1 , ..., H M + k +1 ) , k = 0 , 1 , 2 , ... Just h ydro dynamic c hain ( 5 ) p ossesses an infinite set of conserv atio n la ws giv en by ( 7 ). Suc h hy dro dynamic c hains w e call “triangular” in comparison with all other h ydro dynamic c hains ( 17 ), whose conse rv ation la ws p osses s a deviation fro m this tria ngular case, i.e. first M conserv ation la w fluxes dep end sim ultaneously on first M conserv at io n la w densitie s H k . 14 As it w a s men tio ned in the previous Sec tio n, the compatibility conditions ∂ k ( ∂ q ln B 1 ) = ∂ q ( ∂ k ln B 1 ) , ∂ k ( ∂ n ln B 1 ) = ∂ n ( ∂ k ln B 1 ) lead to the sys tem in in v olution ∂ 1 q k = q 2 k + u 0 q k + u − 1 S , ∂ 1 S = 4 X m =1 (2 α m + 1) q m , (36) ∂ 1 u − 1 = 1 S u − 1 4 X m =1 ( α m + 1) q m − 4 Y k =1 q k 4 X m =1 α m + 1 q m ! , (37) ∂ 0 q k = ( q 2 k + u 0 q k + u − 1 ) ( ∂ 0 S − u − 1 ) q k S − ∂ 0 u − 1 q k − ∂ 0 u 0 , (38) ∂ 0 S = − X m

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