The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERAR CHY REVISITED Y AREM A A. PR YK ARP A TSKY 1 , OREST D. AR TEM OVYCH 2 , MAXIM V. P A VLO V 3 AND ANA TOLIY K. PR YKAR P A TSKY 4 Abstract. A differen tial-algebraic approach to studying the Lax type integ rability of the generalized Riemann t ype h ydro dynamic hier ar c hy is revisited, its new Lax type representation is constructed in ex act form. The related bi-Hamiltonian int egrability and compa tible Poissonian structures of the generalized Riemann t yp e hierarch y are also discussed b y means of the gradien t-holonomic and geometric methods. 1. Introduction Recently new mathema tical approaches, based on differen tial-algebr aic and differential geometric metho ds a nd techniques, w ere a pplied in w o rks [1, 15, 17] for studying the Lax type in tegra bilit y of nonlinear differential equations of Korteweg-de V r ies and Riemann type. In particular , a great deal of analytical studies [2, 3, 5, 7, 1, 8] w ere devoted to finding the corr esp onding Lax-type r epresentations o f the infinite Riema nn type hydrodyna mical hierarchy (1.1) D N t u = 0 , D t := ∂ /∂ t + u ∂ /∂ x, where N ∈ Z + , ( x, t ) ⊺ ∈ R 2 and u ∈ C ∞ ( R / 2 π Z ; R ) . It was found that the related dy namical sy stem (1.2) D t u 1 = u 2 , ..., D t u j = u j +1 , ..., D t u N = 0 , defined on a 2 π -p erio dic infinite-dimensio nal smo oth functiona l manifold M N ⊂ C ∞ ( R / 2 π Z ; R N ) , p os- sesses [3, 17] for an arbitra ry in teger N ∈ Z + a suitable Lax t yp e representation (1.3) D x f = l N [ u ; λ ] f , D t f = q N ( λ ) f with λ ∈ C b eing a complex sp ectral parameter a nd f ∈ L ∞ ( R ; C N ) and matrices l N [ u ; λ ] , q N ( λ ) ∈ E nd C 2 . Here, by definition, u 1 := u ∈ C ∞ ( R 2 ; R ) and the differentiations (1.4) D t := ∂ /∂ t + u 1 D x , D x := ∂ /∂ x satisfy on the manifold M N the following co mm utation rela tionship: (1.5) [ D x , D t ] = ( D x u 1 ) D x . 1991 Mathematics Subje ct Classific ation. 35A30, 35G25, 35N10, 37K35, 58J70,58J72 , 34A34; P ACS: 02.30.Jr, 02.30.Hq. Key wor ds and phr ases. differential-algebraic metho ds, gradient holonomic algorithm, Lax ty p e integrabilit y , compatible Po issonian structures, Lax t yp e represen tation. 1 2 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 In particular , for the cases N = 2 , 3 a nd N = 4 the following ex act matrix express ions l 2 [ u ; λ ] =  λu 1 ,x u 2 ,x − 2 λ 2 − λu 1 ,x  , q 2 ( λ ) =  0 0 − λ 0  , (1.6) l 3 [ u ; λ ] =   λ 2 u 1 ,x − λu 2 ,x u 3 ,x 3 λ 3 − 2 λ 2 u 1 ,x λu 3 ,x 6 λ 4 r (1) 3 [ u ] − 3 λ 3 λ 2 u 1 ,x   , q ( λ ) :=   0 0 0 λ 0 0 0 λ 0   , l 4 [ u ; λ ] =      − λ 3 u 1 ,x λ 2 u 2 ,x − λu 3 ,x u 4 ,x − 4 λ 4 3 λ 3 u 1 ,x − 2 λ 2 u 2 ,x λu 3 ,x − 10 λ 5 r (1) N [ u ] 6 λ 4 − 3 λ 3 u 1 ,x λ 2 u 2 ,x − 20 λ 6 r (2) 4 [ u ] 10 λ 5 r (1) 4 [ u ] − 4 λ 4 λ 3 u 1 ,x      , q 4 ( λ ) :=     0 0 0 0 λ 0 0 0 0 λ 0 0 0 0 λ 0     , po lynomial in λ ∈ C , w ere presented in exa ct form. A similar Lax type r epresentation, as it follows from the statements of the work [17], also ho lds fo r the g eneral ca se N ∈ Z + . These results ar e mainly base d on a ne w differential-algebraic appro ach, devis ed recently in work [1 7] and co ntain complicated enough [3] functional expr essions r ( j ) N , j = 1 , N − 2 , which satisfy the r ecurrent differen tial-functional eq uations (1.7)                  D t r (1) N + r (1) N D x u (1) = 1 , D t r (2) N + r (2) N D x u (1) = r (1) N , ... D t r ( j +1) N + r ( j +1) N D x u (1) = r ( j ) N , j = 1 , N − 3 , ... D t r ( N − 2) N + r ( N − 2) N D x u (1) = r ( N − 3) N on the functional manifold M N . Recently Popowicz [4] h as found a new Lax represent ation for the generalized Riema nn equation (1.2) at arbitra ry N ∈ Z + . In the present article, b eing strongly based on the metho ds devised in works [17, 3], we will r evisit the differe n tial-alge braic approa ch to the Riemann t yp e hydro dynamical hierar ch y (1.1) and construct its new La x type repr esentation l N [ u ; λ ] =         λu N − 1 ,x u N ,x 0 ... 0 0 λu N − 1 ,x 2 u N ,x . . . ... ... . . . . . . . . . 0 0 ... 0 λu N − 1 ,x ( N − 1) u N ,x − N λ N − λ N − 1 N u 1 ,x ... − λ 2 N u N − 2 ,x λ (1 − N ) u N − 1 ,x         , q N ( λ ) =         0 0 0 0 0 − λ 0 0 0 0 0 − λ . . . ... 0 0 0 . . . 0 0 0 0 0 − λ 0         , (1.8) in a very s imple and us eful for applications form for a ny N ∈ Z + , which is scale equiv alent to that found by Pop owicz [4 ]. Moreov er , we prov e that this Riemann type hydro dynamical hier arch y g enerates the bi-Hamiltonian flo ws on the manifold M N and, finally , we analyze for the cases N = 2 , 3 and N = 4 the corresp onding compatible [23, 25, 27] Poissonian structur es, following within the gradient-holonomic scheme fro m these new Lax type representations. A mathematical nature of the Lax type representations (1.3), (1.6) pr esents fro m the differential-algebraic po int of view a very int eresting questio n, an answer to which may be useful for the integrabilit y theory , remains op en and needs additional in vestigations. THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 3 It is also worth to mention that the metho ds devised in this work can b e successfully applied to other int eresting for applications [16, 20, 21, 19, 22] nonlinear dynamical systems, such as Burgers, Korteweg-de V ries and O strovsky-V akhnenko equations (1.9) D t u = D 2 x u, D t u = D 3 x u, D x D t u = − u, on the 2 π -p erio dic functional ma nifold M 1 ⊂ C ( ∞ ) ( R / 2 π Z ; R ) , new infinite hierarchies [6, 9] o f Riemann t yp e hydrodynamic systems (1.10) D N − 1 t u = D 4 x ¯ z , D t ¯ z = 0 , and (1.11) D N − 1 t u = ¯ z 2 x , D t ¯ z = 0 on a smo o th 2 π -p er io dic functional manifold M N ⊂ C ( ∞ ) ( R / 2 π Z ; R N ) for N ∈ N and many others. 2. Differential-algebraic appr oa ch revisiting W e will consider the ring K := R { { x, t }} , ( x, t ) ⊺ ∈ R 2 , o f the conv ergent germs of rea l-v alued smo oth functions from C ∞ ( R 2 ; R ) and construct [1 0, 11, 12, 1 3, 14] the a sso ciated differe n tial p o lynomial ring K{ u } := K [Θ u ] with resp ect to a functiona l v a riable u 1 := u ∈ K , where Θ denotes the standard monoid of the all commuting differentiations ∂ /∂ x and ∂ /∂ t. T he idea l I { u } ⊂ K{ u } is called differential if the condition I { u } = Θ I { u } holds. In the differential ring K{ u } , int erpreted as an inv aria nt differential ideal in K , there ar e naturally defined tw o differentiations (2.1) D t , D x : K{ u } → K{ u } , satisfying the Lie-commutator relationship (1.5). F or a gener al function u ∈ K there exists the only representation of (1.5) in the ideal K{ u } o f form (1.4). Nonetheless, if the additional constra int (1.1) holds, its linear finite dimensio nal matrix repres entation in the corres po nding functional vector space K{ u } N for N ∈ Z + , related with some finitely gener ated in v ar iant differential ideal I { u } ⊂ K{ u } , may exists b eing thereby equiv alent to the corre sp onding Lax type r epresentation for the Riemann type dynamical system (1.2). T o make this sc heme analytically feasible, we consider in det ail the cases N = 1 , 4 and constr uct the corresp o nding linear finite dimensiona l matr ix representations of the Lie-commutator relationship (1.5), p olyno mially dep ending on an arbitra ry spec tral par ameter λ ∈ C . 2.1. The case N = 1 . Aiming to find the corresp o nding representation vector space f or the Lie-algebraic relationship (1.5) at N = 1 , we need firstly to construct [17] a so ca lled inv ariant gene rating Riemann differential ideal R { u } ⊂ K{ u } as (2.2) R { u } := { X j ∈ Z + X n ∈ Z + λ − ( n + j ) f ( j +1) n D j t D n x u ∈ R{ u } : f ( j +1) n ∈ K , j, n ∈ Z + } , where λ ∈ R \{ 0 } is an arbitrary par ameter. The different ial ide al (2.2) is, evidently , in v ar iant a nd characterized by the following lemma. Lemma 2.1. The kernel Ker D t ⊂ R { u } of the differ entiation D t : K{ u } → K { u } is gener ate d by elements f ( j ) ∈ R {{ x, t }} , j ∈ Z + , satisfying the line ar differ ent ial-functional r elationships (2.3) D t f ( j +1) = − λf ( j ) , wher e, by definition, (2.4) f ( j +1) := f ( j +1) ( λ ) = X n ∈ Z + f ( j +1) n λ − n for λ ∈ R \ { 0 } and j ∈ Z + . 4 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 T aking now into a ccount the inv a riant reduction of the differential ideal (2.2) sub ject to the condition D t u = 0 to the ideal (2.5) R (1) { u } := { X n ∈ Z + λ − n f (1) n D n x u ∈ R{ u } : D t u = 0 , f (1) n ∈ K , n ∈ Z + } one can simultaneously reduce the hier arch y of relatio nships (2.3) to the simple expres sion (2.6) D t f (1) ( λ ) = 0 , where f (1) ∈ K 1 { u } := K{ u }} | D t u =0 . Now, to formulate the Lax type int egrability co ndition for the case N = 1 , it is ne cessary to construct the corres po nding D t -inv aria nt Lax differential ideal (2.7) L (1) { u } := n g 1 f (1) ( λ ) ∈ K 1 { u } : D t f (1) ( λ ) = 0 , g 1 ∈ K o and to chec k its in v a riance sub ject to the differentiation D x : K 1 { u } → K 1 { u } . Namely , the following lemma holds. Lemma 2.2. The L ax differ ent ial ide al (2.7) is D x -invariant, if the line ar e quality (2.8) D x f (1) ( λ ) = ( λu x + µu − 2 x u xx ) f (1) ( λ ) holds for arbitr ary p ar ameters λ, µ ∈ R , wher e the subscript ” x ” me ans t he usu al D x -differ entiation. Pr o of. The condition (2.8) makes the Lax ideal (2.7) to b e a lso D x -inv aria nt if the following condition on the linear represe n tation of the D x -differentiation (2.9) D x f (1) ( λ ) = l 1 [ u ; λ ] f (1) ( λ ) holds: (2.10) D t l 1 [ u ; λ ] + l 1 [ u ; λ ] D x u 1 = 0 for any parameter λ ∈ R . Differential-functional e quation (2.1 0) up on substitution (2.11) l 1 [ u ; λ ] := D x a 1 [ u ; λ ] reduces easily to the equation (2.12) D t a 1 [ u ; λ ] = λ for any parameter λ ∈ R . T ak ing into account the co ndition D t u 1 = 0 o ne can easily find that (2.13) a 1 [ u ; λ ] = λx + µ /D x u 1 , where µ ∈ R is arbitrar y . Having now substituted expressio n (2 .13) into (2.1 1) and ta king into account that D t x = u 1 , the res ult (2.8) follows.  Thereby , having natura lly extended the differe n tial ring K to the ring C {{ x, t }} , one can form ula te the following prop ositio n. Prop ositi on 2 .3. The L ax typ e r epr esentation for the Riemann hydr o dynamic al e quation (2.14) D t u = 0 is given by a set of the line ar c omp atible e quations (2.15) D t f (1) ( λ ) = 0 , D x f (1) ( λ ) = ( λu x + µu − 2 x u xx ) f (1) ( λ ) for f (1) ( λ ) ∈ L ∞ ( R 2 ; C ) and for any λ, µ ∈ C . THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 5 2.2. The case N = 2 . F or the ca se N = 2 the res pec tiv ely r educed inv ariant Riema nn differential ideal (2.2) is given by the set R (2) { u } : = { X j = 0 , 1 X n ∈ Z + λ − ( n + j ) f ( j +1) n D j t D n x u : D 2 t u = 0 , f ( j +1) n ∈ K , n ∈ Z + , j = 0 , 1 } (2.16) and characterized by the kernel Ker D t ⊂ R (2) { u } o f the D t -differentiation, which is generated by the following linear differe n tial rela tionships: (2.17) D t f (1) ( λ ) = 0 , D t f (2) ( λ ) = − λf (1) ( λ ) , where f ( j ) ( λ ) ∈ K 2 { u } := K{ u }| D 2 t u =0 , j = 0 , 1 , and λ ∈ R is an arbitrar y parameter. The co ndition (2.17) can be r ewritten in a compact vector for m as (2.18) D t f ( λ ) = q 2 ( λ ) f ( λ ) , q 2 ( λ ) :=  0 0 − λ 0  , where f ( λ ) := ( f (1) ( λ ) , f (2) ( λ )) ⊺ ∈ K 2 { u } 2 . Now we can cons truct the inv aria nt Lax differential ideal L (2) { u } : = { 2 X j =1 g j f ( j ) ( λ ) ∈ K 2 { u } : D t f (1) ( λ ) = 0 , (2.19) D t f (2) ( λ ) = − λf (1) ( λ ) , g j ∈ K , j = 1 , 2 } and to chec k its inv ar iance sub ject to a linea r re presentation of the D x -differentiation in the form: (2.20) D x f ( λ ) = l 2 [ u ; λ ] f ( λ ) for so me matrix l 2 [ u ; λ ] ∈ E nd R 2 . The fo llowing lemma ho lds. Lemma 2 .4. The L ax differ ential ide al (2.19) is D x -invariant if the matrix (2.21) l 2 [ u ; λ ] =  λu 1 ,x u 2 ,x − 2 λ 2 − λu 1 ,x  . Pr o of. The D x -inv aria nt condition for the Lax idea l (2.19) forces the matrix l 2 [ u ; λ ] ∈ End R 2 to satisfy the differential-functional r elationship (2.22) D t l 2 + l 2 D x u 1 = [ q 2 , l 2 ] , where we have put by definition l 2 := l 2 [ u ; λ ] , q 2 := q 2 ( λ ) ∈ End R 2 . Making the substitution (2.23) l 2 = D x a 2 for some matrix a 2 := a 2 [ u ; λ ] ∈ End R 2 , we can eas ily r educe equation (2.22) to the equiv alent one in the for m (2.24) D t a 2 = [ q 2 , a 2 ] . T o solve equation (2.24) it is useful to take into account that (2.25) D 2 t a 2 = k 2 q 2 , D 3 t a 2 = 0 for so me consta nt k 2 ∈ R . T a king into account the relationship (2.25) one eas ily obtains the ex act matrix representation (2.26) a 2 =  λu 1 u 2 − 2 λ 2 x − λu 1  , ent ailing the r esult (2.2 1)  Thu s, having a s ab ov e extended the differential ring K to the ring C { { x, t } } , one can formulate the next pr op osition. 6 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 Prop ositi on 2 .5. The L ax r epr esentation for the Riemann typ e hydr o dynamic al syst em (2.27) D t u 1 = u 2 , D t u 2 = 0 is given by a set of the line ar c omp atible e quations (2.28) D t f ( λ ) = q 2 ( λ ) :=  0 0 − λ 0  f ( λ ) , D x f ( λ ) =  λu 1 ,x u 2 ,x − 2 λ 2 − λu 1 ,x  f ( λ ) for f ( λ ) ∈ L ∞ ( R 2 ; C 2 ) and arbitr ary c omplex p ar ameter λ ∈ C . 2.3. The case N = 3 . Similarly to the ab ov e, for the case N = 3 the respectively r educed inv aria nt Riemann differential ideal (2.2) is g iven b y the set R (3) { u } : = { X j = 0 , 2 X n ∈ Z + λ − ( j + n ) f ( j +1) n D j t D n x u : D 3 t u = 0 , f ( j +1) n ∈ K , n ∈ Z + , j = 0 , 2 } (2.29 ) and characterized by the kernel K er D t ⊂ R (3) { u } , gener ated by the fo llowing differ ent ial relationships : D t f (1) ( λ ) = 0 , D t f (2) ( λ ) = − λf (1) ( λ ) , (2.30) D t f (3) ( λ ) = − λf (2) ( λ ) , where f ( j ) ( λ ) ∈ K 3 { u } := K{ u } | D 3 t u =0 , j = 0 , 2 , and λ ∈ R is a n arbitrar y parameter . The s ystem (2 .30) can b e equiv alently rewritten in the matrix form (2.31) D t f ( λ ) = q 3 ( λ ) f ( λ ) , q 3 ( λ ) =   0 0 0 − λ 0 0 0 − λ 0   , where f ( λ ) := ( f (1) , f (2) , f (3) ( λ )) ⊺ ∈ K 3 { u } 3 . Now we c an constr uct the corre sp onding in v a riant Lax differential ideal L 3 { u } : = { 3 X j =1 g j f ( j ) ( λ ) ∈ K 3 { u } : D t f (1) ( λ ) = 0 , D t f (2) ( λ ) = − λf (1) ( λ ) , (2.32) D t f (3) ( λ ) = − λf (2) ( λ ) , g j ∈ K , j = 1 , 3 } , whose D x -inv aria nce is lo oked for in the linea r matrix form (2.33) D x f ( λ ) = l 3 [ u ; λ ] f ( λ ) for so me matrix l 3 [ u ; λ ] ∈ End R 3 . The latter satisfies the differential-functional equa tion (2.34) D t l 3 + l 3 D x u 1 = [ q 3 , l 3 ] , where we put, by definition, l 3 := l 3 [ u ; λ ] , q 3 := q 3 ( λ ) ∈ End R 3 . Ma king use of the deriv ative represen- tation (2.35) l 3 = D x a 3 , where a 3 := a 3 [ u ; λ ] ∈ End R 2 , eq uation (2.3 4) reduces to a n equiv alent one in the form (2.36) D t a 3 = [ q 3 , a 3 ] . T o so lve matr ix equation (2.36), we tak e into acco unt that (2.37) D 3 t a 3 = k 3 q 3 , D 4 t a 3 = 0 for some constant k 3 ∈ R . Based b oth o n (2.37) and on the component wise form of (2 .36) one easily finds that (2.38) a 3 [ u ; λ ] =   λu 2 u 3 0 0 λu 2 2 u 3 − 3 λ 3 x − 3 λ 2 u 1 − 2 λu 2   , THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 7 ent ailing the matr ix (2.39) l 3 [ u ; λ ] =   λu 2 ,x u 3 ,x 0 0 λu 2 ,x 2 u 3 ,x − 3 λ 3 − 3 λ 2 u 1 ,x − 2 λu 2 ,x   . Thereby , having naturally extended the differential r ing K to the r ing C {{ x, t }} , one can formulate the next pr op osition. Prop ositi on 2.6. The L ax r epr esentation for the Rie mann typ e hydr o dynamic al system (2.40) D t u 1 = u 2 , D t u 2 = u 3 , D t u 3 = 0 is given by a set of line ar c omp atible e quations (2.41) D t f ( λ ) =   0 0 0 − λ 0 0 0 − λ 0   f ( λ ) , D x f ( λ ) =   λu 2 ,x u 3 ,x 0 0 λu 2 ,x 2 u 3 ,x − 3 λ 3 − 3 λ 2 u 1 ,x − 2 λu 2 ,x   f ( λ ) , wher e f ( λ ) ∈ L ∞ ( R 2 ; C 3 ) and λ ∈ C is an arbitr ary c omplex p ar ameter. As o ne c an e asily observe, the scheme of finding the Lax r epresentation for the Riemann type hydro- dynamical equa tions at N = 1 , 3 can b e easily ge neralized for arbitra ry N ∈ Z + , that is a topic of the next s ection. 2.4. The general case N ∈ Z + . Consider a t arbitrary N ∈ Z + the constraint D N t u = 0 and the resp ectively reduced inv a riant Riemann differential ideal (2.2), which is g iven b y the set R ( N ) { u } : = { N − 1 X j =0 X n ∈ Z + λ − ( j + n ) f ( j +1) n D j t D n x u ∈ K{ u } : D N t u = 0 . f ( j +1) n ∈ R {{ x, t }} , n ∈ Z + , j = 0 , N − 1 } . (2.42) The r elated kernel Ker D t ⊂ R ( N ) { u } of the D t -differentiation is, owing to (2.3) and ( 2.42), g enerated by the r elationships (2.43) D t f (1) ( λ ) = 0 , D t f (2) ( λ ) = − λf (1) ( λ ) , ..., D t f ( N ) ( λ ) = − λf ( N − 1) ( λ ) , which can b e rewritten in a matrix form as (2.44) D t f ( λ ) = q N ( λ ) f ( λ ) , q N ( λ ) =         0 0 0 0 0 − λ 0 0 0 0 0 − λ . . . ... 0 0 0 . . . 0 0 0 0 0 − λ 0         , where, by definition, a vector f ( λ ) := ( f (1) , f (2) , ..., f ( N ) ) ⊺ ∈ K N { u } N := K{ u } N | D N t u =0 . The latter generates the inv a riant Lax differential ideal (2.45) L ( N ) { u } := { N X j =1 g j f ( j ) ( λ ) ∈ K N { u } : g j ∈ K , j = 1 , N } , whose D x -inv aria nce holds, if its linear matrix r epresentation in the space K ( N ) { u } N (2.46) D x f ( λ ) = l N [ u ; λ ] f ( λ ) is compatible with (2.43) for so me ma trix l N := l N [ u ; λ ] ∈ E nd R N and arbitr ary λ ∈ R . As a re sult we obtain the co mpatibility condition (2.47) D t l N + l N D x u = [ q N , l N ] . Having represented the matr ix l N ∈ End R N in the deriv a tive form (2.48) l N := D x a N , 8 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 we c an reduce the functional-differential equation (2 .47) to (2.49) D t a N = [ q N , a N ] . Now, ta king into account that (2.50) D N t a N = k N q N , D N +1 t a N = 0 for so me co nstant k N ∈ R , one can easily o btain by mea ns of simple calculations the following solution to equatio n (2.49): (2.51) a N [ u ; λ ] =         λu N − 1 u N 0 ... 0 0 λu N − 1 2 u N . . . ... ... . . . . . . . . . 0 0 ... 0 λu N − 1 ( N − 1) u N − 1 − N λ N x − λ N − 1 N u 1 ... − λ 2 N u N − 2 λ (1 − N ) u N − 1         , ent ailing the exa ct matrix expressio n (2.52) l N [ u ; λ ] =         λu N − 1 ,x u N ,x 0 ... 0 0 λu N − 1 ,x 2 u N ,x . . . ... ... . . . . . . . . . 0 0 ... 0 λu N − 1 ,x ( N − 1) u N ,x − N λ N − λ N − 1 N u 1 ,x ... − λ 2 N u N − 2 ,x λ (1 − N ) u N − 1 ,x         . Thereby , having naturally extended the differential ring K to the ring C {{ x, t } } } , one can formulate the following g eneral pro po sition. Prop ositi on 2 .7. The L ax r epr esentation for the gener alize d Rie mann typ e hydr o dynamic al system (2.53) D t u 1 = u 2 , D t u 2 = u 3 , ..., D t u N − 1 = u N , D t u N = 0 is given for any arbitr ary N ∈ Z + by a set of line ar c omp atible e quations D t f ( λ ) =         0 0 0 0 0 − λ 0 0 0 0 0 − λ . . . ... 0 0 0 . . . 0 0 0 0 0 − λ 0         f ( λ ) , (2.54) D x f ( λ ) =         λu N − 1 ,x u N ,x 0 ... 0 0 λu N − 1 ,x 2 u N ,x . . . ... ... . . . . . . . . . 0 0 ... 0 λu N − 1 ,x ( N − 1) u N ,x − N λ N − λ N − 1 N u 1 ,x ... − λ 2 N u N − 2 ,x λ (1 − N ) u N − 1 ,x         f ( λ ) , wher e f ( λ ) ∈ L ∞ ( R 2 ; C N ) and λ ∈ C is an arbitr ary c omplex p ar ameter. Mor e over, the r elation- ships (2.54 ) r e alize a line ar matrix r epr esentation of the c ommut ator c ondition (1.5 ) in the ve ctor sp ac e K ( N ) { u } N . The general Lax t yp e r epresentation (2.54), obtained ab ov e for ar bitrary N ∈ Z + , lo oks essentially simpler than those obtained b efore for N = 1 , 4 in [3 , 1 7, 5] and given by differ ent ial-matrix express ions (1.6), dep ending at N > 3 on the solutions to the set of differential-functional equations (1.7). As it was already mentioned ab ov e, it was first f ound in a simila r form up to a scaling pa rameter λ ∈ C in [4] by means of another appr oach. T aking this simplicit y in to account, o ne can apply to the Lax t yp e repre sentation (2.54) the gradient-holonomic approa ch a nd find the corres po nding bi- Hamiltonian THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 9 structures, r esp onsible for the Lax type integrability of the Riemann type hier arch y of h ydro dynamica l systems (2.5 3). 3. The bi-Hamil tonian structures for cases N = 2 , 3 and N = 4 T o pro ceed with pro ving t he r elated bi-Hamiltonia n integrability of the generalized Riemann type dynamical sys tem ( 2.53) a t arbitra ry N ∈ Z + and with finding the naturally asso cia ted with the Lax representations (2.54) P oissonia n structures, we need preliminarily to study with resp ect to the gradient- holonomic a pproach [2, 25, 28] the cor resp onding sp ectra l prop er ties of the linear differe n tial sys tem (2.46). Be low will analy ze the nex t ca ses: N = 2 , 3 and N = 4 . 3.1. The case N = 2 . Mak ing use of the exact matr ix expressio n (2.2 1), one can define fo r the se cond linear eq uation of (2 .28) the corresp o nding mono dro my matrix S ( x ; λ ) ∈ End C 3 , x ∈ R , which satisfie s [24, 2 6, 25, 28, 2 9] the Novik ov-Marchenk o co mm utator e quation (3.1) D x S ( x ; λ ) = [ l 2 [ u ; λ ] , S ( x ; λ )] , Since the trace- functional γ ( λ ) := tr S ( x ; λ ) is, owing to the Lax representation (2.41), inv ar iant with resp ect to the R ∋ x, t -ev o lutions on the manifold M 2 , its gra dient ϕ ( x ; λ ) := grad γ ( λ ) ∈ T ∗ ( M 2 ) , x ∈ R , depe nding pa rametrica lly on the sp ectral parameter λ ∈ C , eq uals (3.2) ϕ ( x ; λ ) = tr( l ′∗ 2 S ( x ; λ )) and satisfies the well known gradient r elationship (3.3) ϑϕ ( x ; λ ) = z ( λ ) η ϕ ( x ; λ ) for s ome mer omorphic function z : C → C and x ∈ R , where ϑ, η : T ∗ ( M 2 ) → T ( M 2 ) are compatible [23, 25, 27] Poissonian structures on the manifold M 2 , the dash sign “ ′ “ means the usual F rechet deriv ative a nd “ ∗ “ means the corre sp onding conjugation with resp ect to the s tandard bilinea r form o n T ∗ ( M 2 ) × T ( M 2 ) . The latter naturally follows from the Lie-alge braic theory [27, 23, 25] of Lax t y pe int egrable dynamical sy stems, treating them as the corr esp onding L ie-Poisson flows o n the adjoint s pace G ∗ ( λ ) to a s uitably constr ucted centrally extended affine Lie alg ebra G ( λ ) , λ ∈ C . By simple enoug h calculations one find that (3.4) ϕ ( x ; λ ) := ( ϕ 1 , ϕ 2 ) ⊺ = ( λ ( S (11) x − S (22) x ) , S (21) x ) ⊺ , where we put, by definition, (3.5) S ( x ; λ ) :=  S (11) S (12) S (21) S (22)  . T aking int o ac count tha t equa tion (3.1) ca n b e rewritten as the system S (11) x = − u 2 ,x S (21) + 2 λ 2 S (12) , (3.6) S (12) x = − u 2 ,x ( S (11) − S (22) ) + 2 λu 1 ,x S (12) , S (21) x = − 2 λ 2 u 1 ,x ( S (11) − S (22) ) − 2 λu 1 ,x S (21) , S (22) x = − 2 λ 2 S (12) + u 2 ,x S (21) , by mea ns o f simple enough c alculations, co nsisting in substituting (3.4) into (3.6 ), one e asily finds the gradient relationship (3.3) in the exact differential-matrix form: (3.7)  0 − 1 1 0   ϕ 1 ϕ 2  = 2 λ  ∂ − 1 u 1 ,x ∂ − 1 ∂ − 1 u 1 ,x u 2 ,x ∂ − 1 + ∂ − 1 u 2 ,x   ϕ 1 ϕ 2  , where, by definition, z ( λ ) := 2 λ, λ ∈ C , and (3.8) ϑ :=  0 1 − 1 0  , η :=  ∂ − 1 u 1 ,x ∂ − 1 ∂ − 1 u 1 ,x u 2 ,x ∂ − 1 + ∂ − 1 u 2 ,x  constitute the corresp onding compatible Poissonia n structures on the functiona l ma nifold M 2 . Th us, one can formulate the following pro po sition, b efor e stated in [7 , 1 8, 5, 3] using different appro aches. 10 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 Prop ositi on 3.1. The Riema nn typ e hydr o dynamic al system (1.2) at N = 2 is a L ax inte gr able bi- Hamiltonian flow with r esp e ct to t he c omp atible on the fun ctional manifold M 2 Poissonian p air ( 3.8). As a consequence of P rop osition (3.1 ) eas ily one states that there ex ists [23, 27, 25] an asso ciated infinite hier arch y of commuting to ea ch o ther integrable bi-Hamiltonian flows on M 2 (3.9) d ( u 1 , u 2 ) ⊺ /dt n := − ( η ϑ − 1 ) n ( u 1 ,x , u 2 ,x ) ⊺ , where t n ∈ R , n ∈ Z + , a re the corres po nding evolution parameters . 3.2. The cases N = 3 and 4 . F or the cas e N = 3 there was proved in [1, 3, 5] that the co rresp onding Riemann t yp e hydrodyna mical system (2.4 0) is a Hamilto nian dynamical system on the 2 π - per io dic functional ma nifold M 3 := { ( u 1 , u 2 , u 3 ) ⊺ ∈ C ( ∞ ) ( R / (2 π Z ); R 3 ) } , who se exa ct Poissonia n s tructure is given by the differential-matrix expres sion (3.10) η =   ∂ − 1 u 1 ,x ∂ − 1 0 ∂ − 1 u 1 ,x u 2 ,x ∂ − 1 + ∂ − 1 u 2 ,x ∂ − 1 u 3 ,x 0 u 3 ,x ∂ − 1 0   , acting as a linear mapping η : T ∗ ( M 3 ) → T ( M 3 ) from the cotangent space T ∗ ( M 3 ) to the tangent space T ( M 3 ) . Making use of the exa ct matrix expres sion (2.39), one can define fo r the linear eq uation (2.33) the co rresp onding mono dromy matrix S ( x ; λ ) ∈ End C 3 , x ∈ R , which satisfies the Noviko v- Marchenk o commutator equation (3.11) dS ( x ; λ ) / dx = [ l 3 [ u ; λ ] , S ( x ; λ )] . Since the tra ce-functional γ ( λ ) := tr S ( x ; λ ) is in v a riant with resp ect to the R ∋ x, t -evolutions on the manifold M 3 , its gradient ϕ ( x ; λ ) := grad γ ( λ ) ∈ T ∗ ( M 3 ) , x ∈ R , dep ending parametrically on the sp ectra l parameter λ ∈ C , equals (3.12) ϕ ( x ; λ ) = tr( l ′∗ 3 S ( x ; λ )) = ( − 3 λ 2 S (23) x , λ ( S (11) x + S (22) x − 2 S (23) x , S (21) x + 2 S (32) x ) ⊺ and satisfies the gradie n t relationship (3.13) ϑϕ ( x ; λ ) = z ( λ ) ηϕ ( x ; λ ) for so me meromor phic function z : C → C and all x ∈ R , where ϑ : T ∗ ( M 3 ) → T ( M 3 ) is a seco nd compatible with (3.10) Poisson structure o n the manifo ld M 3 . In this case the sta ndard calculations, consisting in substituting (3 .3) into (3.2) and reducing the r esult to the form (3.3), g ive rise by means of slightly cumbersome calculations to the equiv ale n t to (3.13) relatio nship (3.14) ϕ ( x ; λ ) = λ 2 ϑ − 1 η ϕ ( x ; λ ) , where (3.15) ϑ − 1 =        ∂ u 3 + u 3 ∂ − u 2 ,x − ∂ u 2 − 2 u 1 ∂ + 2 ∂ u 3 u 1 ,x u 3 ,x u 2 ,x − u 2 ∂ ∂ u 1 + u 1 ∂ − 2 − 2 ∂ u 3 u 3 ,x − 2 ∂ u 1 + 2 u 3 u 1 ,x u 3 ,x ∂ 2 − 2 u 3 u 3 ,x ∂ − ( u 3 u 2 1 ,x +2 u 3 u 2 ,x u 2 3 ,x ) ∂ − − ∂ ( u 3 u 2 1 ,x +2 u 3 u 2 ,x u 2 3 ,x )        is the second compatible with (3.10) c o-Poissonian s tructure on the functional manifold M 3 . A completely similar results hold in the case N = 4 . Namely , the cor resp onding Novik ov-Marchenk o equation (3.16) dS ( x ; λ ) / dx = [ l 4 [ u ; λ ] , S ( x ; λ )] . joint ly with the expressio n ϕ ( x ; λ ) = tr( l ′∗ 4 S ( x ; λ )) = ( − 4 λ 3 S (24) x , − 4 λ 2 S (34) x , (3.17) λ ( S (11) x + S (22) x + S (33) x − 3 S (44) x ) , S (21) x + 2 S (32) x + 3 S (43) ) ⊺ , is reduced by means o f simple but c um be rsome calcula tions to the g radient r elationship (3.18) ϕ ( x ; λ ) = λ 3 ϑ − 1 η ϕ ( x ; λ ) , THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 11 where the P o issonian structures η and ϑ : T ∗ ( M 4 ) → T ( M 4 ) are, resp ectively , inverse to the co- Poissonian op erator s (3.19) η − 1 =        0 0 − ∂ ∂ u 3 ,x u 4 ,x 0 ∂ 0 − ∂ u 2 ,x u 4 ,x − ∂ 0 0 ∂ u 1 ,x u 4 ,x u 3 ,x u 4 ,x ∂ − u 2 ,x u 4 ,x ∂ u 1 ,x u 4 ,x ∂ 1 2 [ u − 2 4 ,x ( u 2 2 ,x − 2 u 1 ,x u 3 ,x ) ∂ + + ∂ ( u 2 2 ,x − 2 u 1 ,x u 3 ,x ) u − 2 4 ,x ]        and (3.20) ϑ − 1 =                             0 − 3 u 4 ,x u 3 ∂ − ∂ u 2 − u 2 ∂ − − u 3 ∂ u 3 ,x u 4 ,x − − u 3 ,x u 4 ,x ∂ u 3 3 u 4 ,x 0 u 2 ∂ ∂ u 1 + u 1 ∂ + + u 2 ∂ u 3 ,x u 4 ,x + + u 3 ,x u 4 ,x ∂ u 2 ∂ u 3 ∂ u 2 − u 1 ∂ − u 1 ∂ − 5 + ∂ ( u 4 − u 2 u 2 ,x u 4 ,x ) − − ∂ ( u 1 u 3 ) x u 4 ,x − u 1 ∂ u 3 ,x u 4 ,x − ∂ u 2 − u 2 ∂ − − u 3 ∂ u 3 ,x u 4 ,x − − u 3 ,x u 4 ,x ∂ u 3 ∂ u 1 + u 1 ∂ + + u 2 ∂ u 3 ,x u 4 ,x + + u 3 ,x u 4 ,x ∂ u 2 5 + ( u 4 − u 2 u 2 ,x u 4 ,x ) ∂ − − ( u 1 u 3 ) x u 4 ,x ∂ − u 3 ,x u 4 ,x ∂ u 1 u 4 − u 2 u 2 ,x u 2 4 ,x ∂ + − u 3 ,x u 2 4 ,x ∂ − ∂ u 3 ,x u 2 4 ,x − − ∂ u 4 − u 2 u 2 ,x u 2 4 ,x                             , on the functional ma nifold M 4 . Below we will reanalyze the bi-Hamiltonia n structures of the genera lized Riemann type hier arch y (1.1) from the g eometric p oint of view, des crib ed in detail in [25, 28, 3]. 3.3. Poissonian s tructures: the geom etric approac h . T o co nstruct t he Hamiltonian structur es, related with the gene ral dynamical s ystem (1.2), b y means o f the geometric approach it is fir st necessa ry to pr esent it in the following e quiv alent form: (3.21) du 1 /dt = u 2 − u 1 u 1 ,x du 2 /dt = u 3 − u 1 u 2 ,x ... du j /dt = u j +1 − u 1 u j,x ... du N /dt = − u 1 u N ,x                := K [ u ] , where K : M N → T ( M N ) is the corr esp onding vector field on the functional ma nifold M N for a fixe d N ∈ Z + . Then the following prop osition [3, 25, 28] ho lds. Prop ositi on 3.2. L et ψ ∈ T ∗ ( M N ) b e, in gener al, a quasi-lo c al functional-analytic ve ctor, which satisfies the c ondition ψ ′ 6 = ψ ′ , ∗ and solves the Lie-L ax e quation (3.22) L K ψ := ψ t + K ′ , ∗ ψ = grad L , for some smo oth functional L ∈ D ( M N ) . Then the dynamic al system (3.2 1) al lows the Hamiltonian r epr esentation (3.23) K [ u ] = − η gra d H ( η ) [ u ] , H η = ( ψ ( η ) , K ) − L ( η ) , η − 1 = ψ ′ ( η ) − ψ ′ , ∗ ( η ) , 12 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 wher e η : T ∗ ( M N ) → T ( M N ) is the c orr esp onding Poissonian stru ctur e, invertible on M N . Otherwise, if the op er ator η − 1 : T ( M N ) → T ∗ ( M N ) is not invertible, the fol lowing r elationship (3.24) grad H ( η ) [ u ] = − η − 1 K [ u ] holds on the manifold M N . R emark 3.3 . Concerning a g eneral s olution to the Lie-La x equation (3.2 2) it is easy to observe tha t it po ssesses the following representation: (3.25) ψ = ¯ ψ + g rad ¯ L , where (3.26) L K ¯ ψ := ¯ ψ t + K ′ , ∗ ¯ ψ = 0 , L K ¯ L = L . The rela ted co-Poissonian op erato r (3.27) η − 1 := ψ ′ − ψ ′ , ∗ = ¯ ψ ′ − ¯ ψ ′ , ∗ , owing to the V olter ra symmetry condition (g rad ¯ L ) ′ = (grad ¯ L ) ′ , ∗ , p er sists to b e unchangeable. Based on Prop o sition 3.2, it is enoug h to search for non-symmetric functional-analytic s olutions to (3.22) making use for this, for instance, of sp ecial co mputer-algebr aic algorithms. 3.3.1. The c ase N = 3 . The corres po nding dynamical system (3.28) du 1 /dt = u 2 − u 1 u 1 ,x du 2 /dt = u 3 − u 1 u 2 ,x du 3 /dt = − u 1 u 3 ,x    := K [ u ] is defined on the functiona l ma nifold M 3 . A vector ψ := ( ψ 1 , ψ 2 , ψ 3 ) ⊺ ∈ T ∗ ( M 3 ) sa tisfies owing to (3.22) a set of functional-differential equations dψ 1 /dt + u 1 ψ 1 ,x − u 2 ,x ψ 2 − u 3 ,x ψ 3 = δ L /δ u 1 , dψ 2 /dt + ψ 1 − ( u 1 ψ 2) x = δ L /δ u 2 , (3.29) dψ 3 /dt + ψ 2 + ( u 1 ψ 3 ) x = δ L /δ u 3 for some smo oth functional L ∈ D ( M 3 ) . System (3 .29) can be slightly simplified, if to use the differenti- ation D t : M 3 → T ( M 3 ) : D t ψ 1 − u 2 ,x ψ 2 − u 3 ,x ψ 3 = δ L /δ u 1 (3.30) D t ψ 2 + ψ 1 + u 1 ,x ψ 2 = δ L /δ u 2 , D t ψ 3 + ψ 2 + u 1 ,x ψ 3 = δ L /δ u 3 , which is e quiv alent to the s calar functional-a nalytic ex pression (3.31) D 2 t ( αψ 1 ) = D t ( αδ L /δ u 1 ) + ( D t α ) δ L /δ u 1 + ( D 2 t α ) δ L /δ u 2 + δ L /δ u 3 on the alone function ψ 1 ∈ K{ u } , wher e we have denoted the function α := u − 1 3 ,x , sa tisfying the us eful differential relationships (3.32) D t α = u 1 ,x α, D 2 t α = u 2 ,x α, D 3 t α = 1 , D 4 t α = 0 . The cor resp onding functions ψ 2 , ψ 3 ∈ K{ u } are given by the functional- op erator expres sions ψ 2 = α − 1 D − 1 t ( − αψ 1 + αδ L /δu 2 ) , (3.33) ψ 3 = α − 1 D − 1 t ( − αψ 2 + αδ L /δu 3 , easily following from (3.30). In the sp ecial case L = 0 equa tion (3.31) reduces to (3.34) D 2 t ( αψ 1 ) = 0 , whose functional-ana lytic solutions can b e found analytically b y means of b oth differen tial- algebraic to ols, devised in [17], and mo dern computer-alg ebraic algorithms. In particula r, making use of the THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 13 differential-algebraic a pproach of the work [17 ], one can easily enough to find, among st ma ny other s, the following solutions to (3.31): ψ 1 = u 1 ,x / 2 , L ( η ) = 1 2 Z 2 π 0 (2 u 3 + u 2 u 1 ,x ) dx ; (3.35) ψ 1 = u 1 ,x u 3 − u 1 u 3 ,x , L ( ϑ ) = 0; and ψ 1 = − u 3 ,x / 2 , L ( ϑ 0 ) = 0; (3.36) ψ 1 = u 2 u 3 ,x , L ( ϑ 1 ) = 0; ψ 1 = u 1 ,x u 2 − 2 u 3 − ( u 1 u 2 u 3 − u 3 2 3 u 2 3 ) u 3 ,x , L ( ϑ 2 ) = 0; ψ 1 = u 1 ,x − u 2 2 u 3 ,x 2 u 3 , L ( ϑ 3 ) = 0; ψ 1 = u 1 ,x − u 1 u 3 ,x u 3 , L ( ϑ 4 ) = 0; giving r ise to the generating vectors (3.37) ψ ( η ) = ( u 1 ,x / 2 , 0 , − u − 1 3 ,x u 2 1 ,x / 2 + u − 1 3 ,x u 2 ,x ) ⊺ , L ( η ) = 1 2 R 2 π 0 (2 u 3 + u 2 u 1 ,x ) dx ; ψ ( ϑ ) = ( u 1 ,x u 3 − u 1 u 3 ,x , u 1 u 2 ,x − u 1 ,x u 2 , 2 u 2 − u 1 u 1 ,x + u 3 u 2 1 ,x − 2 u 3 u 2 ,x u 3 ,x ) ⊺ , L ( ϑ ) = 0; ψ ( θ ) = ( u 2 − u 2 u 3 2 ,x 6 u 3 ,x + u 3 u 4 2 ,x 24 u 3 3 ,x , u 2 u 2 ,x u 3 ,x − u 2 u 2 1 ,x 2 u 3 ,x − u 3 u 2 2 ,x 2 u 2 3 ,x − u 3 u 1 ,x u 3 2 ,x 6 u 3 3 ,x + u 3 u 5 2 ,x 30 u 4 3 ,x , − u 2 u 2 2 ,x u 2 3 ,x + u 2 u 2 1 u 2 ,x u 2 3 ,x + u 3 u 2 2 ,x u 2 3 ,x − u 2 u 1 ,x u 3 2 ,x u 3 3 ,x + u 3 u 1 ,x u 4 2 ,x u 4 3 ,x − u 3 u 6 2 ,x 30 u 5 3 ,x ) ⊺ , L ( θ ) = 0; and (3.38) ψ ( ϑ 0 ) = ( − u 3 ,x / 2 , u 2 ,x / 2 , u 1 ,x / 2 − u − 1 3 ,x u 2 2 ,x / 2) ⊺ , L ( ϑ 0 ) = 0; ψ ( ϑ 1 ) = ( u 2 u 3 ,x , − u 1 u 3 ,x , u 1 u 2 u 3 ,x u − 1 3 − u 3 2 u 3 ,x u − 2 3 / 3) ⊺ , L ( ϑ 1 ) = 0; ψ ( ϑ 2 ) = ( u 1 ,x u 2 − 2 u 3 − ( u 1 u 2 u 3 − u 3 2 3 u 2 3 ) u 3 ,x , − u 2 − u 2 1 u 2 ,x 2 u 3 + u 3 ,x u 4 2 12 u 3 3 − u 3 u 2 ,x u 3 ,x + u 3 u 2 1 ,x 2 u 3 ,x , u 1 − u 1 u 1 ,x u 2 ,x u 3 ,x + u 2 1 u 2 u 3 ,x 2 u 2 3 − u 1 u 3 2 u 3 ,x 3 u 3 3 + u 3 u 1 ,x u 3 ,x + u 5 2 u 3 ,x 15 u 4 3 − 2 u 3 u 2 1 ,x u 2 ,x u 2 3 ,x + + u 2 u 2 1 ,x 2 u 3 ,x + u 1 u 3 2 ,x 3 u 2 3 ,x − u 3 u 1 ,x u 2 ,x 2 u 2 3 ,x − u 2 u 4 2 ,x 12 u 3 3 ,x + u 3 u 1 ,x u 3 2 ,x 3 u 3 3 ,x − u 3 u 5 2 ,x 15 u 4 3 ,x ) ⊺ , L ( ϑ 2 ) = 0; ψ ( ϑ 3 ) = ( u 1 ,x − u 2 2 u 3 ,x 2 u 2 3 , u 3 2 u 3 ,x − 6 u 3 3 6 u 3 3 , ( u 4 2 − 6 u 2 1 u 2 3 ) u 3 ,x 12 u 4 3 + u 2 − u 1 u 1 ,x u 3 + u 2 u 2 ,x (3 u 1 u 3 − u 2 2 ) 3 u 3 ) ⊺ , L ( ϑ 3 ) = 0; ψ ( ϑ 4 ) = ( u 1 ,x − u 1 u 3 ,x u 3 , u 1 u 2 ,x − u 1 ,x u 2 u 3 , u 1 u 1 ,x − 2 u 2 u 3 + 2 u 2 ,x − u 2 1 ,x u 3 ,x ) ⊺ , L ( ϑ 4 ) = 0; 14 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 As a result of (3.2 3) and expre ssions (3.37) one easily obtains the corre sp onding co-Poissonian o per ators: (3.39) η − 1 := ψ ′ ( η ) − ψ ′ , ∗ ( η ) =     ∂ 0 − ∂ u 1 ,x u − 1 3 ,x 0 0 ∂ u − 1 3 ,x − u 1 x u − 1 3 ,x ∂ u − 1 3 ,x ∂ 1 2 ( u 2 1 ,x u − 2 3 ,x ∂ + ∂ u 2 1 ,x u − 2 3 ,x ) − − ( u 2 ,x u − 2 3 ,x ∂ + ∂ u 2 ,x u − 2 3 ,x )     , ϑ − 1 := ψ ′ ( ϑ ) − ψ ′ , ∗ ( ϑ ) =        u 3 ∂ + ∂ u 3 − u 2 ,x − ∂ u 2 − 2 u 1 ∂ + 2 ∂ u 3 u 1 ,x u 3 ,x u 2 ,x − u 2 ∂ u 1 ∂ + ∂ u 1 − 2 − 2 ∂ u 3 u 3 ,x − 2 ∂ u 1 + 2 u 3 u 1 ,x u 3 ,x ∂ 2 − 2 u 3 u 3 ,x ∂ − ( u 3 u 2 1 ,x +2 u 3 u 2 ,x u 2 3 ,x ) ∂ − − ∂ ( u 3 u 2 1 ,x +2 u 3 u 2 ,x u 2 3 ,x )        , and (3.40) ϑ − 1 0 := ψ ′ ( ϑ 0 ) − ψ ′ , ∗ ( ϑ 0 ) =   0 0 0 0 ∂ − ∂ u 2 ,x u − 1 3 ,x 0 − u 2 ,x u − 1 3 ,x ∂ 1 2 ( u 2 2 ,x u − 2 3 ,x ∂ + ∂ u 2 2 ,x u − 2 3 ,x )   , ϑ − 1 1 := ψ ′ ( ϑ 1 ) − ψ ′ , ∗ ( ϑ 1 ) =        0 2 u 3 ,x u 2 ∂ − u 2 u 3 ,x /u 3 − 2 u 3 ,x 0 − u 1 ∂ − u 1 u 3 ,x /u 3 + + u 2 2 u 3 ,x /u 2 3 u 2 u 3 ,x /u 3 + ∂ u 2 u 1 u 3 ,x /u 3 − − u 2 2 u 3 ,x /u 2 3 − ∂ u 1 u 1 u 2 ∂ u − 1 3 + u − 1 3 ∂ u 1 u 2 − − u 3 2 3 ∂ 1 u 2 3 − 1 3 u 2 3 ∂ u 3 2        , and s o on. The second expres sion of (3.39) co incides exactly with the co -Poissonian op er ator (3.15), satisfying the gradient relationship (3.14), which prov es the bi-Hamiltonicit y o f the Riemann type equation (3.28). Concerning the next tw o expr essions (3.39) and (3.40) it is easy to observe that they are not strictly in vertible, since the translation v ecto r field d/dx : M 3 → T ( M 3 ) with comp onents ( u 1 ,x , u 2 ,x , u 3 ,x ) ⊺ ∈ T ( M 3 ) b elongs to their kernels, that is η − 1 ( u 1 ,x , u 2 ,x , u 3 ,x ) ⊺ = 0 = ϑ − 1 0 ( u 1 ,x , u 2 ,x , u 3 ,x ) ⊺ . Nonetheless, upon formal inv erting the op erato r expr ession (3.39), w e obtain by means of simple eno ugh, but slightly cumberso me, direct calculations , tha t the Poissonian η -oper ator lo oks as (3.41) η =   ∂ − 1 u 1 ,x ∂ − 1 0 ∂ − 1 u 1 ,x u 2 ,x ∂ − 1 + ∂ − 1 u 2 ,x ∂ − 1 u 3 ,x 0 u 3 ,x ∂ − 1 0   , coinciding with expr ession (3.10), and the corres po nding Hamiltonian function equa ls (3.42) H ( η ) := ( ψ ( η ) , K ) − L ( η ) = Z 2 π 0 dx ( u 1 ,x u 2 − u 3 ) , Concerning the op era tor express ion (3.40) the co rresp onding Hamiltonian function (3.43) H ( ϑ 0 ) := ( ψ ( ϑ 0 ) , K ) = Z 2 π 0 dx ( u 3 u 2 ,x ) It is evident that thes e Ha miltonian functions (3.42) and (3.4 3) ar e conse rv ation laws for the dynamica l system (3.28), which were a lso b efore found in [3]. R emark 3.4 . It is worth to mention her e that as the sum of vectors P 4 j =0 s j ψ ( ϑ j ) ∈ T ∗ ( M 3 ) with arbitrary s j ∈ R , j = 0 , 4 , solves the determining equatio n (3.22), the cor resp onding op er ator (3.44) θ − 1 := 4 X j =0 s j ϑ − 1 j will also b e a co -Poissonian o pe rator for the dynamical system (3.28). THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 15 3.3.2. The c ase N = 4 . The Riemann type hydro dynamical eq uation (1.2) a t N = 4 is equiv alent to the nonlinear dynamical s ystem (3.45) u 1 ,t = u 2 − u 1 u 1 ,x u 2 ,t = u 3 − u 1 u 2 ,x u 3 ,t = u 4 − u 1 u 3 ,x u 4 ,t = − u 1 u 4 ,x        := K [ u 1 , u 2 , u 3 , u 4 ] , where K : M 4 → T ( M 4 ) is a suitable vector field on the smo oth 2 π -p er io dic functional manifold M 4 := C ∞ ( R / 2 π Z ; R 4 ) . T o study its possible Hamiltonian structures, we nee d to find functional so lutions to the deter mining Lie -Lax equation (3.22): (3.46) ψ t + K ′ , ∗ ψ = g r ad L for ψ ∈ T ∗ ( M 4 ) and so me smo oth functional L ∈ D ( M 4 ) , wher e (3.47) K ′ =     − ∂ u 1 1 0 0 − u 2 ,x − u 1 ∂ 1 0 − u 3 ,x 0 − u 1 ∂ 1 − u 4 , x 0 0 − u 1 ∂     , K ′ , ∗ =     u 1 ∂ − u 2 ,x − u 3 ,x − u 4 ,x 1 ∂ u 1 0 0 0 1 ∂ u 1 0 0 0 1 ∂ u 1     are, resp ectively , the F r echet deriv ative of the mapping K : M 4 → T ( M 4 ) and its conjugate. The small parameter metho d [25, 3], applied to equation (3 .46), gives rise to the following its exact s olution: (3.48) ψ ( η ) = ( − u 3 ,x , u 2 ,x / 2 , 0 , − u 2 2 ,x 2 u 4 ,x + u 1 ,x u 3 ,x u 4 ,x ) ⊺ , L ( η ) = Z 2 π 0 ( u 4 u 1 ,x − u 2 u 3 ,x / 2) dx. As a r esult, we obtain right aw ay from (3.2 3) that dynamical sys tem (3.45) is a Hamiltonian sy stem o n the functional manifo ld M 4 , that is (3.49) K = − ϑ g r ad H ( η ) , where the Hamiltonian functional equa ls (3.50) H ( η ) := ( ψ ( η ) , K ) − L ( η ) = Z 2 π 0 ( u 1 u 4 ,x − u 2 u 3 ,x ) dx and the co -implectic op era tor equals (3.51) η − 1 := ψ ′ ( η ) − ψ ′ , ∗ ( η ) =        0 0 − ∂ ∂ u 3 ,x u 4 ,x 0 ∂ 0 − ∂ u 2 ,x u 4 ,x − ∂ 0 0 ∂ u 1 ,x u 4 ,x u 3 ,x u 4 ,x ∂ − u 2 ,x u 4 ,x ∂ u 1 ,x u 4 ,x ∂ 1 2 [ u − 2 4 ,x ( u 2 2 ,x − 2 u 1 ,x u 3 ,x ) ∂ + + ∂ ( u 2 2 ,x − 2 u 1 ,x u 3 ,x ) u − 2 4 ,x ]        . The la tter is degenera te: the r elationship η − 1 ( u 1 ,x , u 2 ,x , u 3 ,x , u 4 ,x ) ⊺ = 0 exa ctly on the whole manifold M 4 , but the inv er se to (3.51) ex ists and can b e calcula ted analytica lly . Pro ceed now to constr ucting other s olutions to the generating equation (3.46) at the reduced case L = 0 , having rewritten it in the following for m: D t ψ 1 − u 2 ,x ψ 2 − u 3 ,x ψ 3 − u 4 ,x ψ 4 = 0 , D t ψ 2 + ψ 1 + u 1 ,x ψ 2 = 0 , D t ψ 3 + ψ 2 + u 1 ,x ψ 3 = 0 , (3.52) D t ψ 4 + ψ 3 + u 1 ,x ψ 4 = 0 , where ψ := ( ψ 1 , ψ 2 , ψ 3 , ψ 4 ) ⊺ ∈ T ∗ ( M 4 ) . The latter is eq uiv alent to the s ystem of differential-functional relationships (3.53) D t ( αψ 1 ) − ψ 1 D t α − ψ 2 D 2 t α − ψ 3 D 3 t α − ψ 4 D 4 t α = 0 , D t ( αψ 2 ) + αψ 1 = 0 , D t ( αψ 3 ) + αψ 2 = 0 , D t ( αψ 4 ) + αψ 4 = 0 , 16 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 where we hav e deno ted the function α := u − 1 4 ,x . F r om (3.5 3) one ensues, by mea ns of s imple ca lculations, the following genera ting differential-functional equation (3.54) D 2 t ( αψ 1 ) = 0 on the manifold M 4 . As ab ov e in the case N = 3 , the obtaine d equatio n (3.54) can b e easily enough solved b y mea ns of the differential-algebraic approa ch, d evised b efo re in [3], which based on the following observ ation: owing to the equality D 5 t α = 0 the s et (3.55) A{ u } := { f 0 α + f 1 D t α + f 2 D 2 t α + f 3 D 3 t α + f 4 D 4 t α ∈ K 4 { u } : f j ∈ K 4 { u } , j = 0 , 4 } generates a finite dimensional in v a riant differential ideal of the differential ring K 4 { u } := K{ u }| D 4 t u =0 . As a result of constructing solutions to the f unctional-differential equation (3 .54), as non-symmetric elemen ts of the ideal (3 .55), one finds the following expres sion: (3.56) ψ ( ϑ ) = ( u 2 ,x u 4 − u 2 u 4 ,x , u 4 ,x u 1 − u 4 u 1 ,x , 2 u 4 − ( u 1 u 3 − u 2 2 / 2) x , 7 u 3 − u 2 u 1 ,x + u 1 u 2 ,x + u 3 ,x u 4 ,x [ u 4 − ( u 1 u 3 − u 2 2 / 2) x ]) ⊺ , L ( ϑ ) = 0 . Now ta king int o account the expression (3.23) one easily obtains the se cond co-Poissonia n op era tor ϑ − 1 = ψ ′ ( ϑ ) − ψ ′ , ∗ ( ϑ ) in the following matrix for m: ϑ − 1 =                             0 − 3 u 4 ,x u 3 ∂ − ∂ u 2 − u 2 ∂ − − u 3 ∂ u 3 ,x u 4 ,x − − u 3 ,x u 4 ,x ∂ u 3 3 u 4 ,x 0 u 2 ∂ ∂ u 1 + u 1 ∂ + + u 2 ∂ u 3 ,x u 4 ,x + + u 3 ,x u 4 ,x ∂ u 2 ∂ u 3 ∂ u 2 − u 1 ∂ − u 1 ∂ − 5 + ∂ ( u 4 − u 2 u 2 ,x u 4 ,x ) − − ∂ ( u 1 u 3 ) x u 4 ,x − u 1 ∂ u 3 ,x u 4 ,x − ∂ u 2 − u 2 ∂ − − u 3 ∂ u 3 ,x u 4 ,x − − u 3 ,x u 4 ,x ∂ u 3 ∂ u 1 + u 1 ∂ + + u 2 ∂ u 3 ,x u 4 ,x + + u 3 ,x u 4 ,x ∂ u 2 5 + ( u 4 − u 2 u 2 ,x u 4 ,x ) ∂ − − ( u 1 u 3 ) x u 4 ,x ∂ − u 3 ,x u 4 ,x ∂ u 1 u 4 − u 2 u 2 ,x u 2 4 ,x ∂ + − u 3 ,x u 2 4 ,x ∂ − ∂ u 3 ,x u 2 4 ,x − − ∂ u 4 − u 2 u 2 ,x u 2 4 ,x                             , (3.57) satisfying jo in tly with (3.51) the g radient r elationship (3.18). 4. Conclusion A differential-algebra ic a pproach, elab or ated in this article for revisiting the integrabilit y a nalysis of generalized Riemann type hydrodyna mical e quation (1.1), made it p ossible to constr uct new and mor e simpler Lax type representation for the general case N ∈ Z + , contrary to those c onstructed b efor e in [17, 3 ]. This representation, obtained by means of the sugg ested differ ent ial-alg ebraic appro ach, prov es also to co incide up to the scaling pa rameter λ ∈ C with that first o btained by Z. Popowicz in [4]. It is worth to mention that the co rresp onding La x t yp e r epresentations for the gener alized Riemann type hierarch y ar e w ell fitting for constructing the related co mpatible Poissonian structur es and proving their corres po nding bi-Hamiltonian integrability . The c orresp onding calculations are fulfilled in details for case N = 1 , 2 and preliminary results ar e obtained for ca ses N = 3 , 4 by mea ns of the ge ometric metho d, devised in [25, 28, 3]. It w as also demonstr ated that the corresp onding differe n tial-functional relationships give rise to the suitable compatible Poissonian structures and prese n t a very interesting mathematical problem fro m the differential-algebraic p oint of view, which is pla nned to b e studied in other work. THE DIFFERENTIAL-ALGEBRAIC AND BI-HAMIL TONIAN INTEGRABILITY ANAL YSIS OF THE RIEMANN TYPE HIERARCHY REVISITED 17 5. Acknow ledgements The authors ar e m uch obliged to P rof. Denis Blackmore (NJIT, New Je rsey , USA) for very ins tru- men tal discussion of t he work, v aluable advices, comments and remarks. Sp ecia l ac knowledgmen t b elong s to the Scientific and T echnological Resear ch Council o f T urkey (T ¨ UBIT AK-201 1) fo r a pa rtial supp or t of the resear ch b y A.K. Pryk ar patsky a nd Y.A. P ryk arpa tsky . M.V.P . was in par t supp orted by RFBR grant 08- 01-00 054 and a gra n t o f the RAS Pr esidium ”F undamental P roblems in Nonlinea r Dynamics”. He also thanks the W r o claw Universit y for the hos pitality . Authors are also gra teful to Prof. Z. P op owicz (W ro c law Universit y , Poland) for sending his rep or t for the Symp osium ” In tegrable Systems: Zielona Gora - May 30- 31, 2011” b efor e its publication. References [1] Golenia J., P a vlov M., Popowicz Z. and Pryk arpatsky A. On a nonlo cal Ostro vsky-Whitham type dynamical system, its Riemann type inhomogenious r egularizations and their integrabilit y . SIGMA, 6, 2010, 1-13 [2] Pr yk arpatsky A. K. and Prytula M. M. The gradient-holonomic i n tegrability analysis of a Whitham-type nonlinear dynamical mo del for a r elaxing medium with spatial memory . Nonlinearity 19 (2006) 2115–21 22 [3] Popowicz Z. and Pryk arpatsky A. K. 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N.Y., Plenum, 1984 [25] Pryk arpatsky A. and Mykytyuk I. Algebraic Integrabilit y of nonlinear dynamical systems on manifolds: classical and quan tum aspects. Kluw er Academic Publishers , the Netherlands, 1998 [26] Mitropolsky Y u., Bogolub ov N. (jr .), Pryk arpatsky A. and Samoy lenk o V . In tegrable dynamical system: sp ectral and differen tial-geometric asp ects. Kiev, ”Nauko v a Dunk a”, 1987. (in Russian) [27] F addeev L.D. , T akht adjan L.A . Hamiltonian methods i n the theory of soli tons. NY, Springer, 1987 [28] Blac kmore D., Pryk arpatsky A.K. and Samo ylenk o V.Hr. Nonlinear dynamical systems of mathematical ph ysics: sp ec- tral and differential-geometrical in tegrability analysis. W orld Scientific Publ., NJ, USA, 2011. 18 Y AREMA A. PR YKARP A TSKY 1 , ORE ST D. AR TEMOVYCH 2 , MAXIM V. P A VLOV 3 AND ANA TOLIY K. PR YKARP A TSK Y 4 [29] Hen tosh O., Prytula M . and Pryk arpatsky A. Differential-geomet ric and Lie-algebraic foundations of inv estigating nonlinear dynamical systems on functional manifolds. The Second edition. Lviv Univ er sity Publ., Lviv, Ukraine, 2006 (in Ukrainian) 1 The Dep art ment of Applied Ma themat ics at the Agrarian University of Krakow , Poland, a n d the Institute of Ma thema tics of NAS, Kyiv, Ukrain e E-mail addr ess : yarpry@gmail.c om 2 The Dep ar tment of Algeb ra and Topology a t the F a cult y of Mat hema tics a n d Informa tics of the V asyl Stef a n yk Pre-Carp a thian Na tional University, Iv an o-Frankivsk, Ukra ine, and the Institute of Mathema tics and In forma tics a t the T a deusz Ko sciuszko University of Technology , Crac ´ ow, Polan d; E-mail addr ess : artemo@usk.pk. edu.pl 3 The Dep ar tment of Ma thema tical Physics, P.N. Lebedev Physical Institute of Russ ian Academy of Sciences, Moscow, 53 Leninskij Pr ospekt, Mosco w, Russia, and Laborato r y of Geometric Methods in Ma thema tical Physics, Dep ar tment of Mech. & Ma th., Mosco w St a te Un iversity, 1 Leninskie gor y, Moscow , R ussia E-mail addr ess : maxim@math.sin ica.edu.tw 4 AGH University of Science and Technology, the Dep ar tment of M in ing Geodesy, Crac ´ ow 300 59, Poland E-mail addr ess : pryk.anat@ua.f m