A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

A VER TEX OPERA TOR REPRESENT A T ION OF SOLUTI ONS TO THE GUREVICH-ZYBIN HYDR OD YNAMICAL EQUA T ION Y AREMA A. PR YKARP A TSKY 1 , 2 , DENIS BLACKMORE 3 , JOLANT A GOLENIA, AND ANA TOLIY K. PR YKARP A TSKY 5 , 6 Abstract. An approac h based on the sp ectral and Lie - algebraic tec hniques for constructing v ertex operator represen tation for solutions to a Ri emann t yp e hy drodynamical hierarch y is devised. A func tional represen tation generating an infinite hierarc hy of dispersive Lax t yp e int egrable flows is obtained. 1. Introduction Nonlinear hydro dynamic equa tions are of constant interest still from class ic al works by B. Rie- mann, w ho had extensively studied them in genera l three-dimensio nal case, having paid sp ecial attent ion to their one-dimensio nal spatial reduction, for w hich he devised the generalized method of character istics and Riemann inv aria n ts. These metho ds app ear e d to be very effective [1] in inv estigating man y t yp es of nonlinea r spatially one-dimensional s ystems o f hydro dynamical type and, in pa r ticular, the characteris tics metho d in the form of a ” recipro cal” transforma tion of v ari- ables ha s b een used rece ntly in studying a so called Gurevich-Zybin system [2, 3] in [8] and a Whitham type system in [6, 5]. Mo reov er, this method w as further effectively applied to studying solutions to a genera lized [5 ] ( owing to D. Holm a nd M. Pavlo v) Riemann type hydrodynamica l system (1.1) D N t u = 0 , D t := ∂ /∂ t + u∂ /∂ x, where N ∈ Z + and u ∈ C ∞ ( R 2 ; R ) is a smo oth function. Making use of nov el metho ds, devised in [24, 7] and base d b oth on the sp ectral theory [9, 16, 1 9, 18] and the differen tia l alg ebra techniques, the Lax type repres en tatio ns for the cases N = 1 , 4 were constructed in explicit for m. In this w o rk w e are int erested in constructing a so calle d vertex o pera tor representation [13, 14, 21, ? ] for solutions to the Gurevich-Zybin hydrody na mical hierar c hy (1.1) at N = 2 : (1.2)  D t u = u t + uu x = v , D t v = v t + uv x = 0 , making use an appr o ach r e cen tly devised in [2 2, 2 3] for the case of the cla ssical AKNS hierarch y of int egrable flows, and whic h can b e easily generaliz e d fo r treating the problem for arbitrar y integers N ∈ Z + . 2. A ver tex opera tor anal ysis W e b egin with a Lax type linear spectral problem [8, 5, 4] for the equation (1.1) at N = 2 : (2.1)  D t u = u t + uu x = v , D t v = v t + uv x = 0 , defined on the space of s moo th real-v alued 2 π -p erio dic functions ( u, v ) ⊺ ∈ M ⊂ C ∞ ( R / 2 π Z ; R 2 ) : (2.2) d f /dx = ℓ [ u, v ; λ ] f , ℓ [ u, v ; λ ] :=  − λu x / 2 − v x λ 2 / 2 λu x / 2  , Date : presen t. 1991 Mathematics Subje ct Classific ation. P r imary 58A30, 56B05 Secondary 34B15; P ACS 02.30.Ik,02.10.Ox,47.35.Fg . Key wor ds and phr ases. Lax type integ rability , vertex op erator represen tation, Lax integrabilit y , Lie-algebraic approac h. 1 2 Y ARE MA A. PR YKARP A TSKY 1 , 2 , DENIS BLA CKMORE 3 , JOLANT A GOLE NIA, AND ANA TOLIY K. PR YKARP A TSKY 5 , 6 where, by definition, v := D t u, f ∈ L ∞ ( R / 2 π Z ; C 2 ) and λ ∈ C is a sp ectral para me- ter. Assume that a vector function ( u, v ) ⊤ ∈ M dep ends parametric ally on the infinite set t := { t 1 , t 2 , t 3 , . . . } ∈ R Z + in such a wa y that the generalized Flo quet spectrum [9 , 15, 18] σ ( ℓ ) := { λ ∈ C : sup x ∈ R || f ( x ; λ ) || ∞ < ∞} of the linear problem (2.2) p ersists in b eing para- metrically is o-sp ectral, that is dσ ( ℓ ) /dt j = 0 for all t j ∈ R . The iso-s p ectrality condition gives rise to a hierarchy of commuting to each other nonlinear bi-Ha milto nia n dynamica l sys tems o n the functional manifold M in the g eneral form (2.3) d dt j ( u ( t ) , v ( t )) ⊤ = − ϑg radH j [ u, v ] := K j [ u ( t ) , v ( t )] , where K j : M → T ( M ) and H j ∈ D ( M ) , j ∈ Z + , are, resp e ctiv ely , vector fields a nd co nserv ation laws on the manifold M , which were b efore describ ed in [5, 4, 7], (2.4) ϑ :=  0 ∂ ∂ 0  is a Poisson structur e on the manifold M and, by definition, (2.5)  u ( t ) v ( t )  :=  u ( x, t 1 , t 2 , t 3 , ... ) v ( x.t 1 , t 2 , t 3 , ... )  for t ∈ R N . It is well kno wn [15, 1 8, 9, 16] that t he Casimir in v aria n ts, determining conserv ation la ws for dynamical systems (2 .3), ar e generated by the suitably norma lized mono dromy matrix ˜ S ( x ; λ ) ∈ E nd C 2 of the linear problem (2 .2) (2.6) ˜ S ( x ; λ ) = k ( λ ) S ( x ; λ ) − k ( λ ) 2 tr S ( x ; λ ) , where F ( y , x ; λ ) ∈ E nd C 2 is the matrix solution to the Cauch y problems (2.7) d dy F ( y, x ; λ ) = ℓ ( y ; λ ) F ( y , x ; λ ) , F ( y , x ; λ ) | y = x = I , for all λ ∈ C and x, y ∈ R , where I ∈ E nd C 2 is the ident it y matrix , S ( x ; λ ) := F ( x + 2 π , x ; λ ) is the usua l mono dromy matrix fo r the equation (2.7). Here the pa rameter k ( λ ) ∈ C is inv aria n t with respect to flo ws (2.3) and is chosen in such a way that the a symptotic co ndition (2.8) ˜ S ( x ; λ ) ∈ ˜ G − as λ → ∞ holds for a ll x ∈ R . Here ˜ G − ⊂ ˜ G , where ˜ G := ˜ G + ⊕ ˜ G − is the natural s plitting int o t wo a ffine suba lgebras of positive and negative λ -ex pa nsions of the cen trally extended [15, 25] affine curren t sl (2)-algebra ˆ G := ˜ G ⊕ C : (2.9) ˜ G := { a = X j ∈ Z , j ≪∞ a ( j ) ⊗ λ j : a ( j ) ∈ C ∞ ( R / 2 π Z ; sl (2; C )) } . The latter is endow ed with the Lie commut ator (2.10) [( a 1 , c 1 ) , ( a 2 , c 2 )] := ([ a 1 , a 2 ] , h a 1 , da 2 /dx i ) , where the sca lar pro duct is defined as (2.11) h a 1 , a 2 i := res λ = ∞ Z 2 π 0 tr( a 1 a 2 ) dx for any t wo elements a 1 , a 2 ∈ ˜ G with ” r es” and ”tr ” b eing the usual residue a nd trace maps, resp ectively . As the sp ectrum σ ( ℓ ) ⊂ C o f the pro blem (2.2) is supp osed to be para metrically independent, flo ws (2.3) are naturally asso ciated with ev olution equations (2.12) d ˜ S /dt j = [( λ j +1 ˜ S ) + , ˜ S ] for all j ∈ R , which are genera ted by the set I ( ˆ G ∗ ) of Casimir inv ariants of the coa djoin t actio n of the cur rent a lgebra ˆ G on a given element ℓ ( x ; λ ) ∈ ˜ G ∗ − ∼ = ˜ G + contained in the space of smo oth functionals D ( ˆ G ) . In particular , a functional γ ( λ ) ∈ I ( ˆ G ) if and only if (2.13) [ ˜ S ( x ; λ ) , ℓ ( x ; λ )] + d dx ˜ S ( x ; λ ) = 0 , THE VER TEX SOLUTION OF A RIEMANN TYPE HYDR ODYNAMICAL EQUA TION 3 where the gradient ˜ S ( x ; λ ) := gr ad γ ( λ )( ℓ ) ∈ ˜ G − is defined with resp ect to the scala r pro duct (2.11) by means of the v ariation (2.14) δ γ ( λ ) := h gr ad γ ( λ )( ℓ ) , δ ℓ i . T o construct the solution to matrix equation (2.13), we find pr eliminary a partial solution ˜ F ( y, x ; λ ) ∈ E nd C 2 , x, y ∈ R , to equation (2.7 ) satisfying the asymptotic Cauch y da ta (2.15) ˜ F ( y, x ; λ ) | y = x = I + O (1 /λ ) as λ → ∞ . It is eas y to c heck that (2.16) ˜ F ( y , x ; λ ) = ˜ e 1 ( y , x ; λ ) − ˜ β ( y ; λ ) λ ˜ e 2 ( y , x ; λ ) − λ ˜ α ( y ; λ ) ˜ e 1 ( y , x ; λ ) ˜ e 2 ( y , x ; λ ) ! , is a n exact functional solution to (2.7) satisfying condition (2.15), wher e we have defined ˜ e 1 ( y , x ; λ ) := exp { λ 2 [ u ( x ) − u ( y )] + λ Z y x ˜ α dv ( s ) } , (2.17) ˜ e 2 ( y , x ; λ ) := exp { λ 2 [ u ( y ) − u ( x )] − λ 2 Z y x ˜ β ds } , with the vector-functions α ± ∈ C ∞ ( R / 2 π Z ; R ) satisfying the fo llowing determining functional relationships: ˜ α = u x + ( u 2 x − 2 v x + ξ ˜ α ) 1 / 2 , ˜ β = u x − ( u 2 x − 2 v x + ξ ˜ β ) 1 / 2 , (2.18) as ξ := 1 /λ → 0 a nd existing when the condition ϕ ( x , t ) := p u 2 x − 2 v x 6 = 0 on the manifold M at t = 0 ∈ R N . The fundamental matrix F ( y , x ; λ ) ∈ E nd C 2 can be represented for all x, y ∈ R in the fo r m (2.19) F ( y, x ; λ ) = ˜ F ( y, x ; λ ) ˜ F − 1 ( x, x ; λ ) . Consequently , if one s ets y = x + 2 π in this formula and defines the expressio n (2.20) k ( λ ) := λ − 1 [ ˜ e 1 ( x + 2 π , x ; λ ) − ˜ e 2 ( x + 2 π , x ; λ )] − 1 , it follo ws from (2.6), (2.16) and (2 .1 9) that the exact fun ctional matrix representation (2.21) ˜ S ( x ; λ ) =   [ ˜ α ( x ; λ )+ ˜ β ( x ; λ )] 2 λ [ ˜ α ( x ; λ ) − ˜ β ( x ; λ )] ˜ α ˜ β λ 2 [ ˜ α ( x ; λ ) − ˜ β ( x ; λ )] − 1 [ ˜ α ( x ; λ ) − ˜ β ( x ; λ )] [ ˜ β ( x ; λ )+ ˜ α ( x ; λ )] 2 λ [ ˜ β ( x ; λ ) − ˜ α ( x ; λ )]   , satisfies the nece ssary co ndition (2.8) as λ → ∞ . R emark 2 .1 . The inv arianc e of the ex pr ession (2.2 0) with res p ect to the genera ting vector field (2.3) on the manifold M derives from the representation (2.19), the equa tio ns (2.13) and (2.22) d dt ˜ F ( y, x 0 ; µ ) = λ 3 µ − λ ˜ S ( x ; λ ) ˜ F ( y , x 0 ; µ ) , which follows natura lly from the determining matr ix flows (2.12) up on applying the transla tion y → y + 2 π . The ma trix expr ession (2.21) gives ris e to the following imp ortant functional rela tio nships: (2.23) 1 − λ ( ˜ s 11 − ˜ s 22 ) 2 ˜ s 21 = ˜ α, − 2 λ 2 ˜ s 12 1 − λ ( ˜ s 11 − ˜ s 22 ) = ˜ β , which allow to in tro duce in a natur a l wa y the v ertex opera tor vector fields (2.24) X ± λ = exp( ± D λ ) , D λ := X j ∈ Z + 1 ( j + 1 ) λ − ( j +1) d dt j +1 , acting on an arbitra ry smo oth function η ∈ C ∞ ( R Z + ; R ) by means of the shifting mappings: (2.25) X ± λ η ( x, t 1 , t 2 , ..., t j , ... ) := η ± ( x, t ; λ ) = = η ( x, t 1 ± 1 /λ, t 2 ± / (2 λ 2 ) , t 3 ± 1 / (3 λ 3 ) ..., t j ± 1 / ( j λ j ) , ... ) as λ → ∞ . Namely , w e following prop osition holds. 4 Y ARE MA A. PR YKARP A TSKY 1 , 2 , DENIS BLA CKMORE 3 , JOLANT A GOLE NIA, AND ANA TOLIY K. PR YKARP A TSKY 5 , 6 Prop osition 2.2. The functional vertex op er ator expr essions ˜ α ( x, t ; λ ) = X − λ α ( x, t ) = α − ( x, t ; λ ) , (2.26) ˜ β ( x, t ; λ ) = X + λ β ( x, t )) = β + ( x, t ; λ ) solve the functional e quations (2.18), that is α − = u x + ( u 2 x − 2 v x + ξ α − ) 1 / 2 , β + = u x − ( u 2 x − 2 v x + ξ β + ) 1 / 2 , (2.27) wher e t ∈ R Z + and ξ = 1 /λ → 0 . Pr o of. T o state this pro positio n it is eno ugh to show that the fo llowing r elationships hold: d dξ  1 − λ ( ˜ s 11 − ˜ s 22 ) 2 ˜ s 21  λ =1 /ξ = d dt  1 − λ ( ˜ s 11 − ˜ s 22 ) 2 ˜ s 21  λ =1 /ξ , d dξ  − 8 λ 2 ˜ s 12 1 − λ ( ˜ s 11 − ˜ s 22 )  λ =1 /ξ = d dt  − 8 λ 2 ˜ s 12 1 − λ ( ˜ s 11 − ˜ s 22 )  λ =1 /ξ (2.28) for an y parameter ξ → 0 , where b y definition (2.29) d dt := d dξ D λ     λ =1 /ξ = X j ∈ Z + ξ j d dt j +1 is a ge nerating evolution vector fie ld. Before doing this w e find the evolution e quation (2.30) d dt ˜ S ( x ; µ ) = [ λ 3 d dλ ˜ S ( x ; µ ) , ˜ S ( x ; λ )] on the matrix ˜ S ( x ; µ ) as µ, λ → ∞ , whic h entails the following differential relations hips: (2.31) d ˜ s 11 /dt = λ 3 ( ˜ s 21 d ˜ s 12 /dλ − ˜ s 12 d ˜ s 21 /dλ ) , d ˜ s 22 /dt = λ 3 ( ˜ s 12 d ˜ s 21 /dλ − ˜ s 21 d ˜ s 12 /dλ ) , d ˜ s 22 /dt = λ 3 [ ˜ s 12 d dλ ( ˜ s 11 − ˜ s 22 ) − ( ˜ s 11 − ˜ s 22 ) d ˜ s 12 dλ ) , d ˜ s 11 /dt = λ 3 [ ˜ s 21 d dλ ( ˜ s 22 − ˜ s 11 ) − ( ˜ s 22 − ˜ s 11 ) d ˜ s 21 dλ ) . Using these relationships (2.31), one can easily obtain b y means of simple, but rather cumbersome calculations, the neede d re la tionships (2.28). As their direc t co nsequences the vertex op er ator r epr esentations (2.26) for the vector functions ˜ α, ˜ β ∈ C ( R Z + ; R ) hold.  Now w e take in to account that, o wing to the determining functional representations (2.18), that the limits ∞ lim λ →∞ α − ( x, t ; λ ) = u x ( x, t ) + ϕ ( x, t ) , (2.32) lim λ →∞ β + ( x, t ; λ ) = u x ( x, t ) − ϕ ( x, t ) , ϕ ( x, t ) := p u 2 x ( x, t ) − 2 v x ( x, t ) , exist on the manifold M . Moreov er, ha ving iterated the functional r elationships (2.18), one can find that X − λ α = α − = u x + ϕ + ξ ( u xx ϕ + ϕ x ϕ )+ + ξ 2 2 ( u 2 xx + 2 u xx ϕ x − u 3 x ϕ ϕ 3 + ϕ xx ϕ + 5 ϕ 2 x ϕ 3 ) + O ( ξ 3 ) , X + λ β = β + = u x − ϕ − ξ ( u xx ϕ − ϕ x ϕ ) − (2.33) − ξ 2 2 ( u 2 xx − 2 u xx ϕ x + u 3 x ϕ ϕ 3 + ϕ xx ϕ + 5 ϕ 2 x ϕ 3 ) + O ( ξ 3 ) , which immediately yield the higher Riema nn type commuting nonlinear Lax integrable disp ersive dynamical systems o n the functional manifold M . F o r instanc e , making use of the relationships (2.34) lim λ →∞ [ α − ( x, t ; λ ) ± β + ( x, t ; λ )] / 2 =  u x ( x, t ) , ϕ ( x, t ) , THE VER TEX SOLUTION OF A RIEMANN TYPE HYDR ODYNAMICAL EQUA TION 5 one easily obta ins that (2.35) d dt 1  u x ϕ  =  − u xx /ϕ − ϕ x /ϕ  , d dt 2  u x ϕ  =  ( u 2 xx + 7 ϕ 2 x ) /ϕ 3 (2 u 3 x ϕ − 4 u x ϕ x ) /ϕ 3  , ..., and s o on, where ϕ = p u 2 x − 2 v x and we to ok in to account that the following asymptotic expan- sions hold (2.36) X − λ α ( x, t ; λ ) = u x + ϕ − ξ ( u x,t 1 + ϕ t 1 )+ + ξ 2 2 ( u x,t 1 ,t 1 + ϕ t 1 ,t 1 − u x,t 2 − ϕ t 2 ) + O ( ξ 3 ) , X + λ β ( x, t ; λ ) = u x − ϕ + ξ ( u x,t 1 − ϕ t 1 )+ + ξ 2 2 ( u x,t 1 ,t 1 − ϕ t 1 ,t 1 + u x,t 2 − ϕ t 2 ) + O ( ξ 3 ) as ξ = 1 /λ → 0 . It is worth here to men tion that the scheme devised a bove for finding the corres p onding vertex op erator repr esen tations for the Riemann type equa tion (2.1) can b e similarly g eneralized for treating others equations of the infinite hierarch y (1.1) when N ≥ 3 , having taking in to acco un t the existence of their suitable Lax type representations found befor e in recent works [24, 5, 4]. 3. Concluding remarks The vertex op erator functional r epresentations of the s olution to the Riema nn type hydro dy- namical equa tion (2.1) in the for m (2 .27) is cr ucially based on the r epresentations (2.23) and evolution e quations (2.28), which provide a very s traightforw ar d and tr a nsparent explanatio n o f many of “mir aculous” v er tex op erator calculations pre s en ted b efore bo th in [13, 14] and in [21]. It should b e noted that the effectiveness of our appr oach to studying the vertex op erato r repres e n- tation of the Riemann type hierar c hy ow es m uch to the imp ortant e x act re presentation (2.21) for the corr espo nding mono dromy matr ix, whos e pro perties are descr ibed by means o f applying the standard [15, 18, 16, 20] Lie-algebraic tec hniques. As an indication of p ossible future research, it should also b e mentioned that it would be interesting to generalize the vertex op erator approa c h devised in this work to o ther linear sp ectral problems suc h as those related to dynamica l systems with a pa rametrical spectr al [11, 17, 18] dependence, s patially tw o-dimensional [10], P avlov’s and heav enly [12] dynamica l systems. 4. Acknow ledgments D. Bla c k mo re wishes to thank the Natio nal Science F o undation for supp ort fr om NSF Gr an t CMMI - 10 2 9809 and his coauthor for enlisting him in the efforts that produce d this paper. A.K. Pryk arpa tsky co rdially thanks Profs. N. Bog o lubov (J r .) for useful dis c us sions of the r esults obtained. References [1] Whitham G.B. Linear and Nonlinear W a ves . Wiley-Int erscience, New Y ork, 1974, 221p [2] Gurevic h A.V. and Zybin K.P . Nondissipative gra vitational turbulence. Sov. Ph ys. –JETP , 67 (1988), p.p. 1–12 [3] Gurevic h A.V . and Zybin K .P . Large-scale structure of the U ni v erse. Analytic theory Sov. Phys. Usp. 38 (1995 ), p.p. 687–722 [4] Golenia J., Bogolubov N. (jr.), P op o wicz Z., P avlo v M. and Pryk ar patsky A. A new Riemann t yp e h ydro dy- namical hi erarc hy and its integ rability analysis. Pr eprint ICTP , IC/2009/095, 2010 [5] Golenia J., Pa vl o v M., Popo wi cz Z. and Pr yk ar patsky A. On a nonlocal Ostrov sky-Whitham type dynamical system, i ts Riemann type inhomogenious r egularizations and their integ rability . SIGMA, 6, 2010 , 1-13 [6] Pryk arpatsky A. K. and Prytula M. M. The gradient-holonomic inte gr abilit y analysis of a Whit ham-t ype non- linear dynamical model for a relaxing medium with spatial memory . Nonlinearity 19 (2006) 2115–21 22 [7] Popo w i cz Z. and Pryk arpatsky A. K. The non-p olynomial conserv ation laws and in tegrabilit y analysis of gen- eralized Riemann type hydrodynamical equations. Nonli nearit y 23 (2010) 2517–2537; arXi v:submit/004484 4 [nlin.SI] 21 May 2010 [8] Pa vlov M. The Gurevich -Zybin system. J. Ph ys. A: M ath. Gen. 38 (2005), p. 3823-384050 [9] Novik ov S.P . (Editor) Theory of Solitons, Moscow, Nauk a Publ., 1980 (in Russian). [10] Zakharo v V.E. , Manako v S.M. Construction of multi-dimensional nonli near integ rable systems and their solu- tions. F unct. Anal. Appl., 1985 , v. 19, N 2, p. 11-25 (in Russian) [11] Burtsev S.P ., Zakhario v V.E., Mikhaylo v A.V. Inn verse scattering problem with changing spectral parameter. Theor. Math. Ph ys., 1987, v. 70, N 3, p. 323-341 (un Russ i an) [12] Manak ov S.M. , Santini P .M. On the solutions of th e second hea venly and Pa vlov equations. J.Phys. A : Math and Theor., 2009, v.42, N 40, p. 404013-10 [13] Newe l l A.C. Solitons in mathemat ics and physics. Ar izona, SIAM Publ., 1985 6 Y ARE MA A. PR YKARP A TSKY 1 , 2 , DENIS BLA CKMORE 3 , JOLANT A GOLE NIA, AND ANA TOLIY K. PR YKARP A TSKY 5 , 6 [14] Dick ey L. A. Soliton equat ions and Hamil tonian systems. NY, W orld Scien tific, 1991 [15] F addeev L.D., T akh tadjan L.A. Hamiltonian methods in the theory of solitons. NY, Springer, 1987 [16] Pryk arpatsky A. and Mykyt yuk I. Algebraic in tegrabilit y of nonlinear dynamical systems on manifolds: classical and quantum asp ects. Klu wer Academic Publishers, the Netherlands, 1998, 553p. [17] Blac kmore D., Pr yk arpatsky A.K., Samoylenk o V. H. Nonlinear dynamical systems of mathematical ph ysics. NY, W orld Scientific, 2011 [18] Hen tosh O.Y e., Prytula M .M. and Pryk arpatsky A.K Different ial-geometric i n tegrability fundamen tals of non- linear dynamical s ystems on functional menifolds. (The second revised edition), Lviv Universit y Publi sher, Lviv, Ukraine, 2006 (in Ukrainian) [19] Mitrop olski Y u. A., B ogoliub ov N.N. (Jr.), Pryk arpatsky A.K., Samoilenk o V.Hr. In tegrable Dynamical Sys- tems. N auk a dumk a, Kiev, 1987 (in Russian) [20] Blaszak M. Multi-Hamil tonian theory of dynamical systems. Springer, Berlin, 1998 [21] Pritula G.M., V ekslerchik V.E. Conserv ation laws for the nonlinear Sc hr ¨ odinger equat ion in Miwa v ar iables. In verse Pr oblems, 2002, 18, p. 1355 (arxiv:nlin.SI/0008034, 2000) [22] Blac kmore D., Pryk arpatsky A.K. and Pryk arpatsky Y.A. Iso-sp ectrally in tegrable dynamical systems on dis- crete m anifolds: analytical aspect. Opuscula Mathematica, 2011, N2. [23] The AKNS hi erarc hy revisited: A v er tex operator approac h and its Lie-algebraic structure. [nlin.SI] [24] Pryk arpatsky A. K., Ar temovyc h O.D., Popo wi cz Z. and Pa vlov M.V. Differential-algebraic integ rability analysis of the generalized R iemann type and Kor tew eg–de V ries h ydro dynamical equations. J. Phys. A: Math. Theor. 43 (2010) 295205 (13pp) [25] Reyman A.G. , Semenov-Tian-Shansky . Int egrable systems. Moscow-Izhe vsk, R&C-Dynamics, 2003 (in Russian) 1 The Dep art ment of Applied Ma thema tics a t the Univ ersity, Krakow 30059, Poland, an d, 2 Dep ar tment of Differential Equa tions of the Institute ma them a tics a t NAS, Kyiv, Ukraine E-mail addr ess : yarpry@gmail.co m 3 Dep ar tment of Ma thema tical Sciences and Center for Applied Ma thema tics and S t a tistics, New Jersey Institute of Technology, New ark, NJ 0710 2, USA E-mail addr ess : deblac@m.njit.e du 4 The Dep ar tmen t of Applied Ma them a tics a t the AGH University of Science a nd Technology, Krakow 30059 , Poland, goljols@tlen.pl 5 The Dep a r tment of Mining Geodesy and Environment Engineering at the A GH University of Science and Technology , Krakow 30059, Poland, an d, 6 Dep ar tment of Economical Cyb ernetics a t the Iv an Franko Pedagogical St a te Un iversity, Drohobych, L viv region, Ukraine E-mail addr ess : pryk.anat@ua.fm , pryka nat@cybe rgal.com