Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

St¨ ac k el systems generating c oupled KdV hierarc hies and their finite-gap and rational solutions Maciej B laszak Departmen t of Phy sics, A. Mic kiewicz Univ ersit y Um ultow sk a 85, 61-614 P ozna ´ n, P oland blaszakm@am u.edu.pl Krzysztof Marciniak Departmen t of Science and T ec hnology Campus Norrk¨ oping, Link¨ oping Univ ersit y 601-74 Norrk¨ oping, Sw eden krzma@itn.liu.se No v ember 20, 2018 Abstract W e sho w how to generate coupled Kd V hierarc hies from St¨ ac kel separable sy stems of Benenti type. W e further show that the solutions of th ese St¨ ac kel systems generate large class of finite-gap and ra tional solutions for cKdV hierarchies. Most of t h ese solutions are new. 1 In tro duction In [1] we pres en ted a s ystematic metho d o f pas sing from St¨ ack el separ able sys tems to infinite hierarchies of commuting nonlinear e v olutionar y PDE’s. W e presented the idea in a concr ete case o f St¨ ac kel systems of Benenti type. In this pap er we recogniz e the obta ined hierarchies as the well known coupled K ortew eg - de V ries (cKdV) hierarchies of An tonowicz a nd F or dy [2] wr itten in a different r epresent a tion. W e also clarify and significantly simplify the appr oac h develop ed in [1]. The ma in idea of the pap er is how ever to present a new metho d o f gener ating so lutions of soliton hierarchies from solutions of the related St¨ ack el systems. F ro m the very b eginning of developmen t of theo ry of integrable systems in the late 60 ’s ma jor efforts hav e b een put into constructing v arious wa ys of finding their solutions. Among many others, a p ossible wa y of finding solutions of integrable systems is through v arious kinds o f symmetry reduction, where one star ts from an infinite-dimensional integrable system and obtains after such r eduction a n integrable ODE. If one then succeeds in so lving this ODE (for example by finding separatio n co ordinates in c ase of Hamiltonian systems) o ne ca n then reconstruct the cor respo nding particular so lutions of the int eg rable PDE. This method or iginated in [3] and has been developed in [2], [4], [5], [6 ], [7] (see also [8]) and later in a lar ge num be r of pap ers. In this pap er we pres en t an opp osite a pproach in the se nse tha t we sta rt with large classes of St¨ ack el sy stems written in separation coor dinates so that their solutions are e xplicitly known and in few steps we co nstruct from these St¨ ack el systems an infinite hiera rc hy of commutin g evolutionary P DE’s while the solutions of the conside red St¨ ackel systems beco me particular multi-time solutions o f the systems of the obta ined hierarch y . In this wa y we pro duce b oth w ell-known and new finite-gap type s olutions of the KdV hierarch y a s well a s new rational and finite-ga p so lutions of cKdV hierarchies. In many cases this metho d also lea ds to implicit solutions of cKdV hier archies. The pap er is o rganized as follows. In Sectio n 2 we briefly describ e the starting p oint of our co nsid- erations, that is St¨ ack el separ able systems of Benenti t yp e, including their general solution. In section 3 we rela te with o ur St¨ ac kel systems a cla ss o f weakly-nonlinear semi-Hamiltonian s ystems (i.e. a class o f 1 hydrodynamic-type systems) that a re reductions of the so ca lled universal hierar c hy [ 9]. These s ystems are in our formulation defined b y Killing tensors o f St¨ ack el metrics. In sectio n 4 we explicitly construct hierarchies of commuting evolutionary PDE’s and present a transfo rmation that ma ps these hiera rc hies onto the well kno wn cKdV hierarchies of An tonowicz and F ordy [2]. This section contains the main result of this pap er, i.e. Theorem 8, that pr oduces lar ge families of solutions to our coupled hier archies. Finally , section 5 is dev o ted to studying specific cla sses of solutions that w e call zero- energy solutions that con tain bo th ra tional and implicit solutions of o ur hier archies. 2 St¨ ac k el systems of Benen ti t yp e Let us consider a set of canonical (Darb oux) co ordinates ( λ, µ ) = ( λ 1 . . . , , λ n , µ 1 , . . . , µ 1 ) on a 2 n - dimensional Poisson manifo ld M . A set of n relatio ns of the for m ϕ i ( λ i , µ i , a 1 , . . . , a n ) = 0, i = 1 , . . . , n , a i ∈ R (1) (each inv olving o nly one pair λ i , µ i of canonical co ordinates) are called separ ation relations [1 0] provided that det  ∂ ϕ i ∂ a j  6 = 0. Reso lving (lo cally) equations (1) with resp ect to a i we obtain a i = H i ( λ, µ ) , i = 1 , . . . , n. (2) with some new functions (Hamiltonians) H i ( λ, µ ) tha t in turn define n cano nical Hamiltonian systems on M : λ t i = ∂ H i ∂ µ , µ t i = − ∂ H i ∂ λ , i = 1 , ..., n (3) (here and in what follows the s ubscript denotes deriv ative with resp ect to the subsc ript v aria ble). F ro m this s etting it follows immediately that the Hamiltonians H i Poisson commute. The c orresp onding Hamilton-Jaco bi eq uations for all Hamiltonians H i are separable in the ( λ, µ )-v aria bles since they a re algebraic ally equiv alent to the separa tion r elations (1). In this article we cons ider a sp ecial but imp ortant class o f separa tion relations: n X j =1 a j λ n − j i = Af ( λ i ) µ 2 i + B γ ( λ i ), i = 1 , . . . , n, (4) where A a nd B are t wo constants. Notice t ha t since the f unctions γ and f do not depend on i the relations (4) can in fact b e co nsidered as n copies of a curve - the so ca lled sep ar ation curve in λ - µ plane. The Hamiltonian s ystems obtained from this clas s of s eparation relatio ns hav e b een widely studied and are also k no wn a s B enen ti systems . Benenti systems constitute the simplest, but still very wide, class of all po ssible St¨ ac kel separable systems. It can b e shown [11] that this class contains all quadratic in momenta St¨ ack el separable sys tems s ince all other systems of this type are constructed from (4) b y appr opriate generalized St¨ ack el transforms and related recipro cal transformatio ns. Below w e remind some established facts ab out Benenti systems. The rela tions (4) are linear in the co efficients a i . Solving these relations with res pect to a i we obtain a i = Aµ T K i G ( f ) µ + B V i ( γ ) ≡ H i , i = 1 , . . . , n , (5) where we use the notation λ = ( λ 1 , . . . , λ n ) T and µ = ( µ 1 , . . . , µ n ) T . F unctions H i defined as the r igh t hand sides o f the solution (5) can b e (lo cally) interpreted as n qua dratic in momenta µ Hamiltonians on the phase spa ce M = T ∗ Q co tangen t to a Riemannian manifold Q equipp ed with the contra v aria n t metric tensor G ( f ) dep ending o n function f only . As mentioned ab ov e, these Hamiltonians a re in in volution with res pect to the ca nonical Poisson bra c ket o n T ∗ Q . Moreov er , they are sepa rable in the sense of Hamilton-Jaco bi theor y since they by the very definitio n sa tisfy St¨ ack el r elations (1). The ob jects K i in (5) can be interpreted as (1 , 1)-type Killing tensors on Q related to the family of metrics G ( f ). The scalar functions V i ( γ ) depend only on the function γ and can be consider ed a s separa ble p otent ia ls. F urther, the metric tensor G and all the K illing tenso rs K i are diagona l in λ -v aria bles. More sp ecifically , in λ -v ar iables they attain the form 2 G ( f ) = diag  f ( λ 1 ) ∆ 1 , . . . , f ( λ n ) ∆ n  with ∆ i = Q j 6 = i ( λ i − λ j ) (6) and K i = − diag  ∂ q i ∂ λ 1 , · · · , ∂ q i ∂ λ n  i = 1 , . . . , n resp ectiv ely . Here a nd b elow q i = q i ( λ ) are Vi` ete p olynomials (signed symmetric p olynomials) in λ : q i ( λ ) = ( − 1) i X 1 ≤ s 1 n . The first p oten tials ar e trivial: V ( k ) i = δ i,n − k for k = 0 , 1 , . . . , n − 1. The first nontrivial p otential is V ( n ) i = − q i , F or k > n the p oten tials V ( k ) i bec ome complicated p olynomial functions of λ . The Lagr angians (9) for a n y sp ecific choice of m and n are denoted as L n,m,k i so that L n,m,k i = 1 4 A λ T t i g ( m ) K − 1 i λ t i − B V ( k ) i . F ro m now o n we will choo se the represe n tation j = 1 in (11) so that the v a riable t 1 plays the ro le o f the spa ce v ariable. W e denote therefor e this v a riable a s x : t 1 = x . The case of higher j is not discussed in this pap er. Thus, we consider the Killing sy stems o f the form λ t i = K i λ x ≡ Z n i ( λ, λ x ), i = 1 , . . . , n. (17) where the new upp er index n in the i th vector field Z i denotes the num b er of its co mponents; note also that the second low er index in Z is now always 1 and can ther efore b e omitted. Also, from now on w e denote the La grangian L n,m,k 1 simply as L n,m,k (to av oid the unnecessa ry index) so tha t L n,m,k = 1 4 A λ T x g ( m ) λ x − B V ( k ) 1 , i = 1 , . . . , n. (18) In order to p erform the aforementioned elimination pr ocedure we will fir st pa ss to Vi` ete co ordina tes (7). The Killing sys tems (17) ar e tensorial so in Vi ` ete co ordinates they have the form q t r = K r ( q ) q x or, explicitly d dt i q j = ( q j + i − 1 ) x + j − 1 X k =1  q k ( q j + i − k − 1 ) x − q j + i − k − 1 ( q k ) x  ≡ ( Z n i [ q ]) j i, j = 1 , . . . , n (19) where q α = 0 as so on as α > n and ( Z n i [ q ]) j denotes the j -th comp onent o f the vector field Z n i [ q ] ≡ Z n i ( q , q x ) (here and below the symbol f [ q ] will denote a differential function of q , that is a function depe nding on q and a finite num b er of its deriv atives). One can see that ( Z n i [ q ]) j =  Z n j [ q ]  i for all i, j = 1 , . . . , n . F urther, in Vi ` ete co ordina tes the Lag rangian (1 8) takes the form L n,m,k = L n,m,k ( q , q x ) = 1 4 A q T x g ( m ) q x − B V ( k ) 1 , i = 1 , . . . , n, (20) where g ( m ) ij = V (2 n − m − i − j ) 1 [1]. The Euler-La grange o perato r E t 1 in (10) will b e s imply denoted a s E , so in Vi` ete co ordinates E = ( E 1 , . . . E n ) , E i = δ δ q i . Theorem 6 L agr angian (20) satisfies t he fol lowing symm et ry r elations: 1. for α = 1 , . . . , n − 1 E i  L n,m,k  = E i − α  L n,m + α,k − α  , i = α + 1 , ..., n, (21) that c an a lso b e wr itten a s E i  L n,m,k  = E i + α  L n,m − α,k + α  , i = 1 , ..., n − α. (22) 2. E l  L n, 0 , 2 n + σ  = E l +1  L n +1 , 0 , 2 n + σ +2  , σ = 1 , ..., n − 1 , l = σ + 1 , ..., n. (23) 5 The pro of o f this theorem can b e found in [1]. This s eemingly technical theo rem gua ran tees tha t the form of E uler-Lagrang e equations sur viv es the pas sage fr om n -comp onent to ( n + 1 )-component Killing system and hence it will b e crucial for the construction of solito n hierar c hies b elow. The index σ will b e related with the num b er of comp onents of the o btained solito n systems. F or o ur further cons iderations we will also ne ed a hier arch y of infinite Killing systems d dt i q j = ( q j + i − 1 ) x + j − 1 X k =1  q k ( q j + i − k − 1 ) x − q j + i − k − 1 ( q k ) x  ≡ ( Z ∞ i [ q ]) j i, j = 1 , . . . ∞ (24) that is known as the universal hierar c hy a nd has b een considered in [9]. 4 Coupled KdV hierarc hies and their solutions W e now sho rtly remind the reader our s pecific elimination procedure from [1] that turns the dispersio nless Killing systems (19) into cKdV hierarchies. More sp ecifically , we sho w how to pro duce s (with s ∈ N ) N - comp onen t ( N ∈ N ) co mm uting vector fields (evolutionary sys tems) by eliminating s ome v ariables from a set of Killing systems (19) with the help of Euler -Lagrange equations for an appr opriate Lagr angian L n,m,k . The crucia l for this pr ocedure is that if applied to s + 1 instead of s it yields the same set of s co mm uting vector fields plus a n extra vector field that commutes with the firs t s fields. That means that this pro cedure lea ds in fact to an infinite hierar c hy of comm uting vector fields in the s ense that for ar bitrary s we c an pr oduce first s vector fields from the s ame infinite sequence of commuting vector fields. Mor eo ver it tur ns out that this w ay w e pr oduce vector fields with dis persion (soliton systems), namely w ell-known coupled KdV hierarc hies of An tonowicz and F ordy [2 ] (in a d iffer en t parametrization). Details ar e a s follows. Firstly , w e cho ose A = 1 a nd B = − 1. This sp ecific choice of A and B is intro duced only for a s moother ident ifica tion of our systems with the aforementioned cKdV hierar c hies; the elimination pro cedure works otherwise for ar bitrary v a lues o f A and B . Consider all N p ossible splittings N = σ + α with σ ∈ { 1 , . . . , N } and α ∈ { 0 , . . . , N − 1 } . Every such splitting leads to a sepa rate hierarch y . Consider als o the Killing systems (1 9), written in a shorthand wa y as: q t r = Z n r [ q 1 , . . . , q n ] , r = 1 , . . . , n (25) where q = ( q 1 , . . . q n ) T . Remark 7 The first s = n − N + 1 e quations in (25) ar e such that t hei r first N c omp onents c oincide with the c orr esp onding c omp onents of the infinite Kil ling hier ar chy (24). The r emaining n − s e quations in (25) ar e inc omplete with r esp e ct t o the infinite hier ar chy (24) sinc e b e ginning with the flow s + 1 systems (24) c ontain at its first N c omp onent s also the variables q n +1 , . . . , q n + N − 1 . Let us now choos e m = − α and k = 2 n + N in (14) so tha t f = λ − α and γ = λ 2 n + N and co nsider the last n − N E uler-Lagrang e eq uations asso ciated with L n, − α, 2 n + N . One can show [1] that they ha ve the form: E N +1  L n, − α, 2 n + N  ≡ 2 q n + ϕ ( α ) n − N +1 [ q 1 , ..., q n − 1 ] = 0 , E N +2  L n, − α, 2 n + N  ≡ 2 q n − 1 + ϕ ( α ) n − N [ q 1 , ..., q n − 2 ] = 0 , . . . E n  L n, − α, 2 n + N  ≡ 2 q N +1 + ϕ ( α ) 1 [ q 1 , ..., q N ] = 0 . (26) Due to their s tructure, equa tions (2 6) can b e ex plicitly solved with res pect to the v ar iables q N +1 , . . . , q n which yields q N +1 , . . . , q n as some differential functions of q 1 , . . . , q N : q N +1 = f ( α ) 1 [ q 1 , . . . , q N ] . . . q n = f ( α ) n − N +1 [ q 1 , . . . , q N ] . (27) 6 Naturally , the s olutions (8) (with o ur choice of f and γ ) solve b oth (2 5) and (26). Thus, within the class (8) of solutions (13) , we ca n use the E uler-Lagrang e equations (26) o r r ather their solved form (27) to successively express (eliminate) the v ar iables q N +1 , . . . , q n as differential functions o f q 1 , . . . , q N in (25). Plugging (27) into (2 5) pro duces n vector fields with N = σ + α comp onen ts: q t r = Z n,N ,α r [ q ] r = 1 , . . . , n , α ∈ { 0 , . . . , N − 1 } (28) (with q = ( q 1 , . . . , q N ) T ) . The higher comp onent s of (25) disapp ear after this elimination within our class (8) of solutions. Mor eo ver, since the first s equations in (25) ar e complete in the sense of Rema rk 7 it can b e shown that Z n,N ,α r [ q ] = Z N ,α r [ q ] r = 1 , . . . , s , meaning that the first s = n − N + 1 equa tions in (28) do not dep end on n . O bserv e a lso that equa tions (26) do not dep end on n. Actually , if n increases to n ′ the last n − N equations in (2 6) with this new n ′ will re main unalter ed. This means that we can r epeat this elimination pro cedure by taking n ′ = n + 1 instead of n (so that s incre ases to s + 1 and k = 2 n + N incr eases to 2( n + 1) + N = k + 2 while σ and α are kept unaltered. This new pro cedure (with n ′ = n + 1 instead of n ) will therefore lead to a s equence of s + 1 autonomo us N = ( σ + α )-compo nen t systems in which the firs t s systems will coincide with the corres ponding systems obtained fro m the prev ious pro cedure (with s ). This wa y we can obtain arbitrar y long sequences o f the same infinite set of commuting vector fields (soliton hiera rc hy): q t r = Z N ,α r [ q ] r = 1 , 2 , . . . ∞ , α ∈ { 0 , . . . , N − 1 } . (29) The second index α in (29) denotes different hierarchies. It can be sho wn [1] that vector fields Z N ,α r commute h Z N ,α i , Z N ,α j i = 0 fo r a n y i , j ∈ N . Note also that functions f ( α ) i in (27) dep end on α so that indeed the pr ocedure leads to N differe n t hierarchies. No w, the n functions λ i ( t 1 , . . . , t n ) given implicitly by the system o f equations t i + c i = ± 1 2 n X r =1 Z λ n − i + α/ 2 r p ∆ N r dλ r i = 1 , . . . , n (30) with ∆ N r = λ 2 n + N r + P n j =1 a j λ n − j r = Q 2 n + N i =1 ( λ r − E i ) are solutions of the firs t n equations of the N -comp onent hiera rc hy (29) with N = σ + α . The reas on is that equations (30) ar e just equations (8 ) with f = λ − α and γ = λ 2 n + N so they clearly solve all equations (2 8). More o ver, it can be shown, that these solutions are zer o on q n +1 , . . . , q n + N − 1 expressed as differential functions of q 1 , . . . , q N through an appropr iate system (26) (with n ′ = n + N − 1 ). It means that (30) indeed s olv e the first n eq uations in (29). Consider now the following infinite multi-Lagrang ian ” ladder” of E uler-Lagra nge equations of the form E 1  L n,m + j − 1 ,k − j +1  = E 2  L n,m + j − 2 ,k − j +2  = · · · = E n ( L n,m + j − n,k − j + n ) (31) with fixed m, k ∈ Z and with j = . . . , − 1 , 0 , 1 , . . . (the m ulti-La grangian form of (31) is due to Theor em 6). The equations (26) that we us e for v aria ble elimination are then a part of this infinite ladder with m = − α and k = 2 n + σ + α and with j = 1 , . . . , n . The equations (26) are the only eq uations in the ladder (31) (for this sp ecific choice of m and k ) that allow for the eliminatio n descr ibed a bov e. All o ther equations are co mplicated p olynomial differential equations with no o b vious structure tha t do not allow for any elimination pro cedure. As a c onsequence, ther e exists mor e so lutions of the type (30) asso ciated with all the La grangians L n,β − α, 2 N − β for β = 1 , . . . , n − 1 . How ever, one can show that for β = 2 the v ar iable q n + N − 1 is no t zero on these solutions so that these relations solve only first n − 1 equatio ns in (29). Mor e g enerally , for any β > 1 the obtained solutions λ i ( t 1 , . . . , t n ) will only satisfy first n − β + 1 equations of the hier arc hy (29). W e can thus formulate the following theorem. 7 Theorem 8 F or any β ∈ { 0 , . . . , n − 1 } and any N = σ + α < n the n functions λ i ( t 1 , . . . , t n ) given implicitly by the system of e quations t i + c i = ± 1 2 n X r =1 Z λ n − i + α/ 2 − β / 2 r q ∆ ( N ,β ) r dλ r i = 1 , . . . , n (32) and with ∆ ( N ,β ) r given by ∆ ( N ,β ) r = λ 2 n + N − β r + P n j =1 a j λ n − j r = Q 2 n + N − β i =1 ( λ r − E i ) ar e solut ions of t he fi r s t n − β + 1 (al l n for β = 0 , 1 ) e quations of t he N -c omp onent hier ar chy (29). The v ariables t 1 = x, t 2 , . . . , t n − β +1 in (3 2) are ”dyna mical times” (evolution para meters) o f the hierarch y (29 ) while the v ariables t n − β +2 , . . . , t n are just fr ee par ameters (i.e. the solutions (32) do not solve flows higher than the flow n umber n − β + 1). Note als o that, due to the str ucture of (7), all n functions λ i ( t 1 , . . . , t n ) o btained in (3 2) are nec essary in or der to co mpute N functions q i ( t 1 , . . . , t n − β +1 ) that so lv e (29). In the cas e N = 1 the so lutions (32) ar e finite-gap solutions fo r the KdV equation with the par ameters E i playing the role of endpo in ts of fo rbidden zones. F or β > 1 thes e solutions are up to our knowledge new. F or N > 1 all the so lutions (32) ar e new. Remark 9 F or a fixe d α ∈ { 0 , . . . , N − 1 } , the fol lowing map u r = ∂ V ( N , 2 N ) 1 ∂ q N +1 − r , r = 1 , . . . , N − α (33) u r = E N +1 − r  L N ,N − α, 2 N  , r = N − α + 1 , . . . , N (wher e V ( N , 2 N ) 1 denotes the sep ar able p otent ial V (2 N ) 1 in the dimension N ) tr ansforms the hier ar chy (29) to the hier ar chy gener ate d by the sp e ctr al pr oblem  λ α ∂ 2 x + P N i =1 u i λ N − i  Ψ = λ N Ψ . (34) This is the wel l known sp e ctr al pr oblem of A n t onowi cz and F or dy le ading to N -c omp onent cKdV hier ar- chies, one for e ach α ∈ { 0 , . . . , N − 1 } . Thus, N hier ar chies (29) ar e nothing else as (re p ar ametrize d) N cKdV hier ar chies fr om [2]. 5 Zero-energy solutions Let us now inv estigate - on few chosen exa mples - the nature of so lutions (32) in the case tha t all a i v anish (zer o-energy s olutions). In this cas e the solutio ns (3 2) ca n easily b e integrated yielding t i + c i = ± 1 2 − 2 i − σ n X r =1 λ 1 − i − σ/ 2 r i = 1 , . . . , n (35) and co n tain therefore no β (are the same for a ll β = 0 , . . . , n − 1) and no N except in n = s + N − 1. Therefore, we obta in the following cor ollary: Corollary 10 F or any N = σ + α < n , the n functions λ i ( t 1 , . . . , t n ) given implicitly by the system of e qu atio ns (35) ar e solutions of the fi rst n e quations of the N -c omp onent cKdV hier ar chy (29). Naturally , the solutions (35) also so lv e (on the s urface H i = 0 for all i ) all St¨ ack el systems (5) fo r which f ( λ ) γ ( λ ) = λ 2 n + σ . Observe also that the solutions (35) s olv e all the Euler -Lagrange eq uations for the infinite sequence o f Lag rangians L n, − α + j, 2 n + N − j where j ∈ Z . As a co nsequence, the second part of the map (33) is ze ro o n solutions (35). The reas on for this is that due to Theor em 6 the expr essions E N +1 − r  L N ,N − α, 2 N  , r = N − α + 1 , . . . , N 8 can b e wr itten as E r  L n, 0 , 2 n + σ  , r = n − α + 1 , . . . , n whereas all the ab ov e expr essions ar e members o f the sequence L n, − α + j, 2 n + N − j , j ∈ Z , so tha t they are zero o n s olutions (35). This implies that as so on as α > 0 the solutions (35) in the representation o f An tonowicz and F ordy , reduce to the solutions o f the c orresp onding hierar c hy with the same σ but with α = 0. Example 1. W e start with the case N = 1. W e wish to obtain the firs t s = 3 flows in (29). There is now only one splitting N = σ + α po ssible, namely σ = 1 and α = 0. This choice leads to the usua l K dV hierarch y . W e hav e to take n = s + N − 1 = 3. The Killing systems (25) hav e in this case the form d dt 1   q 1 q 2 q 3   =   q 1 ,x q 2 ,x q 3 ,x   = Z 3 1 d dt 2   q 1 q 2 q 3   =   q 2 ,x q 3 ,x + q 1 q 2 ,x − q 2 q 1 ,x q 1 q 3 ,x − q 3 q 1 ,x   = Z 3 2 (36) d dt 3   q 1 q 2 q 3   =   q 3 ,x q 1 q 3 ,x − q 3 q 1 ,x q 2 q 3 ,x − q 3 q 2 ,x   = Z 3 3 The Lagr angian L n, − α, 2 n + σ + α = L 3 , 0 , 7 is L 3 , 0 , 7 = 1 4  q 2 1 − q 2  q 2 1 ,x − 1 2 q 1 q 1 ,x q 2 x + 1 2 q 1 ,x q 3 ,x + 1 4 q 2 2 ,x + 2 q 2 q 3 − 3 q 2 1 q 3 − 3 q 1 q 2 2 + 4 q 3 1 q 2 − q 5 1 and the Euler- Lagrange equations (26) atta in the fo rm E 2  L 3 , 0 , 7  ≡ 2 q 3 − 6 q 1 q 2 + 4 q 3 1 + 1 4 q 2 1 ,x + 1 2 q 1 q 1 ,xx − 1 2 q 2 ,xx = 0 , E 3  L 3 , 0 , 7  ≡ 2 q 2 − 3 q 2 1 − 1 2 q 1 ,xx = 0 . Due to their str ucture, these equations can b e solved with r espect to q 2 , q 3 yielding (27) of the form q 2 = 1 4 q 1 ,xx + 3 2 q 2 1 , q 3 = 1 16 q 1 ,xxxx + 5 4 q 1 q 1 ,xx + 5 8 q 2 1 ,x + 5 2 q 3 1 . (37) Substituting (37) to the Killing systems (36) yields the three one-comp onent flows (29): q 1 ,t 1 = q 1 ,x = Z 1 , 0 1 q 1 ,t 2 = 1 4 q 1 ,xxx + 3 q 1 q 1 ,x = Z 1 , 0 2 (38) q 1 ,t 3 = 1 16 q 1 ,xxxxx + 5 2 q 1 ,x q 1 ,xx + 5 4 q 1 q 1 ,xxx + 15 2 q 2 1 q 1 ,x = Z 1 , 0 3 which are just the first three flows of the KdV hiera rc hy . By taking larger s we ca n pr oduce an a rbitrary nu mber o f flows fro m the KdV hierarchy . Now, accor ding to Coro llary 10, the formula (35) yields so me sp ecific solutions o f all three flows in (38). Explicitly , this formula reads (with x = t 1 , c i = 0 a nd + sign in (35)): x = − P 3 i =1 z i = − ρ 1 t 2 = − 1 3 P 3 i =1 z 3 i = − 1 3  ρ 3 1 − 3 ρ 1 ρ 2 + 3 ρ 3  (39) t 3 = − 1 5 P 3 i =1 z 5 i = − 1 5  ρ 5 1 − 5( ρ 1 ρ 2 − ρ 3 )( ρ 2 1 − ρ 2 )  where z i = λ − 1 / 2 i , i = 1 , 2 , 3 and where ρ 1 = P 3 i =1 z i , ρ 2 = z 1 z 2 + z 1 z 3 + z 2 z 3 and ρ 3 = z 1 z 2 z 3 are elementary symmetric p olynomials in z i . The rig h t hand sides of (39) follow from Newton formulas: P n i =1 z m i = X α 1 +2 α 2 + ... + nα n = m ( − 1) a 2 + α 4 + α 6 + ··· m ( α 1 + α 2 + · · · + α n − 1)! α 1 ! . . . α n ! ρ α 1 1 ρ α 2 2 . . . ρ α n n , m = 1 , . . . , n (40) 9 that can b e easily ex tended to the case m ≥ n by taking larger n and putting all higher ρ i equal to zero . The system (39) ca n b e solved explicitly y ielding the 3 -time solutions ρ 1 = − x ρ 2 = 15 t 3 + 2 x 5 − 15 x 2 t 2 5 ( x 3 − 3 t 2 ) (41) ρ 3 = − 15 t 2 x 3 − 45 t 2 2 + x 6 + 45 xt 3 15 ( x 3 − 3 t 2 ) On the other hand, it is ea sy to see that q 1 =  ρ 2 ρ 3  2 − 2 ρ 1 ρ 3 . (42) Plugging (41) in to (42) we finally obtain a 3-time solution of the first three flows (38) of the KdV hier arch y . It has a ra ther complicated, r ational form: q 1 ( x, t 2 , t 3 ) = − 3  675 t 2 3 − 270 t 3 x 5 + 2 x 10 + 675 x 4 t 2 2 − 1350 xt 3 2  ( − 15 t 2 x 3 − 45 t 2 2 + x 6 + 45 xt 3 ) 2 , (43) By taking larger s we c an in this way obtain s -time solutions of first s flows o f the KdV hierarch y . Finally , the map (33) is in this case triv ial and rea ds u 1 = 2 q 1 . Example 2. Let us consider the t wo field case: N = 2. There are now tw o splittings p ossible: N = σ + α = 2 + 0 a nd N = σ + α = 1 + 1. W e consider only the fir st tw o flows s = 2 (i.e. only the fir st nontrivial flow) so that n = s + N − 1 = 3 as b efore and therefore the or iginal Killing systems are as b efore, i.e (36) - we just consider the first tw o of them. Let us first consider the splitting N = σ + α = 2 + 0. The Lagr angian L n, − α, 2 n + σ + α = L 3 , 0 , 8 is L 3 , 0 , 8 = 1 4  q 2 1 − q 2  q 2 1 ,x − 1 2 q 1 q 1 ,x q 2 x + 1 2 q 1 ,x q 3 ,x + 1 4 q 2 2 ,x + + q 2 3 − 6 q 1 q 2 q 3 + 4 q 3 q 3 1 − q 3 2 + 6 q 2 2 q 2 1 − 5 q 2 q 4 1 + q 6 1 . (note that its kinetic energy par t is the same as in L 3 , 0 , 7 ab o ve). The formulas (26) contain only one equation that ca n b e s olv ed with r espect to q 3 yielding (27) of the form q 3 = 3 q 1 q 2 − 2 q 3 1 + 1 4 q 1 ,xx . Substituting this to (25) (with s = 2 now) yields the fir st tw o flows of the firs t (i.e with α = 0) 2- comp onen t cKdV hier arch y : d dt 1  q 1 q 2  =  q 1 ,x q 2 ,x  = Z 3 , 0 1 d dt 2  q 1 q 2  =  q 2 ,x 2 q 2 q 1 ,x + 4 q 1 q 2 ,x − 6 q 2 1 q 1 ,x + 1 4 q 1 ,xxx  = Z 3 , 0 2 (44) The zero -energy solutions (35) (ag ain with a ll c i = 0 and with the plus sig n only) attain now the form: x = − 1 2 P 3 i =1 z i = − ρ 1 t 2 = − 1 4 P 3 i =1 z 2 i = − 1 3  ρ 3 1 − 3 ρ 1 ρ 2 + 3 ρ 3  (45) t 3 = − 1 6 P 3 i =1 z 3 i = − 1 5  ρ 5 1 − 5( ρ 1 ρ 2 − ρ 3 )( ρ 2 1 − ρ 2 )  10 where ρ i again denote element a ry symmetric p olynomials in z i but where now z i = λ − 1 i . Solving (45) yields the following 3 -time solutions : ρ 1 = − 2 x ρ 2 = 2 x 2 + 2 t 2 (46) ρ 3 = − 4 3 x 3 − 4 xt 2 − 2 t 3 where the v ariable t 3 plays the role of a free parameter for eqs.(4 4) . Moreover, we have q 1 = − ρ 2 ρ 3 , q 2 = ρ 1 ρ 3 . (47) Plugging (46) into (4 7) we obtain the fo llo wing solutio ns for (44) q 1 ( x, t 2 , t 3 ) = 3( t 2 + x 2 ) 3 t 3 + 2 x 3 + 6 xt 2 , q 2 ( x, t 2 , t 3 ) = 3 x 3 t 3 + 2 x 3 + 6 xt 2 (48) Note tha t Cor ollary 10 implies that the functions (48) solve first n = 3 flows of the hierarchy (29) with N = 2 and α = 0. In order to compute this third flow we just need to take s = 3 in the elimination pro cedure. The re sult is d dt 3  q 1 q 2  =  3 q 2 q 1 ,x + 3 q 1 q 2 ,x − 6 q 2 1 q 1 ,x + 1 4 q 1 ,xxx 6 q 1 q 2 q 1 ,x − 18 q 3 1 q 1 ,x + 6 q 2 1 q 2 ,x + 3 4 q 1 q 1 ,xxx + 3 q 2 q 2 ,x + 1 4 q 2 ,xxx  = Z 3 , 0 3 . Let us finally pa ss to Antono wicz -F ordy v ar iables. The map (33) a ttains the for m u 1 = 2 q 1 , u 2 = 2 q 2 − 3 q 2 1 and it transfor ms b oth systems in (44) to the representation of Antonowicz and F o rdy . Explicitly: d dt 1  u 1 u 2  =  u 1 ,x u 2 ,x  = Z 3 , 0 1 [ u ] d dt 2  u 1 u 2  =  u 2 ,x + 3 2 u 1 u 1 ,x u 2 u 1 ,x + 1 2 u 1 u 2 ,x + 1 4 u 1 ,xxx  = Z 3 , 0 2 [ u ] (49) In the u - v a riables the solutions (4 8) yield s olutions for (4 9) and a ttain the form u 1 ( x, t 2 , t 3 ) = 6( t 2 + x 2 ) 3 t 3 + 2 x 3 + 6 xt 2 , u 2 ( x, t 2 , t 3 ) = 3(6 xt 3 − 5 x 4 − 6 x 2 t 2 − 9 t 2 2 ) (3 t 3 + 2 x 3 + 6 xt 2 ) 2 . Example 3. Let us now co nsider the case N = σ + α = 1 + 1. Again, we lo ok for the first s = 2 flows in (29) with N = 2 , a = 1. W e have to take n = s + N − 1 = 3 and the r ather lenght y La grangian L n, − α, 2 n + σ + α = L 3 , − 1 , 8 = − 1 4  q 3 1 − 2 q 1 q 2 + q 3  q 2 1 ,x + 1 2  q 2 1 − q 2  q 1 ,x q 2 ,x − 1 2 q 1 q 1 ,x q 3 ,x − 1 4 q 1 q 2 2 ,x + 1 2 q 2 ,x q 3 ,x + q 2 3 − 6 q 1 q 2 q 3 + 4 q 3 q 3 1 − q 3 2 + 6 q 2 2 q 2 1 − 5 q 2 q 4 1 + q 6 1 (the p otent ia l part is of course the same as in L 3 , 0 , 8 ab o ve). The elimination equa tions (2 6) y ield again only one equation that solved with res pect to q 3 reads q 3 = − 1 8 q 2 1 ,x − 2 q 3 1 + 3 q 1 q 2 − 1 4 q 1 q 1 ,xx + 1 4 q 2 ,xx . Plugging this into tw o first flows in (36) we o btain the first tw o flows o f the second (i.e. with α = 1) 2-field cKdV hier arc hy: d dt 1  q 1 q 2  =  q 1 ,x q 2 ,x  = Z 3 , 1 1 d dt 2  q 1 q 2  =  q 2 ,x 2 q 2 q 1 ,x + 4 q 1 q 2 ,x − 1 2 q 1 ,x q 1 ,xx − 6 q 2 1 q 1 ,x − 1 4 q 1 q 1 ,xxx + 1 4 q 2 ,xxx  = Z 3 , 1 2 (50) 11 Now, the solutions (35) in this case attain precisely the form (39), or (41) in s olv ed form, since b oth n and σ ar e the same in b oth cases. How ever, (50) ar e tw o -component, so in this case we need to express bo th q 1 and q 2 as functions of ρ i : q 1 =  ρ 2 ρ 3  2 − 2 ρ 1 ρ 3 , q 2 =  ρ 1 ρ 3  2 − 2 ρ 2 ρ 2 3 . (51) Substituting (41) into (5 1) we fina lly arr iv e at a 3-time so lution of (50) with t 3 as a free para meter: q 1 ( x, t 2 , t 3 ) = − 3  675 t 2 3 − 270 t 3 x 5 + 2 x 10 + 675 x 4 t 2 2 − 1350 xt 3 2  ( − 15 t 2 x 3 − 45 t 2 2 + x 6 + 45 xt 3 ) 2 , (52) q 2 ( x, t 2 , t 3 ) = 45( x 3 − 3 t 2 )( x 5 + 15 x 2 t 2 − 30 t 3 ) ( − 15 t 2 x 3 − 45 t 2 2 + x 6 + 45 xt 3 ) 2 . As in the previous example, the ab o ve solutions solve first n = 3 flows of this cK dV hierarch y whic h means that they also solve the next flow in the hierar c hy (with the dynamical time t 3 ). W e w ill how ever not write it her e. Finally , let us pass to the An tonowicz-F o rdy r epresen tatio n.The map (33) is now u 1 = 2 q 1 , u 2 = 2 q 2 − 3 q 2 1 − 1 2 q 1 ,xx so that (50) in An tonowicz-F o rdy v a riables rea ds d dt 1  u 1 u 2  =  u 1 ,x u 2 ,x  = Z 3 , 1 1 [ u ] d dt 2  u 1 u 2  =  u 2 ,x + 3 2 u 1 u 1 ,x + 1 4 u 1 ,xxx u 2 u 1 ,x + 1 2 u 1 u 2 ,x  = Z 3 , 1 2 [ u ] (53) In the u - v a riables the solutions (5 2) yield s olutions for (5 3) and a ttain the form u 1 ( x, t 2 , t 3 ) = − 6  675 t 2 3 − 270 t 3 x 5 + 2 x 10 + 675 x 4 t 2 2 − 1350 xt 3 2  ( − 15 t 2 x 3 − 45 t 2 2 + x 6 + 45 xt 3 ) 2 , u 2 ( x, t 2 , t 3 ) = 0 which is nothing else tha n the solution (43) of the first three flows of the KdV hiera rc hy (with N = 1), in accor dance with the obser v ation a t the b eginning of this s ection. Remark 11 It c an b e shown that our metho d yield s r ational solutions only for σ = 1 and σ = 2 . F or σ > 2 our m et ho d le ads to new implicit solutions of our cKdV hier ar chies . Example 4. Let us th us finally inv estigate the case N = 3 = σ + α = 3 + 0 that will lead to implicit solutions. W e take a gain s = 2 so that n = s + N − 1 = 4. The firs t s = 2 K illing systems in (2 5) hav e now the form d dt 1     q 1 q 2 q 3 q 4     =     q 1 ,x q 2 ,x q 3 ,x q 4 ,x     = Z 4 1 (54) d dt 2     q 1 q 2 q 3 q 4     =     q 2 ,x q 3 ,x + q 1 q 2 ,x − q 2 q 1 ,x q 4 ,x + q 1 q 3 ,x − q 3 q 1 ,x q 1 q 4 ,x − q 4 q 1 ,x     = Z 4 2 The Lagr angian L n, − α, 2 n + N = L 4 , 0 , 11 yields one elimination e quation (26): E 4 ( L 4 , 0 , 11 ) = 2 q 4 + 12 q 2 q 2 1 − 6 q 1 q 3 − 3 q 2 2 − 5 q 4 1 − 1 2 q 1 ,xx = 0 . 12 Solving this with r espect to q 4 we obtain q 4 = − 6 q 2 q 2 1 + 3 q 1 q 3 + 3 2 q 2 2 − 5 2 q 4 1 + 1 4 q 1 ,xx . Substituting this into (54) yields tw o first flows of the 3-co mponent cK dV hiera rc hy (29 ) with α = 0. d dt 1   q 1 q 2 q 3   =   q 1 ,x q 2 ,x q 3 ,x   = Z 3 , 0 1 (55) d dt 2   q 1 q 2 q 3   =   q 2 ,x q 3 ,x + q 1 q 2 ,x − q 2 q 1 ,x 2 q 3 q 1 ,x + 4 q 1 q 3 ,x − 6 q 2 1 q 2 ,x − 12 q 1 q 2 q 1 ,x + 3 q 2 q 2 ,x + 10 q 3 1 q 1 ,x + 1 4 q 1 ,xxx   = Z 3 , 0 2 . The solutions (35) are now (again with all c i = 0 and with the plus sign only): x = − 1 3 P 4 i =1 z 3 i = − 1 3  ρ 3 1 − 3 ρ 1 ρ 2 + 3 ρ 3  t 2 = − 1 5 P 4 i =1 z 5 i = − 1 5  ρ 5 1 − 5( ρ 1 ρ 2 − ρ 3 )( ρ 2 1 − ρ 2 )  (56) t 3 = − 1 7 P 4 i =1 z 7 i = − 1 7  ρ 7 1 − 7( ρ 1 ρ 2 − ρ 3 )  ( ρ 2 1 − ρ 2 ) 2 + ρ 1 ρ 3  − 7 ρ 4 ( ρ 3 1 − 2 ρ 1 ρ 2 + ρ 3 )  t 4 = − 1 9 P 4 i =1 z 9 i = − 1 9 P 9 with z i = λ − 1 / 2 i where P 9 is a complicated p olynomial of degree 9 in ρ i that can be obtained from Newton formulas (40). This system can no t b e algebr aically solved with resp ect to ρ i . How ever, if we embed the system (56) in the sy stem α = − P 5 i =1 z i = − ρ 1 x = − 1 3 P 5 i =1 z 3 i = − 1 3  ρ 3 1 − 3 ρ 1 ρ 2 + 3 ρ 3  t 2 = − 1 5 P 5 i =1 z 5 i = − 1 5  ρ 5 1 − 5( ρ 1 ρ 2 − ρ 3 )( ρ 2 1 − ρ 2 )  + 5 ρ 1 ρ 4 + 5 ρ 5 (57) t 3 = − 1 7 P 5 i =1 z 7 i = − 1 7  ρ 7 1 − 7( ρ 1 ρ 2 − ρ 3 )  ( ρ 2 1 − ρ 2 ) 2 + ρ 1 ρ 3  − 7 ρ 4 ( ρ 3 1 − 2 ρ 1 ρ 2 + ρ 3 )  + 7 ρ 2 1 ρ 5 t 4 = − 1 9 P 5 i =1 z 9 i = − 1 9 Q 9 where α is a par ameter, and wher e Q 9 is a p olynomial o f deg ree 9 in ρ 1 , . . . , ρ 5 such that Q 9 | ρ 5 =0 = P 9 , then obviously the solutio n of (57 ) with the co ndition ρ 5 = 0 will y ield the solution for (56). The system (57) is alg ebraically solv able and yields ρ i = R i ( α, x, t 2 , t 3 , t 4 ) (58) where R i are complicated ra tional functions of their ar gumen ts. The p olynomial equation ρ 5 = 0 yields then implicitly a (m ultiv alued) function α = f ( x, t 2 , t 3 , t 4 ). Now, using the fact tha t q 1 = −  ρ 3 ρ 4  2 + 2 ρ 2 ρ 4 , q 2 =  ρ 2 ρ 4  − 2 ρ 1 ρ 3 ρ 2 4 + 2 ρ 4 , q 3 = − ( ρ 4 1 + 2 ρ 2 2 − 4 ρ 2 1 ρ 2 + 4 ρ 1 ρ 3 − 4 ρ 4 ) ρ 2 4 we arrive at an implicit solution of (5 5) of the form ρ 5 ( α, x, t 2 , t 3 , t 4 ) = 0 , q i = r i ( α, x, t 2 , t 3 , t 4 ) , i = 1 , 2 , 3 , that is thus determined up to the implicitly expr essed function α = f ( x, t 2 , t 3 , t 4 ). The concr ete formulas hav e b een obtained with the help of Maple and are to o complicated to present them here. The map (33) do es not simplify these so lutions. 13 6 Conclusions In t his article we pr esen ted a metho d o f cons tructing coupled Korteweg - de V ries h ier archies from Be nen ti class of St¨ ac kel separable systems. Our method allows for pro ducing certain cla ss of solutions of these hierarchies from solutions of corr esponding St¨ ack el systems (Theor em 8 and C orollary 1 0). F or N = 1 and β = 0 , 1 these solutions a re known but for lar ger β and for la rger N they all se em to b e new. Also, for σ > 2 all our solutions a re implicit in the sense describ ed in Example 4 ab o ve. It has to b e stresse d that our metho d is g eneral in the sense that other separ ation relations lead to other hierarchies like for example coupled Har ry-Dym hierar c hies. These p ossibilities will be studied in a s eparate pap er. Considering the class of s olutions c onstructed in the pap er one arrives at natural question whether these solutins are somehow related with some class of symmetry reductions. So me hint is given in the case N = 1: the finite g ap solutions (32) for β = 0 and β = 1 for the KdV ca n be obta ined from its stationary flows constructed with the help of firs t t wo lo cal Hamiltonian repr esen tations of the K dV hierarchy . This question is b eyond the sco pe of this pa per but it certainly deserves a sepa rate study . 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