Lagrangian Approach to Dispersionless KdV Hierarchy

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 3 (2007), 096, 11 pages Lagrangian Approac h to Di sp ersionl ess KdV Hierarc h y Amita va CHOUDHURI † 1 , B. T ALUKDAR † 1 † 2 and U. DAS † 3 † 1 Dep artment of Physics, Visva-Bhar ati University, Santiniketan 731235, India E-mail: amitava ch26@yaho o.c om † 2 Dep artment of Physics, Visva-Bhar ati University, Santiniketan 731235, India E-mail: binoy123@bsnl.in † 3 Abhe danand a Mahavidyala ya, Sainthia 73123 4, India Received June 05, 20 07, in f ina l form Septem b er 1 6, 2007; P ublished o nline September 30, 2007 Original article is av a ilable at http: //www. emis. de/journals/SIGMA/2007/096/ Abstract. W e deriv e a Lagrangian based appro a ch to study the compatible Hamiltonian structure of the disp ersio nless KdV and sup er s ymmetric KdV hierar chies and claim that our treatmen t of the pr oblem serves as a very useful supplement o f the so-called r -matrix metho d. W e suggest spec if ic wa ys to cons tr uct results for conserved densities and Hamil- tonian operato rs. The Lag rangia n form ulation, via No ether ’s theorem, provides a method to ma ke the relation b etw een symmetries and conserved quantit ies more precise. W e have exploited this fact to study the v ar iational sy mmetr ies o f the disp er sionless K dV e quation. Key wor ds: hierarch y of disp ersio nle s s KdV equa tions; Lagra ng ian approach; bi-Hamiltonian structure; v ar iational sy mmetr y 2000 Mathematics Subje ct Classi fic ation: 3 5A15; 3 7K05 ; 37 K10 1 In tro duction The equat ion of Korteweg and de V ries or the so- called KdV equation u t = 1 4 u 3 x + 3 2 uu x in th e disp ersionless limit [1] ∂ ∂ t → ǫ ∂ ∂ t and ∂ ∂ x → ǫ ∂ ∂ x with ǫ → 0 reduces to u t = 3 2 uu x . (1.1) Equation (1.1), often called the Riemann equation, serves as a protot ypical nonlinear partial dif feren tial equation for the r ealization of man y phenomena exhibited by h yper b olic systems [2]. This migh t b e one of the reasons why , during the last decade, a num ber of w orks [3] w as en visag ed to stud y the p rop erties of disp ersionless KdV and other related equations with sp ecial emp hasis on their Lax represen tati on and Hamil tonian structure. The complete inte grabilit y of the KdV equation yields the existence of an inf inite family of conserv ed functions or Hamiltonian densities H n ’s that are in in v olution. All H n ’s that generate f lo ws whic h commute with the KdV f lo w giv e rise to the Kd V hierarc hy . The equations of th e hierarc h y ca n b e constructed using [4] u t = Λ n u x ( x, t ) , n = 0 , 1 , 2 , . . . (1.2) 2 A. Choudh uri, B. T alukdar and U. Da s with the recursion op erator Λ = 1 4 ∂ 2 x + u + 1 2 u x ∂ − 1 x . In the disp er s ionless limit the recur s ion op erator b ecomes Λ = u + 1 2 u x ∂ − 1 x . (1.3) According to (1.2), the pseu d o-dif feren tial op erator Λ in (1.3) def ines a disp ersionless KdV hierarc h y . Th e f irst few member s of the hierarc h y are giv en b y n = 0 : u t = u x , (1.4a) n = 1 : u t = 3 2 uu x , (1.4b) n = 2 : u t = 15 8 u 2 u x , (1.4c) n = 3 : u t = 35 16 u 3 u x , (1.4d) n = 4 : u t = 315 128 u 4 u x . (1.4e) Th us the equations in the disp ersionless hierarc hy ca n b e written in the general form u t = A n u n u x , (1.5) where the v alues of A n should b e computed using (1.3) in (1.2). W e can also ge nerate A 1 , A 2 , A 3 etc r ecur siv ely using A n =  1 + 1 2 n  A n − 1 , n = 1 , 2 , 3 , . . . and A 0 = 1 . The Hamiltonian structure of the d isp ersionless Kd V hierarc h y is often stud ied by taking recourse to the use of Lax op erators expressed in th e semi-classical limit [5 ]. In this work w e shall follo w a dif ferent viewp oint to deriv e Hamiltonian structure of the equations in (1.5). W e shall construct an expression for th e Lagrangian d en sit y and use the time-honoured metho d of classical mec hanics to rederiv e and reexamine the corresp onding canonica l formulatio n. A single ev olution equation is nev er the Euler–Lagrange equation of a v ariational problem. On e common tric k to put a single ev olution equation in to a v ariational form is to replace u b y a p oten tial function u = − w x . In terms of w , (1.5) w ill b ecome an Euler–Lagrange equation. W e can, ho w ev er, couple a n on lin ear ev olutio n equation with an asso ciated one and deriv e the action principle. Th is allo ws one to wr ite the Lagrangia n d ensit y in terms of the original f ield v ariables rather th an the w ’s, often ca lled the Casimir p oten tial. In Section 2 w e a dapt both these approac hes to obtain the Lagrangian and Hamiltonian densities of the Riemann typ e equations. In Section 3 we stud y the b i-Hamiltonian structure [6]. One of the add ed adv an ta ge of the Lagrangian description is that it allo ws one to establish, via No ether’s theorem, the relationship b et w een v ariational symmetries and asso ciated conserv ation la ws. The concept of v ariatio nal symmetry resu lts from the application of group metho ds in the calculus of v ariations. Here one deals with the symmetry group of an action f u nctional A [ u ] = R Ω 0 L  x, u ( n )  dx with L , the so-calle d L agrangian density of the f ield u ( x ). The groups considered will b e lo cal group s of transformations acting on an op en sub set M ⊂ Ω 0 × U ⊂ X × U . T he symbols X and U denote the space of ind ep endent and dep endent v ariables resp ectiv ely . W e dev ote Section 4 to stud y this classical problem. Finally , in Sectio n 5 we make some concluding remarks. 2 Lagrangian and Hamiltonian densities F or u = − w x (1.5) b ecomes w xt = A n ( − 1) n w n x w 2 x . (2.1) Lagrangian Ap proac h to Disp ersionless KdV Hierarch y 3 The F r ´ ec het der iv ativ e of the righ t side of (2.1) is self-adjoint . Thus we ca n u se the homotop y form ula [7] to obtai n the Lagrangian densit y in the form L n = 1 2 w t w x + A n ( − 1) n +1 ( n + 1)( n + 2) w n +2 x . (2.2) In writing (2.2) we ha v e subtracted a gauge term which is harmless at the classical level . Th e subscript n of L m erely indicates th at it is the Lagrangian density for the n th member of the disp ers ionless KdV hierarc h y . The corresp ond ing canonical Hamiltonian dens ities ob tained by the use of Leg endre map are giv en b y H n = A n ( n + 1)( n + 2) u n +2 . (2.3) Equation (1. 5) can b e written in the form u t + ∂ ρ [ u ] ∂ x = 0 (2.4) with ρ [ u ] = − A n ( n + 1) u n +1 . (2.5) There exist s a p r olongation of (1.5) or (2.4) in to another equation v t + δ ( ρ [ u ] v x ) δ u = 0 , v = v ( x, t ) (2.6) with the v aria tional deriv ativ e δ δ u = m X k =0 ( − 1) k ∂ k ∂ x k ∂ ∂ u k x , u k x = ∂ k u ∂ x k suc h th at the coupled system of equations follo ws from the action principle [8] δ Z L c dxdt = 0 . The Lag rangian densit y for the co upled equations in (2.4) and (2.6) is giv en b y L c = 1 2 ( v u t − uv t ) − ρ [ u ] v x . F or ρ [ u ] in (2.5), (2 .6) reads v t = A n u n v x . (2.7) F or the system represen ted b y (1.5 ) and (2.7 ) we hav e L c n = 1 2 ( v u t − uv t ) + A n ( n + 1) u n +1 v x . (2.8) The result in (2.7) could also b e obtained u sing the method of Kaup and Malomed [9]. Referring bac k to th e sup ersymmetric KdV equation [10] w e iden tify v as a ferm ionic v ariable associated with the b osonic equation in (1.5). I t is of in terest to note that the sup ersymmetric system is complete in the sense of v ariational principle while neither of the partner s is. T he Hamiltonian densit y obtained from the Lagrangian in (2.8 ) is giv en by H c n = − A n ( n + 1) u n +1 v x . (2.9) It r emains an in teresting curiosit y to demonstr ate that the results in (2.3) and (2.9) represent the conserv ed densities of the disp ersionless KdV and sup ersymmetric KdV f lo ws. W e demon- strate th is by examinning the appropriate b i-Hamiltonian structures of (1.5) and the pair (1 .5) and (2 .7). 4 A. Choudh uri, B. T alukdar and U. Da s 3 Bi-Hamiltonian structure Zakharo v and F addeev [11] develo p ed the Hamiltonian appr oac h to in tegrabilit y of n onlinear ev olution equations in one spatial an d one temp oral (1+1) d imensions and Gardner [12], in particular, in terpreted the KdV equation as a completely in tegrable Hamiltonia n system with ∂ x as the relev an t Hamiltonian op erator. A signif icant deve lopmen t in the Hamiltonian theory is due to Magri [6] who realized that integ rable Hamiltonian systems ha v e an additional structure. They are bi-Hamiltonian, i.e., th ey are Hamiltonian with resp ect to tw o d if ferent compatible Hamiltonian op erators. A similar consideration will also hold go o d for the disp ersionless KdV equations and we ha v e u t = ∂ x  δ H n δ u  = 1 2 ( u∂ x + ∂ x u )  δ H n − 1 δ u  , n = 1 , 2 , 3 . . . . (3.1) Here H = Z H dx. (3.2) It is easy to verify that for n = 1, (2. 3), (3.1) and (3.2) giv e (1 .4b). The o ther equations of the hierarc h y can b e ob tained for n = 2 , 3 , 4 , . . . . The op erators D 1 = ∂ x and D 2 = 1 2 ( u∂ x + ∂ x u ) in (3.1) are sk ew-adjoin t and satisfy the Jacobi identit y . The disp ersionless KdV equation, in particular, ca n b e wr itten in the Hamilto nian form as u t = { u ( x ) , H 1 } 1 and u t = { u ( x ) , H 0 } 2 endo w ed with the P oisson structures { u ( x ) , u ( y ) } 1 = D 1 δ ( x − y ) and { u ( x ) , u ( y ) } 2 = D 2 δ ( x − y ) . Th us D 1 and D 2 constitute t wo compatible Hamiltonian op erators suc h that the equations obtained from (1.5) are in tegrable in Liouville’s sense [6]. Th us H n ’s in (2.3) via (3.2) giv e the conserv ed dens ities of (1.5). In other w ords, H n ’s generate f lo ws whic h comm ute with the disp ers ionless Kd V f lo w and giv e rise to an appropriate hierarc h y . It will b e quite in teresting to examine if a similar analysis could also b e carried o ut for the sup ersymmetric disp ersionless KdV equati ons. The pair of sup ersymmetric equ ations u t = u n u x and v t = u n v x can b e written as η t = J 1  δ H s n δ η  = J 2  δ H s n − 1 δ η  , (3.3) where η =  u v  , H s n = H c n A n and H c n = R H c n dx . In (3.3) J 1 and J 2 stand for the matrices J 1 =  0 1 − 1 0  and J 2 =  0 u − u 0  . (3.4) Since H c n for dif ferent v alues of n represent the conserv ed Hamiltonian densities obtained by the use of action principle, the sup ersymmetric d isp ersionless KdV equations will b e bi-Hamiltonian pro vided J 1 and J 2 constitute a pair of compatible Hamiltonian op erators. Clearly , J 1 and J 2 are sk ew-adjoint. Thus J 1 and J 2 will b e Hamiltonian op erators pr o vided we can sho w that [5] pr v J i θ (Θ J i ) = 0 , i = 1 , 2 . (3.5) Lagrangian Ap proac h to Disp ersionless KdV Hierarch y 5 Here pr stands for the prolongation of the ev olutionary v ector f ield v of the c haracte ristic J i θ . The quan tit y pr v J i θ is calculated by using pr v J i θ = X µ,j D j X ν ( J i ) µν θ ν ! ∂ ∂ η µ j , D j = ∂ ∂ x j , µ, ν = 1 , 2 . (3.6) In our case the column matrix θ =  φ ψ  represent s the basis univ ectors asso ciated with th e v ariables η =  u v  . Und erstandably , θ ν and η µ denote the comp onents of θ and η and ( J i ) µν carries a similar m eaning. The functional biv ectors corresp onding to the op erators J i is giv en b y Θ J i = 1 2 Z θ T ∧ J i θ dx (3.7) with θ T , th e transp ose of θ . F rom (3.4) , (3 .6) and (3.7) we found th at b oth J 1 and J 2 satisfy (3.5) suc h th at eac h of them constitutes a Hamiltonian op erator. F urther, one can chec k th at J 1 and J 2 satisfy the compatibilit y condition pr v J 1 θ (Θ J 2 ) + pr v J 2 θ (Θ J 1 ) = 0 . This sho w s th at (3.3) giv es the bi-Hamiltonian form of sup ersymmetric disp ersionless KdV equations. Th e recursion oper ator def ined b y Λ = J 2 J − 1 1 =  u 0 0 u  repro du ces the hierarc h y of sup ersymm etric disp ers ionless Kd V equation according to η t = Λ n η x . for n = 0 , 1 , 2 , . . . . This v erif ies that H c n A n ’s as conserved densities generate f lo ws which commute with the sup ersymmetric disp ersionless Kd V f lo w. 4 V ariational symmetries The Lagrangian and Hamiltonian formulations of dynamical systems giv e a wa y to mak e the re- lation b etw een symmetries and conserved quant ities more p recise and thereby pro vide a metho d to deriv e exp ressions for th e conserved qu an tities from the symm etry transformations. In its general form this is referred to as Noether’s theorem. More precisely , this theorem asserts that if a giv en sy s tem of dif ferenti al equations follo ws from the v ariational principle, then a con tin u- ous symmetry transformation (p oin t, con tact or higher order) that lea v es the actio n functional in v ariant to within a diverge nce yields a conserv atio n la w. The p ro of of this theorem requires some knowledge of dif ferentia l forms, Lie deriv ativ es and pull-bac k [5 ]. W e shall, ho w ev er, carr y out the symmetry analysis for the disp ersionless KdV equation using a relati v ely simp ler mathe- matical framew ork as compared to that of the alge bro-geometric theories. In fact, we shall mak e use of some point transform ations that dep end on time and spatial coordinates. The app r oac h to b e follo w ed by us has an old ro ot in the classical-mec h anics literature. F or example, as e arly as 1951, Hill [13] p ro vided a simplif ied account of No ether’s theorem by considering inf in itesimal transformations of the dep end en t and indep endent v ariables c haracte rizing the classical f ield. W e shall f irst pr esen t our ge neral sc heme for symmetry analysis and then study the v ariational or No ether’s symmetries of the disp ers ionless KdV equatio n. 6 A. Choudh uri, B. T alukdar and U. Das Consider the inf initesimal transformations x i ′ = x i + δ x i , δx i = ǫξ i ( x, f ) (4.1a) and f ′ = f + δ f , δ f = ǫη ( x, f ) (4.1b) for a f ield v ariable f = f ( x, t ) w ith ǫ , an arbitrary s mall quan tit y . Here x = { x 0 , x 1 } , x 0 = t and x 1 = x . Understandably , our treatmen t for the symmetry analysis will b e applicable to (1 + 1) dimensional cases. Ho w ev er, the result to b e presented here can easily b e generalized to deal with (3 + 1) dimensional problems. F or an arbitrary analytic function g = g ( x i , f ), it is straigh tforw ard to sho w that δ g = ǫX g with X = ξ i ∂ ∂ x i + η ∂ ∂ f , (4.2) the generato r of the inf initesimal transformations in (4.1). A similar consideration when applied to h = h ( x i , f , f i ) with f i = ∂ f ∂ x i giv es δ h = ǫX ′ h (4.3) with X ′ = X +  η i − ξ j i f j  ∂ ∂ f i . (4.4) Understandably , X ′ stands for the f irst prolongation of X . T o arriv e at the statemen t for the No ether’s theorem we consider among the general set of transformations in (4.1) only those that lea v e the f ield-theoretic action in v arian t. W e th us write L ( x i , f , f i ) d ( x ) = L ′ ( x i ′ , f ′ , f i ′ ) d ( x ′ ) , (4.5) where d ( x ) = dxdt . In order to satisfy the condition in (4.5) w e allo w the Lagrangian den sit y to c hange its f unctional form L to L ′ . If the equations of motion, expr essed in terms of the new v ariables, are to b e of precisely th e same functional form as in the old v ariables, the t w o densit y fu nctions m ust b e related b y a d iv ergence transf orm ation. W e th us express the relation b et w een L ′ and L by in tro ducing a gauge function B i ( x, f ) such that L ′ ( x i ′ , f ′ , f i ′ ) d ( x ′ ) = L ( x i ′ , f ′ , f i ′ ) d ( x ′ ) − ǫ dB i dx i ′ d ( x ′ ) + o ( ǫ 2 ) . (4.6) The general form of (4.6 ) for the def inition of symmetry transformations will allo w the scale and d iv ergence tr an s formations to b e considered as symmetry trans formations. Understand ably , the scale transformations giv e rise to No ether’s symmetries while the scale transformations in conjunction with the dive rgence term lead to No ether’s div ergence symmetries. T raditionally , the concept of d iv ergence symmetries and concommitan t conserv ation la ws are int ro duced b y replacing No ether’s inf initesimal criterion for inv ariance b y a diverge nce condition [14]. Ho w ev er, one ca n directly wo rk with th e conserv ed d ensities that follo w from (4.6) b ecause nature of the v ector f ields will determine the con tributions of the gauge term. F or some of th e vec tor f ields the contributions of B i to conserv ed quantiti es will b e equ al to zero. These vect or f ields are Lagrangian Ap proac h to Disp ersionless KdV Hierarch y 7 No ether’s symmetries else we ha v e No ether’s diverge nce sym m etries. In view of (4.5), (4.6) can b e written in the form L ( x i ′ , f ′ , f i ′ ) d ( x ′ ) = L ( x i , f , f i ) d ( x ) + ǫ dB i dx i d ( x ) . (4.7) Again u sing L for h in (4.3), we ha v e L ( x i ′ , f ′ , f i ′ ) d ( x ′ ) = L ( x i , f , f i )  d ( x ) + ǫdξ i ( x, f i )  + ǫ X ′ L ( x i , f , f i ) d ( x ) . (4.8) F rom (4.7) and (4.8), we write dB i dx i = dξ i dx i L + X ′ L . (4.9) Using the v alue of X ′ from (4 .4) in (4.9), dB i dx i is obtained in the f inal form dB i dx i = dξ i dx i L + ξ i ∂ L ∂ x i + η ∂ L ∂ f +  η i − ξ j i f j  ∂ L ∂ f i . (4.10) Th us w e f ind that the action is in v ariant und er those transformations wh ose co nstituent s ξ and η satisfy (4.10). The te rms in (4.10) can be rearranged to write d dx i  B i − ξ i L +  ξ j f j − η  ∂ L ∂ f i  +  ξ j f j − η   ∂ L ∂ f − d dx i  ∂ L ∂ f i  = 0 . (4.11) The expression ins ide the squared brac k et stand s for the Eu ler–Lagrange equati on for the clas- sical f ield under consideration. In view of th is, (4 .11) leads to the conserv ation la w d I i dx i = 0 (4.12) with the conserve d densit y giv en b y I i = B i − ξ i L +  ξ j f j − η  ∂ L ∂ f i . (4.13) In the case of t w o indep end en t v ariables ( x 0 , x 1 ) ≡ ( t, x ), (4.12) can b e written in the explicit form d I 0 dt + d I 1 dx = 0 . (4.14) F rom (2.2) the Lagrangian d ensit y for the disp ersionless K dV e quation is obtained as L = 1 2 w t w x + 1 4 w 3 x . (4.15) Iden tifying f with w w e can com bine (4. 13), (4.14) and (4.15) to get B 0 t + w t B 0 w − 1 4 ξ 0 t w 3 x − 1 4 ξ 0 w w t w 3 x + 1 2 ξ 1 t w 2 x + 1 2 ξ 1 w w t w 2 x − 1 2 η t w x − η w w t w x + B 1 x + w x B 1 w + 1 2 ξ 1 x w 3 x + 1 2 ξ 1 w w 4 x + 1 2 ξ 0 x w 2 t + 1 2 w x w 2 t ξ 0 w + 3 4 ξ 0 x w 2 x w t − 3 4 η x w 2 x − 1 2 η x w t − 3 4 η w w 3 x + 3 4 ξ 0 w w 3 x w t = 0 . (4.16) 8 A. Choudh uri, B. T alukdar and U. Das In writing (4 .16) w e ha v e m ad e use of (2 .1) with n = 1. Eq u ation (4.1 6) can b e globally satisf ied if f the co ef f icients of the fol lo wing terms v anish separately w 0 x or w 0 t : B 0 t + B 1 x = 0 , (4.17 a) w t : B 0 w − 1 2 η x = 0 , (4.1 7b) w 2 t : 1 2 ξ 0 x = 0 , (4.17c ) w x : B 1 w − 1 2 η t = 0 , (4.1 7d) w 2 x : 1 2 ξ 1 t − 3 4 η x = 0 , (4.17e ) w 3 x : − 1 4 ξ 0 t − 3 4 η w + 1 2 ξ 1 x = 0 , (4.17f ) w 4 x : 1 2 ξ 1 w = 0 , (4.17g) w t w x : − η w = 0 , (4 .17h) w t w 2 x : 1 2 ξ 1 w + 3 4 ξ 0 x = 0 , (4.17i) w t w 3 x : 1 2 ξ 0 w = 0 , (4.17j) w 2 t w x : 1 2 ξ 0 w = 0 . (4.17k) Equations in (4.17) will lead to f inite n umber of sym m etries. This num b er a pp ears to b e disapp ointingly small sin ce we ha v e a disp ersionless K dV hierarch y giv en in (1.5). F urther, symmetry p rop erties r ef lecting the existence of inf initely many conserv ation la ws will require an appr opriate develo pment for the theory of generalized sym m etries. In this w ork, ho w ev er , w e shall b e concerned with v ariat ional symmetries only . F rom (4.1 7c) , (4.17j) and (4.1 7k) w e see that ξ 0 is only a fu nction of t . W e, therefore, w rite ξ 0 ( x, t, w ) = β ( t ) . (4.18) Also from (4.17g), (4.17i) and (4.18) we see th at ξ 1 is not a fun ction of w . In view of (4.17h) and (4 .18), (4.17 f ) give s ξ 1 x − 1 2 β t = 0 whic h can b e solv ed to get ξ 1 = 1 2 β t x + α ( t ) , (4.19) where α ( t ) is a constan t of integrati on. Usin g (4.19) in (4.17e) w e hav e η x = 1 3 β tt x + 2 3 α t . (4.20) The sol ution of (4.2 0) is g iv en b y η = 1 6 β tt x 2 + 2 3 α t x + γ ( t ) ( 4.21) with γ ( t ), a constant of in tegrat ion. In view of (4.21), (4.1 7b) and (4 .17d) yield B 0 = 1 6 β tt xw + 1 3 α t w (4.22) and B 1 = 1 12 β ttt x 2 w + 1 3 α tt xw . (4.23) Equations (4.2 2) and (4.23) can b e combined w ith (4.17a) to ge t f inally β ttt = 0 and α tt = 0 . (4.24) Lagrangian Ap proac h to Disp ersionless KdV Hierarch y 9 F rom (4.2 4) w e wr ite β = 1 2 a 1 t 2 + a 2 t + a 3 (4.25) and α = b 1 t + b 2 , (4.26) where a ’s and b ’s are arb itrary constan ts. Sub stituting the v alues of β and α in (4.18), (4.19), (4.21) w e obtain the inf initesimal transformation, ξ 0 , ξ 1 and η , as ξ 0 = 1 2 a 1 t 2 + a 2 t + a 3 , (4.27a ) ξ 1 = 1 2 ( a 1 t + a 2 ) x + b 1 t + b 2 , (4.27b) η = 1 6 a 1 x 2 + 2 3 b 1 x + b 3 . (4.27c ) In writing (4.27c) w e ha v e treate d γ ( t ) as a constan t and replaced it b y b 3 . Implicati on of this c hoice will b e made clear while considering the symmetry algebra. In terms of (4.27), (4.2) b ecomes X = a 1 V 1 + a 2 V 2 + a 3 V 3 + b 1 V 4 + b 2 V 5 + b 3 V 6 , where V 1 = 1 2 t 2 ∂ ∂ t + 1 2 xt ∂ ∂ x + 1 6 x 2 ∂ ∂ w , V 2 = t ∂ ∂ t + 1 2 x ∂ ∂ x , V 3 = ∂ ∂ t , V 4 = t ∂ ∂ x + 2 3 x ∂ ∂ w , V 5 = ∂ ∂ x , V 6 = ∂ ∂ w . (4.28) It is ea sy to c hec k that the v ector f ields V 1 , . . . , V 6 satisfy th e closure p rop erty . The comm u tation relations b et w een these v ect or f ields are giv en in T able 1. T able 1. Commut ation re la tions for the genera tors in (4.2 8). Each element V ij in the T able is repres e nt ed by V ij = [ V i , V j ]. V 1 V 2 V 3 V 4 V 5 V 6 V 1 0 − V 1 − V 2 0 − 1 2 V 4 0 V 2 V 1 0 − V 3 1 2 V 4 − 1 2 V 5 0 V 3 V 2 V 3 0 V 5 0 0 V 4 0 − 1 2 V 4 − V 5 0 − 2 3 V 6 0 V 5 1 2 V 4 1 2 V 5 0 2 3 V 6 0 0 V 6 0 0 0 0 0 0 The symmetries in (4.28) are expr essed in terms of the v elocit y f ield and d ep end explicitly on x and t . Lo oking from this p oint of view the symmetry v ectors obtained b y u s b ear some similarit y with the so called ‘add ition symmetries’ suggested indep enden tly by Chen , Lee and Lin [15] and by Orlo v and Shulman [16]. It is easy to see that V 2 to V 6 corresp ond to scaling, time translation, Galilean b o ost, space translation and translation in v el o cit y sp ace resp ectiv ely . The vecto r f ield V 1 do es n ot admit such a simple ph ysical realization. Ho w ev er, we can write V 1 as V 1 = 1 2 tV 2 + 1 4 xV 4 . Making use of (4.15), (4.22), (4.23), (4.25) and (4.26) we can write the expressions for the conserv ed quan titi es in (4.13 ) as I 0 = 1 6 a 1 xw + 1 3 b 1 w − 1 4 ξ 0 w 3 x + 1 2 ξ 1 w 2 x − 1 2 η w x , (4.29a ) I 1 = 1 2 ξ 0 w 2 t + 3 4 ξ 0 w t w 2 x + 1 2 ξ 1 w 3 x − 1 2 η w t − 3 4 η w 2 x . (4.29b) 10 A. Choudh uri, B. T alukdar and U. Das The expressions for I 0 and I 1 are c haracte rized by ξ i and η , the v alues of whic h c hange as w e go fr om one vect or f ield to th e other. The f irst t w o terms in I 0 stand for the con tribution of B 0 and there is no con tribution of the gauge term in I 1 since from (4.23) and (4.24) B 1 = 0. F or a particular v ector f ield a 1 and b 1 ma y either b e zero or non zero. One can ve rify that except for v ector f ields V 1 and V 4 , a 1 = b 1 = 0 suc h t hat V 2 , V 3 , V 5 and V 6 are simple Noether’s symmetries while V 1 and V 4 are No ether’s d iv ergence symm etries. Coming down to d etails we ha v e found the follo wing conserve d quan titi es from (4 .29a) and (4.29b) I 0 V 1 = 1 6 xw − 1 8 t 2 w 3 x + 1 4 xtw 2 x − 1 12 x 2 w x , (4.30a ) I 1 V 1 = 1 4 xtw 3 x + 3 8 t 2 w t w 2 x + 1 4 t 2 w 2 t − 1 12 x 2 w t − 1 8 x 2 w 2 x , (4.30b) I 0 V 2 = − 1 4 tw 3 x + 1 4 xw 2 x , (4.30c ) I 1 V 2 = 1 4 xw 3 x + 3 4 tw t w 2 x + 1 2 tw 2 t , (4.30d) I 0 V 3 = − 1 4 w 3 x , (4.30e ) I 1 V 3 = 3 4 w t w 2 x + 1 2 w 2 t , (4.30f ) I 0 V 4 = 1 3 w + 1 2 tw 2 x − 1 3 xw x , (4.30g) I 1 V 4 = 1 2 tw 3 x − 1 3 xw t − 1 2 xw 2 x , (4.30h) I 0 V 5 = 1 2 w 2 x , (4.30i) I 1 V 5 = 1 2 w 3 x , (4.3 0j) I 0 V 6 = − 1 2 w x , (4.30k) I 1 V 6 = − 1 2 w t − 3 4 w 2 x . (4.30l) It is easy to c hec k that the results in (4.30) is consisten t with (4.14). The pair of conserv ed quan tities corresp onding to time translation, space translation and v elocit y sp ace tran s lation, namely , { (4.3 0e ),(4.30f) } , { (4.3 0i ),(4.30j) } and { (4 .30k),(4.30l) } d o not in volv e x and t explicitly . Eac h of the pair in conjunction with (4.14) giv e the disp ersionless KdV equation in a rather straigh tforw ard manner . As exp ected (4.30e) s tands for the Hamilto nian densit y or energy of (1.4b). 5 Conclusion Compatible Hamiltonian structures of the d isp ersionless KdV h ierarc h y are traditionally ob- tained with sp ecial attent ion to their L ax represen tati on in the semiclassical limit. The deriv a- tion inv olve s judicious us e of the so-called r -matrix metho d [1 7]. W e ha ve sho wn that th e com bined Lax representa tion– r -matrix metho d ca n b e sup plemen ted by a Lagrangian appr oac h to the problem. W e foun d th at the Hamiltonian densities corresp ondin g to our Lag rangian rep- resen tations stand for the conserve d densities for the d isp ersionless KdV f low. W e could ea sily construct the Hamiltonian op erators from the recursion op erator w h ic h generates th e hierarc h y . W e ha v e deriv ed the bi-Hamilt onian structures for b oth disp ersionless KdV and sup ersymmetric KdV hierarc hies. As an added realism of the Lagrangian approac h we stu d ied the v ariational symmetries of equation (1.4b). W e b eliev e that it will b e quite in teresting to carry out similar analysis for the sup ersymm etric KdV pair in (1. 4b) and for n = 1 limit of (2.7). Ac kno wledgemen ts This work is supp orted by the Univ ersit y Gran ts Commission, Go v ernment of India, through gran t No. F.32-39/ 2006(SR). 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