Hydrodynamic type integrable equations on a segment and a half-line

Hydro dynamic t yp e in tegrable equations on a segmen t and a half-line. Metin G ¨ urses a , Ismagi l Habibul lin ∗ b and Kost yan tyn Zheltukhin c a Departmen t of Mathemat ics, F acult y of Sciences Bilk en t Universit y , 06800 Ank ara, T urkey e-mail gur ses@fen.bilk en t.edu.tr b Departmen t of Mathematics, F acult y of Sciences Bilk en t Universit y , 06800 An k ara, T ur k ey e-mail h abib@fen.bilk ent. edu .tr c Departmen t of Mathematics, F acult y of Sciences Middle East T ec hn ical Universit y 0653 1 Ank ara, T urkey e-mail zheltukh@m etu.edu .tr No vem b er 18 , 201 8 Abstract The concept of int egrable b oun dary cond iti ons is app lie d to h y- dro dynamic t yp e sy s te ms . Examples of suc h b oundary conditions for disp ersionless T o da systems are obtained. The close relation of int e- grable b oundary conditions w ith inte grable reductions of m u lti-field systems is observ ed. The pr oblem of consistency of b oundary con- ditions with th e Hamiltonian form u lat ion is discus s ed. Examples of Hamiltonian in tegrable h yd r odyn amic t yp e systems on a segmen t and a semi-line are presen ted. Keywor ds: h ydro dynamic t yp e equations, in tegrable boundar y condi- tions, symmetries, T o da systems, Hamilto nian represen tation. ∗ e-mail: habibullin i@mail.r b.ru, (On leav e from Ufa Institute o f Mathematics, Russia n Academy of Science, Che r n yshevsk ii Str., 112 , Ufa, 450 077, Russ ia) 1 1 In tro du c tion. The theory of the integrable h ydro dynamic t yp e of systems u i t = v i j ( u ) u j x , i, j = 1 , 2 , ...N , (1) w as initia ted b y S.P .Novik o v and B.A.D ubro vin [1], and S.P .Tsarev [2]. Here in equation (1) summation o v er the rep eated indices is assumed and u is an N-comp onen t column v ector of the form u = ( u 1 , u 2 , ...u N ) t . Suc h systems ha v e a v ariet y of applications in gas dynamics, fluid mec hanics [6, 7, 8, 11], c hemical kinetics, Whitham a ve rag ing pro cedure [9, 10, 4, 5], differen tial ge- ometry and top ological field theory . W e refer to [3, 1 2] fo r further discussions and references. In the presen t art icle, a problem of finding b oundary conditions for hy - dro dynamic type equations consisten t with the in tegrabilit y prop erty is stud- ied for a sp ecial case of the system (1) called disp ersionless T o da lattices ([17, 19, 2 0]). Actually we assume that equation (1) admits a Lax represen- tation on the algebra A = ( ∞ X −∞ u i ( x ) p i : u i deca y sufficien tly rapidly as x → ±∞ ) (2) with the f o llo wing P oisson brac ket { f , g } = p  ∂ f ∂ p ∂ g ∂ x − ∂ f ∂ x ∂ g ∂ p  f , g ∈ A. Suc h equations, for example, app ear in the fluid mec hanics as reductions of Benn y momen t equations [6]-[12]. Our definition o f consistency o f b oundary conditions with the in tegrability (see [13 , 14, 15 ]) is ba se d on the notion of symmetries. A constrain t of the form f ( t, u, u [1] , ..., u [ k ] ) | x = x 0 = 0 , (3) where u [ j ] = ∂ j ∂ x j u , imp osed at some p oint x 0 is called a b oundary condition at this p oin t. Boundary v alue pro blem (1), (3) (or simply b oundary condition (3)) is called consisten t with the symmetry u i τ = σ ( t, x, u, u [1] , ..., u [ m ] ) (4) 2 if (1) and (4) are compatible under the constrain t (3). More precisely , w e mean the f ollo wing: D ifferen tiation of (3 ) with resp ect to τ yields X ∂ f ∂ u i [ n ] ( u i [ n ] ) τ = 0 , (5) where τ -deriv ativ es are replaced b y means of equation (4). Definition 1 . B oundary v a lue pr oblem ( 1), (3) is c onsistent with the sym- metry (4) if (5) holds identic al ly by me ans of (3) and i ts differ ential c onse- quenc es o btaine d by differ entiating with r esp e ct to t . Note that, since constraint ( 3 ) is v alid only for x = x 0 it canno t b e differen tiated with resp ect to x . F o r this reason it is conv enien t t o exc lude the x -deriv ativ es of dep enden t v ariable u from our sc heme. By solving equation (1) for u i x one gets u x = v − 1 u t , where v − 1 is the ma t r ix in v erse to v i j ( u ). Similarly , u xx = ( v − 1 u t ) x = ( v − 1 ) x u t + v − 1 u tt is expresse d thro ugh u, u t , u tt and so on. As a result one can rewrite b oundary condition (3 ) and symmetry (4) tak en at the p oin t x 0 as f 1 ( t, u, u t , ... ) = 0 (6) and u τ = σ 1 ( t, x 0 , u, u t , ... ) . (7) No w the consistency r eq uiremen t can b e reformulated as follow s. Boundary condition ( 3) is consisten t with (4) if differen tial constrain t (6) is consis- ten t with the asso ciated τ -dynamics (7). W e call the b oundary condition consisten t with integrabilit y if it is consisten t with an infinite dimensional subspace of symmetries. Hydro dynamic type system giv en in (1) defines an N -dimensional dynamical system and the b oundary condition (6) defines an h yp ersurface in N -dimensional space of functions. Th us, due to t he remark ab o v e an integrable b oundary conditio n is closely connected with reductions of the asso ciated system (7) compatible with in tegrability [17]. Belo w we use this imp ortant observ ation in order to find symmetry consisten t b oundary conditions. Boundary conditions passed the symmetry test ar e then tested for consis- tency with the conserv ed quan tities, Hamiltonian structures and the complete in tegrability pro perty of system (1). It is remark able that some of the b ound- ary conditions satisfy also these additio nal requiremen ts and th us allow one 3 to r educ e (1) to a completely integrable Hamiltonian system on a segmen t and a ha lf-line. The pap er is organized as follows. In Section 2 some integrable b oundary conditions for T o da system are deriv ed a nd it is show en that these b oundary conditions are compatible with infinite n um b er of symmetries. The relation b et ween the in tegrable reductions of N- sy stem and the integrable b ound- ary conditions is considered in Section 3. It is observ ed that some integrable b oundary conditions lead to in tegrable reductions. In section 4 w e discuss the compatibilit y of the integrable b oundary conditions, found in the previous sections, with t he Hamiltonia n formulation. W e show that some b oundary conditions are indeed compatible with the Hamiltonian formu lat ion and also with an infinite class of symmetries. In all sections up to Section 4 only N = 2 systems are considered. In Section 5 w e study N = 3 systems whic h giv e other examples of hy dro dynamic t yp e equations. F or this case, some in- tegrable b oundary conditions compatible with infinite num b er of symmetries and b oundary conditions compatible with the Hamiltonian form ulation are found. 2 In tegrable B oundary c onditions for the T o da system. In this section w e study the w ell kno wn example of integrable mo del [16] S t = P x , P t = P S x (8) called T o da system, admitting the Lax represen tation on the algebra of Lau- ren t series (2) L t = { ( L ) ≥ 0 , L } , (9) where L = p + S + P p − 1 . (10) The corresp onding hierarch y of symmetries of t he T o da system (8) is L t n = { L, ( L n ) ≥ 0 } . (11) 4 Recursion op erator corresp onding to the ab o ve hierarc hy is (fo r calculatio n of recursion op erator see [17], [18]) R =  S 2 + P x D − 1 x · P − 1 2 P S + S x P D − 1 x · P − 1  . (12) In some cases it is con venie nt to consider the T o da system(8) in other v ar i- ables. W e write the Lax function (10) as L = p − 1 ( p + u )( p + v ) that is S = u + v , P = uv . (13) Then the T o da system (9) giv es u t = u v x v t = v u x . (14) Let us find b oundary conditions compatible with an infinite n umber of sym- metries f r om the hierarch y (11). As a b oundary we tak e x = 0. First w e find b oundary conditions compatible with the first symmetry of the hierar- c h y (11). Assume that the b oundary condition dep ends on P , S and can b e solv ed with resp ect to S . So the b oundary condition can b e written as S = F ( P ) , x = 0 . (15) Lemma 1 On the b oundary x = 0 , the b oundary c ondition of the form (15) c omp atible with the first symmetry of the hier ar chy (11) S t 1 = 2 S P x + 2 P S x P t 1 = 2 P P x + 2 S P S x (16) is given by P = ( S + c ) 2 4 , x = 0 , (17) Pro of. The b oundary condition (15) is compatible with the symmetry (16) if on t he b oundary x = 0 S t 1 = F ′ ( P ) P t 1 (18) for all solutions of the equation (8). Let us find functions F fo r whic h the ab o v e equalit y holds. W e rewrite the symmetry (16) in terms of v ariables S , P and their t deriv ativ es using the equation (8) S t 1 = 2 S S t + 2 P t P t 1 = 2 P S t + 2 S P t . (19) 5 Then w e substitute S t 1 and P t 1 in (18), so 2 S S t + 2 P t = F ′ ( P )(2 P S t + 2 S P t ) (20) F rom (15) it follow s that S t = F ′ ( P ) P t , so 2 S F ′ ( P ) P t + 2 P t = F ′ ( P )(2 P F ′ ( P ) P t + 2 S P t ) . (21) Hence F ′ 2 ( P ) = 1 P . (22) The ab o v e equation has a solution (17).  It is con v enien t to write the b oundary condition (17) as P = S 2 4 , x = 0 . (23) b y shifting S , the T o da system(8) is in v ariant with r esp ect to suc h shift. Lemma 2 A l l the symmetries of the h i e r ar chy (11) ar e c omp atible wi th the b oundary c ondition (23). Pro of. The b oundary condition ( 23) is compatible with an ev olution sym- metry  S P  τ =  σ π  (24) if π = 1 2 S σ for P = 1 4 S 2 . That is under the constraint (2 3 ) the symmetry (24) should t a k e the form  S τ 1 2 S S τ  =  σ 1 2 S σ  (25) Eviden tly the first symmetry of the hierarc hy (11) has suc h a fo rm. Let us sho w that the recursion op erator (12) preserv es the prop ert y (25). On the b oundary x = 0 we rewrite the recursion op erator (1 2 ) in terms of t deriv ativ es using the T o da system(8) as follow s R =  S + S t D − 1 t 2 2 P + P t D − 1 t S  . (26) 6 Applying the recursion op erator ( 2 6) to a symmetry (25) w e obtaine a sym- metry  S P  ˜ τ =  ˜ σ 1 2 S ˜ σ  . (27)  W e a lso hav e the follo wing b oundary condition compatible with the hier- arc hy (11). Lemma 3 On the b oundary x = 0 , the b oundary c ondition P = 0 (28) is c omp atible with al l symm etries of the hie r ar chy (11) The ab ov e lemma is prov ed in the same wa y as lemma (2). Another b oundary condition comes from considering o dd and ev en solu- tions of the T o da sys tem(8 ). This b oundary conditio n is not compatible with all symmetries of the hierar c hy (11) but only with eve n ones. Lemma 4 On the b oundary x = 0 , the b oundary c ondition S = 0 (29) is c omp atible with al l ev e n numb er e d symmetries of the hier a r chy (11). The ab o v e lemma is prov ed in the same w ay as lemma (2) using the square of the recursion op erator (26). 3 In tegrable r e ductions. Let us consider other equations admitting a Lax represen tation on the algebra (2). F or a Lax function L = p − 1 ( p − u N )( p − u N − 1 ) . . . ( p − u 1 ), where N > 2, w e consider the Lax equation L t = { L, ( L ) ≥ 0 } . (30) and an infinite hierarch y of symmetries L t n = { L, ( L n ) ≥ 0 } n = 1 , 2 , . . . . (31) F or suc h equations w e can not directly find b oundary conditions compatible with symmetries (see section (5)). So we use integrable reductions [17 ]. 7 Definition 2 A r e duction o f an inte gr able e quation is c al le d inte gr able if r e duc e d e quation is also inte gr able. Tha t is the r e duc e d e quation adm its an infinite hier ar chy of symm e tries. In [17] it w as sho wn that the following reductions u N = u N − 1 = · · · = u i = 0 , i ≥ 2 , u N = u N − 1 = · · · = u j , j ≥ 1 (32) of the a bov e equations are integrable. W e note that for these reductions the symmetries of the reduced equation are obtained b y t he reduction of the symmetries of t he original system. If w e ha v e an integrable reduction suc h that symmetries of the reduced system are obtained b y the reduction of the symmetries of the original system then the reduction can b e tak en as in tegrable b oundary conditions. Indeed, the original system is inv ar ian t under the hierarch y of symmetries a nd the reduced system is inv aria nt under the symmetries. Since reduction can b e reco v ered fro m original system and the reduced system it is a lso in v a r ian t under the symme tr ies. So, taking the reductions (32) as b oundary conditions w e obta in symmetry in v ariant b oundary conditions. Theorem 1 F or a system ( 3 0 ) the b oundary c onditions ( u N = u N − 1 = · · · = u i ) | x = a = 0 , or ( u N = u N − 1 = · · · = u j ) | x = a (taking x = a as the b oundary) ar e inte gr able. Let us take b oundary conditio ns obtained in section 2. The condition ( P = S 2 4 ) | x =0 in u, v v ariables (13) is ( u − v ) | x =0 = 0. It corresp onds to a reduc- tion u = v . The condition P | x =0 = 0 in u, v v ariables is ( uv ) | x =0 = 0. It corresp onds to a reduction u = 0 (or v = 0 ). The condition S | x =0 = 0 in u, v v ariables is ( u + v ) | x =0 = 0 . It do es not corresp ond to reductions considered ab o v e. Remark. If w e tak e a reduction men tioned ab o v e as a b oundary con- dition then w e can consider the cor r es p onding reduced system. Solutions of the reduced system o b viously satisfy the main system equations and the b oundary condition. F o r T o da system the reduction P = 0 leads to the equation S t = 0 . (33) Its solution S = f ( x ), for any differen tiable function f , giv es the solution of T o da system (8) satisfying the corresp onding b oundary condition (28). The 8 reduction P = S 2 4 leads to the Hopf equation S t = 1 2 S S x . (34) Its solution S = h (2 x + tS ) give s the solution o f T o da system (8) satisfying the corresp onding b oundary condition. Here h is an y differentiable func- tion of x a nd t . T o find a solution o f N − system satisfying the integrable b oundary condition the metho d describ ed ab o v e is v ery effectiv e. W e tak e the corresp onding reduction and the corresp onding reduced ( N − 1) system. Solving the reduced system giv es automatically the solution of the N − sys- tem satisfying the inte g r a ble b oundary condition. 4 Hamiltonian repres en tatio n of the in te g rable b oundary v alue p roblems. T o obtain the Hamiltonian form ulatio n of the T o da system (8) w e use its Lax represen tation on the algebra (2). W e define, for the alg ebra of Louren t series (2), a trace functional tr f = Z ∞ −∞ u 0 dx f ∈ A, f = ∞ X −∞ u i ( x ) p i (35) and a no n-degenerate ad-inv ariant pa ir ing ( f , g ) = tr( f · g ) f , g ∈ A. (36) Th us w e hav e a P oisson algebra with a comm utativ e m ultiplication and unity , the multiplication satisfies the deriv ation prop ert y with resp ect to t he P ois- son brac k et, and the algebra is equipp ed with a non- degenerate ad- in v arian t pairing. F o llo wing [2 0] we can define an infinite family of Pois son structures for smo oth functions on the algebra A . A function F on A is smo oth if there is a map dF : A → A suc h that F ′ | t =0 ( f + tg ) = ( dF ( f ) , g ) , f , g ∈ A. The following theorem ([20], see also [21]) holds 9 Theorem 2 L et A b e a Poisson alg ebr a w i th c ommutative multiplic ation and unity, Poisson br acke t { ., . } and non-de gen er a te, a d -invariant p airin g ( ., . ) . L et the multiplic ation satisfies the de riv ation pr op erty w i th r esp e ct to the Poisson br acket and symmetric with r esp e ct to the p airing, ( f g , h ) = ( f , g h ) . Assume that R : A → A is a cla s s ic a l r -matrix. Then for smo oth functions F and G on A a. { F , G } ( n ) ( f ) = ( f , { R ( f n +1 dF ( f )) , dG ( f ) } ) + ( f , { d F ( f ) , R ( f n +1 dG ( f )) } ) , (37) defines a Poisson structur e for e ach inte ger n ≥ − 1 . b. The structur es { ., . } ( n ) ar e c omp atible with e ach other (their sum is again a Poisson structur e). A linear op erator R : A → A is a classical r -matrix if the brac k et [ f , g ] = 1 2 ( { Rf , g } + { f , Rg } ) is a Lie brac k et. T o apply the ab o ve t heorem w e tak e an r -matrix R = 1 2 ( P ≥ 0 − P ≤− 1 ) (38) where P ≥ 0 and P ≤− 1 are pro j ec to rs on P oisson subalgebras A ≥ 0 = { u = ∞ X 0 u i p i : u ∈ A } and A ≤ 0 = { u = − 1 X −∞ u i p i : u ∈ A } resp ec tively . Not e that the Lax equation (9) is L t = { R ( L ) , L } (39) where L = p + S + P p − 1 . Using the P oisson structures giv en b y the Theorem (2) we obtain bi- Hamiltonian fo rm ulatio n of the T o da lat t ice. The submanifold M = { L ∈ A : L = p + S + P p − 1 } is a P oisson submanifold for the Poiss o n structure (37) with n = − 1. Restricting this structure on M w e obtain the following Hamiltonian o perator D − 1 =  0 P D x + P x P D x 0  (40) 10 W e hav e first Hamiltonian formu lat ion for (8)  S P  t = D − 1  δ H − 1 /δ S δ H − 1 /δ P  , (41) where H − 1 = 1 2 tr L 2 that is H − 1 = 1 2 Z ∞ −∞ ( S 2 + 2 P ) d x. (42) The second Hamiltonian op erator can b e obtained b y restricting the P ois- son structure (37) with n = 0 on the submanifold M or b y a pplication of the recursion o perator ( 12) to the Hamiltonian op erator (4 0). The second Hamiltonian op erator is D 0 =  2 P D x + P x S P D x + S P x S P D x + S x P P 2 D x + P P x  . (43) The corresp onding Hamiltonia n functional is H 0 = tr L that is H 0 = Z ∞ −∞ S dx (44) Since Ha milto nian op erators D − 1 and D 0 are compatible w e ha v e a bi- Hamiltonian represen tatio n of the equation ( 8 ). In u, v v ariables (13) the Hamiltonian op erators and functionals t a k e form B − 1 = uv ( u − v ) 2  − 2 u u + v u + v − 2 v  D x + (45) 1 ( u − v ) 3  2 uv 2 u x − u 3 v x − u 2 v v x u 2 v v x + u 3 v x − 2 u v 2 u x 2 uv 2 v x − uv 2 u x − v 3 u x v 3 u x + uv 2 u x − 2 u 2 v v x  (46) and G − 1 = Z ∞ −∞ ( u 2 + v 2 + 4 u v ) dx. (47) B 0 =  0 uv x + uv D x v u x + uv D x 0  (48) and G 0 = Z ∞ −∞ ( u + v ) dx. (49) 11 A differen t approac h w as used in [19] to obtain the Hamiltonian op erator B 0 (see also references in [1 9]). The explicit expressions of an infinite n um b er o f conserv a tion la ws for the T o da system(14) w as give n in [19 ] Q n,t = F n,x n = 1 , 2 . . . , (50) where Q n = n X j =0  n j  2 u j v n − j n = 1 , 2 , 3 . . . (51) and F n = n X j =0 n − j j + 1  n j  2 u j +1 v n − j n = 1 , 2 , 3 . . . . (52) The conserv ed quan tities Q n = R ∞ −∞ Q n dx are in in volution with resp ect to the Hamilto nian op erators B − 1 and B 0 . One can easily c heck if the b o undary conditions preserv e the conserv ed quantities. Lemma 5 F or the T o da s ystem( 1 4)with the b oundary c o ndition a. ( u − v ) | x =0 = 0 ( ( P = S 2 4 ) | x =0 ) the ab ove c onserv a tion law s ar e not pr eserve d; b. uv | x =0 = 0 ( P | x =0 = 0 ) the quantities Z ∞ 0 Q n dx n = 1 , 2 , 3 . . . (53) ar e c onserve d; c. ( u + v ) | x =0 = 0 ( S | x =0 = 0 ) the quantities Z ∞ 0 Q n dx n = 2 , 4 , 6 . . . (54) ar e c onserve d. W e can use the ab o v e Hamilto nian o perators to obtain the Hamiltonian represen ta t io n of some of the b oundary v alue problems. Theorem 3 The T o da system(14) on a se gment [0 , 1] with b oundary c ondi- tions uv | x =0 = 0 and uv | x =1 = 0 . (55) 12 admits the bi-Hami ltonian r epr e s entation with Hamiltonia n op er ators B ( n ) , n=-1,0, and Hamiltonian s ¯ G − 1 = Z 1 0 ( u 2 + v 2 + 4 u v ) dx = Z ∞ −∞ ( u 2 + v 2 + 4 u v ) θ ( x ) θ (1 − x ) dx (56) and ¯ G 0 = Z 1 0 ( u + v ) dx, = Z ∞ −∞ ( u + v ) θ ( x ) θ (1 − x ) dx, (57) r esp e ctively, whe r e θ ( x ) is the He aviside step function. Pro of. The Hamiltonian equations  u v  t = B n  δ ¯ G n /δ u δ ¯ G n /δ v  n = − 1 , 0 (58) are for n = − 1 u t = uv x − uv u − v ( δ ( x ) − δ (1 − x )) , v t = v u x + uv u − v ( δ ( x ) − δ (1 − x )) (59) and for n = 0 u t = u v x + uv ( δ ( x ) − δ (1 − x )) , v t = v u x + uv ( δ ( x ) − δ (1 − x )) , (60) where x ∈ [0 , 1]. Under the b oundary conditions uv | x =0 = 0 and uv | x =1 = 0 w e ha v e the T o da system(14 ) on [0 , 1]. Note that the Poisson brac kets are giv en b y {K , N } = Z ∞ −∞  δ K /δ u δ K /δ v  B ( n )  δ N /δ u δ N /δ v  (61) where n = − 1 , 0.  5 In tegrable b oundary c onditions for the t hree field s ystems. Let us consider a three field h ydro dynamic type system on the alg ebra (2). W e take a Lax function L = p 2 + S p + P + Qp − 1 . (62) 13 W e can construct t wo in tegrable hierarc hies with this Lax function. The first hierarc h y is given b y L t = { ( L n + 1 2 ) ≥ 0 , L } n = 0 , 1 , 2 , . . . , (63) the first equation of the hierarc h y is S t = P x − 1 2 S S x , P t = Q x , Q t = 1 2 QS x . (64) The second hierarch y is giv en b y L t = { ( L n ) ≥ 0 , L } n = 1 , 2 , 3 . . . , (65) the first equation of the hierarc h y is S t = 2 Q x P t = S Q x + QS x Q t = QP x (66) W e also ha v e a recursion op erator [17] o f the hierarc hies (63) and (65).   P − 1 4 S 2 + ( 1 2 P x − 1 4 S S x ) D − 1 x 1 2 S 3 + 2 Q x D − 1 x Q − 1 3 2 Q + 1 2 Q x D − 1 x P 2 S + ( S Q ) x D − 1 x Q − 1 1 4 S Q + 1 4 S x QD − 1 x 3 2 Q P + QP x D − 1 x Q − 1   (67) The bi- Hamiltonian represen tation of equations (64) and ( 6 6) is obtained b y restricting the P oisson structure (37) with n = − 1 and n = 0 on the submanifold M = { L ∈ A : L = p 2 + S p + P + Qp − 1 } . So w e hav e Hamiltonian op erators C − 1 =   2 D x 0 0 0 0 QD x + Q x 0 QD x 0   (68) and C 0 =   (2 P − 1 2 S 2 ) D x + P x − 1 2 S S x 3 QD x + 2 Q x 1 2 QD x + 1 2 S Q x QD x + Q x 2 S QD x + S Q x + QS x P QD x + P D x 1 2 S QD x + 1 2 QS x P D x + P x Q 3 2 Q 2 D x + 3 2 QQ x   . (69) 14 The equation (64) can b e written as   S P Q   t = C − 1   δ ¯ H − 1 /δ S δ ¯ H − 1 /δ P δ ¯ H − 1 /δ Q   = C 0   δ ¯ H 0 /δ S δ ¯ H 0 /δ P δ ¯ H 0 /δ Q   , (70) where ¯ H − 1 = 2 3 tr L 3 2 that is ¯ H − 1 = Z ∞ −∞  Q + 1 2 S P − 1 24 S 3  dx (71) and ¯ H 0 = 2tr L 1 2 that is ¯ H 0 = Z ∞ −∞ S dx . (72) The equation (66) can b e written as   S P Q   t = C − 1   δ ˜ H − 1 /δ S δ ˜ H − 1 /δ P δ ˜ H − 1 /δ Q   = C 0   δ ˜ H 0 /δ S δ ˜ H 0 /δ P δ ˜ H 0 /δ Q   , (73) where ˜ H − 1 = 1 2 tr L 2 that is ˜ H − 1 = Z ∞ −∞  S Q + 1 2 P 2  dx (74) and ˜ H 0 = tr L that is ˜ H 0 = Z ∞ −∞ P dx . (75) W e can giv e b oth hierarc hies in mo dified v ariables, writing the Lax f unc- tion (62) as L = p − 1 ( p − u )( p − v )( p − w ) that is S = u + v + w , P = uv + uw + v w , Q = uv w . (76) T o find in tegrable b oundary condition directly for three field systems is quite difficult. F or example, consider hierarc hy (63). In the follow ing lemmas w e use P , Q, R v a riables since symmetries and recursion op erator ha ve a simple form in t hese v ariables. 15 Lemma 6 L et x = 0 b e the b oundary. The b ounda ry c onditions of the form P = F ( S ) and Q = G ( S ) ar e c o mp atible with the first symmetry of the hier- ar chy (63) if the functions F and G satisfy the fol lowing differ en tial e quations 3 2 S ( F ′ ) 2 + 3 F ′ G ′ − 3 4 F ′ S 2 − 3 G ′ S − 3 2 G = 0 , (77) 3 2 S F ′ G ′ + 3( G ′ ) 2 − 3 2 F ′ G − 3 4 G ′ S 2 − 3 4 S G = 0 . (78) Pro of. The first symmetry of t he hierarc hy (63) is S t 1 = 3 2 ( P − 1 4 S 2 )( P x − 1 2 S S x ) + 3 2 S Q x + 3 2 S x Q, P t 1 = 3 2 P Q x + 3 2 P x Q 3 4 QS S x + 3 8 S 2 Q x , Q t 1 = 1 4 S Q ( P x − 1 2 S S x ) + 1 4 Q ( P − 1 4 S 2 ) + 3 2 QQ x + 1 2 QP S x + 1 2 QS P x . (79) Differen tiating the b oundary conditions P = F ( S ) , Q = G ( S ) with resp ect to the ab o ve symmetry and expressing all the x deriv a tiv es in terms of t deriv ativ es using the equation (6 4) w e o btain the equations (7 7) and (7 8 ).  Lemma 7 L et x = 0 b e the b oundary. The b ounda ry c ondition of the fo rm S = F ( P , Q ) is c omp atible w i th the first symmetry of the hier ar chy (63) if function F satisfies the fol low ing differ ential e quations 3 2 ( P − 1 4 F 2 ) F P + 3 2 F = 3 2 P F P + 3 2 QF 2 P + 3 8 QF 2 P + 3 8 F 2 F P + 1 4 QF F Q + 3 2 QF Q + 1 2 F F P F Q , (80) 3 2 ( P − 1 4 F 2 ) F Q + 3 2 = 3 2 QF P F Q + 3 2 F F P + 1 2 ( P − 1 4 S 2 ) F Q + P F Q + 1 2 QF F 2 Q + 1 2 F 2 F Q . (81) Pro of. W e differen tiate the b oundary condition S = F ( P , Q ) with resp ect to the symmetry (63) and express all the x deriv a tiv es in t erms of t deriv a t iv es using the equation (64). Then separating terms containing P t and Q t w e obtain the equations (80) a nd (81).  The differential equations obta ined in the ab o v e lemmas are nonlinear par tial differen tial equations whic h are ra ther complicated. So, to obtain integrable 16 b oundary conditions it is easy to use in tegra ble reductions discusse d in sec- tion 3. Let x = 0 b e a b oundary . a. Integrable reduction u = v giv es inte g r a ble b oundary condition u | x =0 = v | x =0 or ( S 3 Q − S 2 P 2 +4 Q 3 − 18 S P Q + 27 Q 2 ) | x =0 = 0 (condition on co efficien ts of cubic equation to ha v e at least to equal ro ots) in S , P , Q v ariables. b. In tegra ble reduction u = v = w giv es in tegrable b oundary conditions u | x =0 = v | x =0 = w | x =0 or P | x =0 = 1 3 S 2 | x =0 , Q | x =0 = 1 27 S 3 | x =0 (condition on co efficien ts of cubic equation to ha ve all ro ots equal. c. In tegra ble reduction u = 0 give s in tegrable b oundary condition u | x =0 = 0 or Q | x =0 = 0 . d. In tegrable reduction u = 0, v = 0 giv es in tegrable b oundary conditions u | x =0 = 0, v | x =0 = 0 or P | x =0 = 0, Q | x =0 = 0. T o o btain b oundary v alue problems that admit bi-Hamiltonian represen tation w e mo dify Hamiltonia n functions, as in the case o f T o da system. W e use S, P, Q v a riables, the Hamiltonian op erators hav e simpler form in this v ar ia bles . F or equation (64) w e hav e Theorem 4 The e quation (64) on a se gment [0 , 1] with b oundary c onditions  P − 1 4 S 2  | x =0 = 0 , Q | x =0 = 0 and  P − 1 4 S 2  | x =1 = 0 , Q | x =1 = 0 . (82) admits the bi-Hamil ton i a n r epr e sentation with Hamiltonian op er ators (68), (69) and Hamiltoni ans ¯ ¯ H − 1 = Z ∞ −∞  Q + 1 2 S P − 1 24 S 3  θ ( x ) θ (1 − x ) dx (83) and ¯ ¯ H 0 = Z ∞ −∞ S θ ( x ) θ (1 − x ) dx, (84) r esp e ctively, whe r e θ ( x ) is the He aviside step function. Pro of. The Hamiltonian equations   S P Q   t = C n   δ ¯ ¯ H n /δ S δ ¯ ¯ H n /δ P δ ¯ ¯ H n /δ Q   n = − 1 , 0 (85) 17 are for n = − 1 S t = P x − 1 2 S S x + ( P − 1 4 S 2 )( δ ( x ) − δ (1 − x )) , P t = Q x + 1 2 S Q ( δ ( x ) − δ (1 − x )) , Q t = 1 2 QS x + Q ( δ ( x ) − δ (1 − x )) (86) and for n = 0 S t = P x − 1 2 S S x + (2 P − 1 2 S 2 )( δ ( x ) − δ (1 − x )) , P t = Q x + Q ( δ ( x ) − δ (1 − x )) , Q t = 1 2 QS x + 1 2 S Q ( δ ( x ) − δ (1 − x )) , (87) where x ∈ [0 , 1]. Under the b oundary conditions ( 82) we hav e the equation (64) on [0 , 1].  The b oundary conditions (82) are symmetry in tegra ble. Lemma 8 A l l the symmetries of the h i e r ar chy (63) ar e c omp atible wi th the b oundary c ondition (82). Pro of. The b oundary condition ( 82) is compatible with an ev olution sym- metry   S P Q   τ =   σ π κ   (88) if π = 1 2 S σ and κ = 0 f or P = 1 4 S 2 and Q = 0 on the b oundary x = 0. That is under the conditions ( 82) the symmetry (88) should take the form   S τ 1 2 S S τ 0   =   σ 1 2 S σ 0   (89) One can che ck that the first symmetry o f the hierarch y (63) has suc h a form. Let us sho w that the recursion op erator (67) preserv es the form (89) . On the b oundary x = 0 we rewrite the recursion op erator (6 7 ) in terms of t deriv ativ es using the equation (6 4) as follows   P − 1 4 S 2 − 1 4 S t D − 1 t S 1 2 S + 1 2 S t D − 1 t 3 + P t D − 1 t 3 2 Q − 1 4 P t D − 1 t S P + 1 2 P t D − 1 t 2 S + ( 1 2 S P t + Q t ) D − 1 t 1 4 S Q − 1 4 Q t D − 1 t S 3 2 Q + 1 2 Q t D − 1 t P + 1 2 QP t D − 1 t   (90) 18 Applying the recursion op erator ( 9 0) to a symmetry (89) w e obtaine a sym- metry   S ˜ τ 1 2 S S ˜ τ 0   =   ˜ σ 1 2 S ˜ σ 0   . (91)  F or equation (66) w e hav e Theorem 5 The e quation (66) on a se gment [0 , 1] with b oundary c onditions Q | x =0 = 0 and Q | x =1 = 0 . (92) admits the bi-Hamil ton i a n r epr e sentation with Hamiltonian op er ators (69), (69) and Hamiltoni ans ˜ ˜ H − 1 = Z ∞ −∞  S Q + 1 2 P 2  θ ( x ) θ (1 − x ) dx (93) and ˜ ˜ H 0 = Z ∞ −∞ P θ ( x ) θ (1 − x ) dx, (94) r esp e ctively, whe r e θ ( x ) is the He aviside step function. Pro of. The Hamiltonian equations   S P Q   t = C n    δ ˜ ˜ H n /δ S δ ˜ ˜ H n /δ P δ ˜ ˜ H n /δ Q    n = − 1 , 0 (95) are for n = − 1 S t = 2 Q x + 2 Q ( δ ( x ) − δ (1 − x )) , P t = S Q x + QS x + P Q ( δ ( x ) − δ (1 − x )) , Q t = QP x + S Q ( δ ( x ) − δ ( 1 − x )) (96) and for n = 0 S t = 2 Q x + Q ( δ ( x ) − δ (1 − x )) , P t = S Q x + QS x + 2 S Q ( δ ( x ) − δ (1 − x ) , Q t = QP x + P Q ( δ ( x ) − δ (1 − x )) , (97) 19 where x ∈ [0 , 1]. Under the b oundary conditions ( 92) we hav e the equation (66) on [0 , 1].  In the same w a y as in lemma 8 one can sho w that the b oundary condition (92) is symmetry integrable. This case is similar to the case of T o da system (the b oundary condition Q | Γ = 0 in mo dified v ar iables is uv w | Γ = 0 ) . 6 Conclus ion In this article w e studied the problem of integrable b oundary conditions for h ydro dynamic type in t egr a ble systems. T o our knowle dge the problem has neve r b een discuss ed in the literature b efore. Since the term integrabil- it y has v arious meanings the notion of in tegrable b oundary conditions has also sev eral definitions. As basic ones w e take three definitions, namely , consistency with infinite set of symmetries, consistency with infinite set of conserv ed quan tities, and consistency with the Ha milto nian integrabilit y (or bi-Hamiltonian structure). Comparison of these t hree kinds of in tegrable b oundary conditions sho ws that the consistency with the bi-Hamiltonian structure is a v ery sev ere restriction. Only v ery special kind of b oundary conditions passes this test. The class o f symmetry consisten t b oundary con- ditions seems to b e relativ ely larger. As a n example w e studied disp ersionles s T o da system as a n example of N = 2 system. W e found all symmetry com- patible b oundary conditions of this system and show ed that only a sub class of these b oundary conditions are compatible with the Hamiltonian form u- lation of the system. W e p oin ted out that the inte g r a ble reductions of t he N − system of hydrodynamical ty p e of equations are dir ectly related to the in tegrable b oundary conditions of the same systems. Using this prop ert y , a metho d for constructing exact solutions satisfying the in t egr a ble b ound- ary conditio ns is giv en. W e considered also an N = 3 sys tem. In tegrable b oundary conditions compatible with symmetries and compatible with the Hamiltonian fo rm ulatio n of this system w ere f ound. 7 Ac kno wle d gemen t This w ork is partially supp orted b y the Scien tific and T ec hnolog ical Researc h Council of T urk ey (TUBIT AK) and T urkish Academ y of Sciences (TUBA). One of the authors (I.H.) thanks also RFBR grant ♯ 06-01-9 2051 KE-a. 20 References [1] B.A.Dubrov in, S.P .Nov iko v, Hamilto nia n formalism of one-dimensional systems of hydrodynamic type and t he Bogolyub ov -Whitham a v erag- ing metho d. Soviet Math. D okl. 2 7 , 66 5-669(1983). [2] S.P .Tsarev, On P oisson brac k ets and one-dimensional Hamiltonian sys- tems of h ydro dynamic t yp e. So viet Math. D okl. 3 1, 4 88-491(1985). [3] E.V.F erap on tov , Nonlo cal Hamiltonia n op erators of hy dro dynamic t yp e: differen tial geometry and applications, Amer. Math. So c. T ransl.(2), 170(1995 ) 33-58 . 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