Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierarchy with self-consistent sources

In tegrable Roso c hatius deformations of hig her-orde r constrain e d flo ws and the solit on hierarc h y with self-cons istent sources Y uqin Y ao 1 and Y un b o Zeng 2 Dep art ment of Mathematics, Tsingh ua U niversity, Beijing 10 0084 , PR China Abstract W e prop ose a systematic metho d to generalize the in tegrable Roso c hatius defor - mations for finite dimensional in tegrable Hamiltonian system s to integrable Roso c hatius deformations for infinite dimensional in tegrable equations. Infinite n umber of the in- tegrable Roso c hatius deformed higher-order constrained flow s of some soliton hierar- c hies, which includes the generalized integrable H ´ e non-Heiles system, and the in tegrable Roso c hatius deformations of the K dV hierarc h y with self-consisten t sources, of the AKNS hierarc h y with self-consisten t sources a nd of the mKdV hierarc h y with self-consisten t sources as w ell as their La x represen tations are presen ted. 1 In tro duction Roso c hatius found that it would still k eep the in tegrability to a dd a p ot ential of the sum of in v erse squares of the co ordina t es to that of the Neumann system [1, 2]. The deformed system is called Neumann-Roso c hatius system. W o jciec ho wski obtained an analo gy sys- tem for the Garnier system as a statio na r y KdV flo w in 1985 [3, 4]. In 1997, Kub o et al. [5] constructed the analogy system fo r the Jacobi system [6] and the geo desic flow equation on the ellipsoid based up on the D eift techniq ue and a theorem that t he Gauss map transfor ms the Neumann system to the Jacobi system [7, 8]. All of these systems ha v e the same c haracter tha t they are in tegrable Hamiltonian systems containing N arbi- trary pa r a meters and the original finite dimensional in tegrable Hamiltonian systems are reco v ered when all these pa rameters v anish . In fact, these systems are a sort of in tegrable deformations of the corresp onding in tegrable Hamiltonian systems, whic h are called in te- grable Roso c hatius deformation. The resulting systems are called the Ro so c hatius-t yp e in tegrable systems . These systems hav e imp ortant ph ysical applications. F or exam- ples, Neumann-Roso chatius system can b e used to describe t he dynamics of ro t a ting closed string solutions in AdS 5 × S 5 and the mem branes o n AdS 4 × S 7 [9]- [17]. The Garnier-Roso c hatius system can b e used to solv e the m ulticomp onent coupled nonlinear Sc hr ¨ o dinger equation [3, 4, 18]. It is not difficult to see tha t eac h of the ab ov e system has its o wn origin. In Ref. [19], Z hou generalize the Roso c hatius metho d to study the integrable Roso c hatius deformations of some explicit constrained flows of soliton equations. Ho w ev er, so far the Roso c hatius defo r ma t io ns a re limited to few finite dimensional in tegrable Hamiltonian systems(FDIHSs). It is natural to ask whether t here exist inte- grable Roso c hatius deformat io ns for infinite dimensional integrable equations. The main 1 yqyao@math.tsinghua.edu.cn 2 yzeng@math.tsinghua.edu.cn 1 purp ose of this pap er is to generalize the Roso c hatius deformation from FDIHSs t o in- finite dimensional in tegrable equations. W e will in v estigate the integrable Roso chatius deformations firstly for infinite n um b er of higher-order constrained flows of some soliton equations, then for some soliton hierarchies with self-consisten t sources. In recen t y ears the constrained flo ws of soliton equations obt a ined from the symmetry reduction of soliton equations, whic h can b e transfor med in to a FD IHS, attracted a lo t of attention [20]- [3 3 ]. Many w ell-know n F DIHSs are r eco v ered by means of constrained flo ws. F urthermore, soliton equation can b e factorized into tw o comm uting constrained flo ws, whic h pro vides an effectiv e w ay to solv e the soliton equations through solving the constrained flo ws. T o mak e the pap er self-contained, w e first briefly recall the higher-order constrained flo ws o f the soliton hierarc h y and the soliton hierarch y with self-consisten t sources. Con- sider a hierar c h y of solito n equation whic h can b e form ulated a s an infinite dimensional Hamiltonian systems u t n = J δ H n δ u . (1) The a uxiliary linear problems asso ciated with (1) are give n by φ x = U ( λ, u ) φ, φ = ( φ 1 , φ 2 ) T (2a) φ t n = V ( n ) ( λ, u ) φ . (2b) Then the higher-or der constrained flow s of (1) consist of the equations obtained from the sp ectral problem (2a) for N distinct λ j and the restriction o f the v ariational deriv ativ es for the conserv ed quantities H n and λ j [23] J [ δ H n δ u + N X j =1 δ λ j δ u ] = 0 , (3a) φ j,x = U ( λ j , u ) φ j , φ j = ( φ 1 j , φ 2 j ) T , j = 1 , 2 , · · · , N . (3b) By in tro duction the so called Jacobi-Ostrog r adsky co ordinates [34], (3) can b e trans- formed into a FDIHS. The Lax represen tation of (3) can b e deduced from the adjoint represen tation of ( 2) [24] N ( n ) x = [ U, N ( n ) ] . (4) Mainly , N ( n ) has the following forms N ( n ) ( λ ) = V ( n ) ( λ ) + N 0 =  A ( λ ) B ( λ ) C ( λ ) − A ( λ )  , where N 0 = N X j =1 1 λ − λ j  φ 1 j φ 2 j − φ 2 1 j φ 2 2 j − φ 1 j φ 2 j  , or N 0 = N X j =1 1 λ 2 − λ 2 j  λφ 1 j φ 2 j − λ j φ 2 1 j λ j φ 2 2 j − λφ 1 j φ 2 j  . 2 The solito n hierarc hy with self-consisten t sources is defined b y [25, 26] u t n = J [ δ H n δ u + N X j =1 δ λ j δ u ] , (5a) φ j,x = U ( λ j , u ) φ j , j = 1 , 2 , · · · , N . (5b) Since the higher-order constrained flows (3) a re just the stationary equations of ( 5 ), the zero-curv ature represen tation f o r (5) can b e induced from (4) as follows U t n − N ( n ) x + [ U, N ( n ) ] = 0 (6) whic h implies that (5) is Lax in tegrable. In fact, (5) can also b e form ulated as an infinite- dimensional inte gra ble Hamiltonian system with t-type Hamiltonian op era t o r b y taking t as the ’spatial’ v ariable and x as the evolution parameter as w ell as in tro ducing the Jacobi-Ostrogradsky co ordinates [35, 36]. Then w e can find the P oisson brac ke t defined b y the t-ty p e Hamiltonian op erator, and the conserv ed densit y when considering x a s the ev olution parameter. The t-type bi-Hamiltonian description for KdV equation with self-consisten t sources and for Jaulen t-Mio dek equation with self-consisten t sources w ere presen ted in [35] and [36], resp ectiv ely . The soliton equation with self-consisten t sources ha v e imp ortant ph ysical application, for example, the K dV equation with self-consisten t sources describ es the in teraction of long and short capillary-gravit y wa v e [3 7]- [3 9 ]. In this pap er, in the same w a y [19 ], we construct the Ro so c hatius deformation ˜ N ( n ) of Lax matrix N ( n ) b y replacing φ 2 2 j in the matr ix N 0 with φ 2 2 j + µ j φ 2 1 j , namely the en tries of ˜ N ( n ) are giv en by ˜ A ( λ ) = A ( λ ) , ˜ B ( λ ) = B ( λ ) , ˜ C ( λ ) = C ( λ ) + N X j =1 µ j ( λ − λ j ) φ 2 1 j , or ˜ C ( λ ) = C ( λ ) + N X j =1 λ j µ j ( λ 2 − λ 2 j ) φ 2 1 j (7) Then Lax represen tation (4) with N ( n ) replaced b y ˜ N ( n ) giv es rise to the Roso c hatius defor- mations of (3). The fact that suc h a substitute kee ps the relations of the P oisson brac k ets of A ( λ ) , B ( λ ) and C ( λ ) guarantees the in tegrability of Roso chatius deformations of (3). In this w a y w e can obtain infinite num ber of in tegrable Roso c hatius deformed F D IHSs con- structed from the Roso c hatius deformed higher-order constrained flows of KdV hierarc h y , AKNS hierarch y and mKdV hierarc h y , resp ectiv ely . Among these Roso c hatius deformed FDIHSs, it needs to p oin t out that the R oso c hatius deformation of the first higher-order constrained flow of KdV hierarc h y con tains the w ell-kno wn generalized in tegrable H ´ e non- Heiles system, and can b e regarded as the in tegrable m ultidimensional extension of H ´ e non- Heiles system. Then, based on the R oso c hatius deformed higher-order constrained flows , the Roso c hatius deformat ions of the soliton hierarc hy with self-consisten t sources (RDSH- SCS) can b e constructed through (6) with N ( n ) replaced b y ˜ N ( n ) . The inte gra bility of the 3 RDSHSCS can b e explained b y t he fact that the RDSHSCS p ossesses the zero-curv ature represen tation (6) with N ( n ) replaced by ˜ N ( n ) and its stationary reduction is in tegrable Roso c hatius deformations of (3). In these w a y , w e construct Roso c hatius deformations of KdV hierarc h y with self-consisten t sources (RDKdVHSCS), of AKNS hierarc h y with self- consisten t sources (RDAKNSHSC S) and of mKdV hierarc h y with self-consisten t sources (RDmKdVHSCS), a s w ell as their zero-curv ature represen tations. W e not ice that there are t wo kinds of RD SHSCS. F or RDKdVHSCS and RDmKdVHSCS, the Roso c hatius formed terms only app ear in (5b). Ho w ev er for RDAKNSHS CS Roso c hatius deformed terms o ccur in b oth (5a) and (5 b). The structure of the pap er is as f ollo ws. In Sec.2, w e presen t the Roso c ha t ius de- formation o f the higher-order constrained flo ws of KdV hierarc h y and RD KdVHSCS as w ell as their Lax represen tations. In Sec.3, we obtain the Ro so c hatius deformation of higher-order constrained flow s o f AKNS hierar ch y and RDA KNSHSCS as w ell as their Lax represen tations. In Sec.4, we obtain the Roso c ha tius deformation of higher-order constrained flows of mKdV hierarc hy and RDmKdVHSCS and their Lax represen tations. In Sec.5, a conclusion is made. 2 The Roso c hatius deformed KdV h i erarc h y with sel f- consis t en t source s Consider t he Sc hr ¨ o dinger equation [40] φ 1 xx + ( λ + u ) φ 1 = 0 , (8) whic h can b e written in the matrix form  φ 1 φ 2  x = U  φ 1 φ 2  , U =  0 1 − λ − u 0  . (9) The a djoin t represen tation of (9) reads V x = [ U, V ] . (10) Set V = ∞ X i =1  a i b i c i − a i  λ − i . (11) Solving (1 0) yields a k = − 1 2 b k ,x , b k +1 = Lb k = − 1 2 L k − 1 u, c k = − 1 2 b k ,xx − b k +1 − b k u, a 0 = b 0 = 0 , c 0 = − 1 , a 1 = 0 , b 1 = 1 , c 1 = − 1 2 u, a 2 = 1 4 u x , (12) b 2 = − 1 2 u, c 2 = 1 8 ( u xx + u 2 ) , b 3 = 1 8 ( u xx + 3 u 2 ) , · · · 4 where L = − 1 4 ∂ 2 − u + 1 2 ∂ − 1 u x , ∂ = ∂ ∂ x . Set V ( n ) = n X i =1  a i b i c i − a i  λ n − i +  0 0 b n +1 0  , (13) and t a k e  φ 1 φ 2  t n = V ( n ) ( u, λ )  φ 1 φ 2  . (14) Then t he compatibility of the Eqs.(9) and (14) g iv es rise to the KdV hierarch y u t n = − 2 b n +1 ,x ≡ ∂ δ H n δ u , n = 0 , 1 , · · · , (15) where H n = 4 b n +2 / 2 n + 1. W e ha v e δ λ δ u = φ 2 1 , Lφ 2 1 = λφ 2 1 . (16) The hig her-order constrained flows of t he KdV hierarc hy is giv en b y [23], δ H n δ u − α N X j =1 δ λ j δ u ≡ − 2 b n +1 − α N X j =1 φ 2 1 j = 0 , (17a) φ 1 j,x = φ 2 j , φ 2 j,x = − ( λ j + u ) φ 1 j , j = 1 , 2 , · · · , N . (17b) According to (1 2), (16) and (17 ), w e find the Lax represen tation (4) for (17) with N ( n ) = n X k =0  a k b k c k − a k  λ n − k + α 2 N X j =1 1 λ − λ j  φ 1 j φ 2 j − φ 2 1 j φ 2 2 j − φ 1 j φ 2 j  . (18) By taking the so-called Jacobi-Ostrogradsky co ordinates [34] q i = u ( i − 1) , i = 1 , · · · , n − 1 , p i = δ H n δ u ( i ) = X l ≥ 0 ( − ∂ ) l ∂ H n ∂ u ( i + l ) and setting Φ 1 = ( φ 11 , φ 12 , · · · , φ 1 N ) T , Φ 2 = ( φ 21 , φ 22 , · · · , φ 2 N ) T , Q = ( φ 11 , φ 12 , · · · , φ 1 N , q 1 , · · · , q n − 1 ) T , P = ( φ 21 , φ 22 , · · · , φ 2 N , p 1 , · · · , p n − 1 ) T , Λ = diag ( λ 1 , λ 2 , · · · , λ N ) . Eq.(17) with α = 2 4 n can b e transfor med in to a FDIHS [2 3] Q x = ∂ H ∂ P , P x = − ∂ H ∂ Q , (19) 5 with H = n − 1 X i =1 q i,x p i − H n + 1 2 h Φ 2 , Φ 2 i + 1 2 h ΛΦ 1 , Φ 1 i + 1 2 q 1 h Φ 1 , Φ 1 i , where hi denotes the inner pro duction in R N . F or example, (17) fo r n = 0 , α = − 2 give s rise t o the Neumann sys tem [20], (17) for n = 1 , α = 1 leads to the Garnier system [3, 2 0]. When n = 2, the Eq.(17) for α = 1 8 giv es the first higher-or der constrained flow [23] u xx + 3 u 2 = − 1 2 N X j =1 φ 2 1 j = − 1 2 h Φ 1 , Φ 1 i , (20a) φ 1 j,x = φ 2 j , φ 2 j,x = − ( λ j + u ) φ 1 j , j = 1 , 2 , · · · , N . (20b) Let q 1 = u, p 1 = u x , ( 2 0) b ecomes a FDIHS (19 ) with H = 1 2 h Φ 2 , Φ 2 i + 1 2 h ΛΦ 1 , Φ 1 i + 1 2 q 1 h Φ 1 , Φ 1 i + 1 2 p 2 1 + q 3 1 , and ha s the Lax represen tation (4) with the en tries of N (2) giv en by A ( λ ) = 1 4 p 1 + 1 16 N X j =1 φ 1 j φ 2 j λ − λ j , B ( λ ) = λ − 1 2 q 1 − 1 16 N X j =1 φ 2 1 j λ − λ j , C ( λ ) = λ 2 − q 1 2 λ − q 2 1 4 − 1 16 h Φ 1 , Φ 1 i + 1 16 N X j =1 φ 2 2 j λ − λ j . With resp ect to the standard P oisson brac k et it is found that { A ( λ ) , A ( µ ) } = { B ( λ ) , B ( µ ) } = 0 , { C ( λ ) , C ( µ ) } = A ( λ ) − A ( µ ) 4 , { A ( λ ) , B ( µ ) } = B ( λ ) − B ( µ ) 8( λ − µ ) , { A ( λ ) , C ( µ ) } = C ( λ ) − C ( µ ) 8( µ − λ ) − B ( λ ) 8 , (21) { B ( λ ) , C ( µ ) } = A ( λ ) − A ( µ ) 4( λ − µ ) . It f o llo ws f rom (21) that { A ( λ ) 2 + B ( λ ) C ( λ ) , A ( µ ) 2 + B ( µ ) C ( µ ) } = 0 . (22) When n = 3, (17 ) for α = 1 32 yields the second higher-order constrained flo w [23] u xxxx + 5 u 2 x + 10 uu xx + 10 u 3 = 1 2 h Φ 1 , Φ 1 i , (23a) φ 1 j,x = φ 2 j , φ 2 j,x = − ( λ j + u ) φ 1 j , j = 1 , 2 , · · · , N . (23b) 6 Let q 1 = u, q 2 = u x , p 1 = − u xxx − 10 uu x , p 2 = u xx , ( 2 3) b ecomes a FDIHS (19 ) with H = 1 2 h Φ 2 , Φ 2 i + 1 2 h ΛΦ 1 , Φ 1 i + 1 2 q 1 h Φ 1 , Φ 1 i + 1 2 p 2 2 + q 2 p 1 + 5 q 1 q 2 2 − 5 2 q 4 1 . W e now consider the Ro so c hatius defor mation ˜ N (2) of the La x matrix N (2) with ˜ A ( λ ) = A ( λ ) , ˜ B ( λ ) = B ( λ ) , ˜ C ( λ ) = C ( λ ) + 1 16 N X j =1 µ j ( λ − λ j ) φ 2 1 j . It is not difficult to find that ˜ A ( λ ) , ˜ B ( λ ) and ˜ C ( λ ) k eep the r elatio ns of the P oisson brac k ets (2 1) and (22). A direct calculation gives ˜ A 2 ( λ ) + ˜ B ( λ ) ˜ C ( λ ) = − λ 3 + P 0 + N X j =1 P j λ − λ j − 1 256 N X j =1 µ j ( λ − λ j ) 2 , (24) where P 0 = 1 16 ( h Φ 2 , Φ 2 i + h ΛΦ 1 , Φ 1 i + q 1 h Φ 1 , Φ 1 i + 2 q 3 1 + p 2 1 + N X j =1 µ j φ 2 1 j ) P j = p 1 32 φ 1 j φ 2 j + 1 16 ( λ j − q 1 2 )( φ 2 2 j + µ j φ 2 1 j ) + 1 16 ( λ 2 j + q 1 2 λ j + 1 16 h Φ 1 , Φ 1 i + q 2 1 4 ) φ 2 1 j + 1 256 X k 6 = j 1 λ j − λ k [2 φ 1 j φ 1 k φ 2 j φ 2 k − φ 2 1 j ( φ 2 2 k + µ k φ 2 1 k ) − φ 2 1 k ( φ 2 2 j + µ j φ 2 1 j )] , j = 1 , · · · , N . (25) Cho osing 8 P 0 = ˜ H as a Hamiltonian function, we get the follow ing Hamiltonian system q 1 x = p 1 , p 1 x = − 1 2 h Φ 1 , Φ 1 i − 3 q 2 1 , (26a) φ 1 j x = φ 2 j , φ 2 j x = − λ j φ 1 j − q 1 φ 1 j + µ j φ 3 1 j , (26b) whic h is the Roso chatius deformation of the first higher-order constrained flo w (20). F rom (4) , w e hav e d dx tr ˜ ( N (2) ( λ )) 2 = d dx [ ˜ A 2 ( λ ) + ˜ B ( λ ) ˜ C ( λ )] = tr [ U, ˜ ( N (2) ( λ )) 2 ] = 0 , (27) whic h implies that P 0 , P 1 , · · · , P N are N + 1 indep enden t first integrals of the Hamiltonia n system (26). The equalit y (22) of P oisson brac k et for ˜ A ( λ ) , ˜ B ( λ ) and ˜ C ( λ ) indicates that { P i , P j } = 0 , i, j = 0 , 1 , · · · , N . So the Roso ch atius deformation (26) of the first higher- order constrained flo w (2 0) is a FD IHS in the Lio uville’s sense [41 ]. Remark 1. F or N = 1 , λ 1 = 0 , (2 6) yields q 1 xx = − 1 2 φ 2 1 − 3 q 2 1 , 7 φ 1 xx = − q 1 φ 1 + µ 1 φ 3 1 , whic h is the w ell-kno wn generalized integrable H ´ e non-Heiles syste m [42]- [44 ]. In f act (26) can b e regarded as the inte gra ble m ultidimensional extension of H ´ e non-Heiles system. Similarly , c ho osing ˜ H = 1 2 h Φ 2 , Φ 2 i + 1 2 h ΛΦ 1 , Φ 1 i + 1 2 q 1 h Φ 1 , Φ 1 i + 1 2 p 2 2 + q 2 p 1 + 5 q 1 q 2 2 − 5 2 q 4 1 + 1 2 N X j =1 µ j φ 2 1 j , w e get the Roso chatius deformation of t he second higher-order constrained flow (23) q 1 x = q 2 , q 2 x = p 2 , p 1 x = 1 0 q 3 1 − 5 q 2 2 − 1 2 h Φ 1 , Φ 1 i , p 2 x = − 10 q 1 q 2 − p 1 , (28a) φ 1 j x = φ 2 j , φ 2 j x = − λ j φ 1 j − q 1 φ 1 j + µ j φ 3 1 j , (28b) whic h, in the same w ay , can b e sho wn to b e a F D IHS. In general, the in tegrable R o so c hatius deformation of the higher-order constrained flo w (19) is generated b y the follow ing Hamiltonian function ˜ H = n − 1 X i =1 q i,x p i − H n + 1 2 h Φ 2 , Φ 2 i + 1 2 h ΛΦ 1 , Φ 1 i + 1 2 q 1 h Φ 1 , Φ 1 i + 1 2 N X j =1 µ j φ 2 1 j . (29) The K dV hierarch y with self-consisten t sources is defined by [25, 26, 29], [37 ]- [39] u t n = ∂ [ δ H n δ u − α N X j =1 δ λ j δ u ] ≡ ∂ [ − 2 b n +1 − α N X j =1 φ 2 1 j ] , (30a) φ 1 j x = φ 2 j , φ 2 j,x = − ( λ j + u ) φ 1 j , j = 1 , 2 , · · · , N . (30b) Since the higher-or der constrained flo ws (17) are j ust the stationary equation of the KdV hierarc h y with self-consisten t sources (30), it is ob viously that the zero-curv ature repre- sen ta tion for the KdV hierarc hy with self-consisten t sources (30) is given by (6) with [29] N ( n ) = n X i =1  a i b i c i − a i  λ n − i +   0 0 b n +1 + α 2 N P j =1 φ 2 1 j 0   + α 2 N X j =1 1 λ − λ j  φ 1 j φ 2 j − φ 2 1 j φ 2 2 j − φ 1 j φ 2 j  . (31) Eq.(30) for n = 2 , α = 1 8 giv es rise to the KdV equation with self-consisten t sources [29, 37] u t = − 1 4 ( u xxx + 6 uu x ) − 1 8 N X j =1 ( φ 2 1 j ) x , (32a) φ 1 j x = φ 2 j , φ 2 j x = − ( λ j + u ) φ 1 j , j = 1 , 2 , · · · , N . (32 b) 8 Based o n (26), the Roso c hatius deformation of KdV equation with self-consisten t sources is given b y u t = − 1 4 ( u xxx + 6 uu x ) − 1 8 N X j =1 ( φ 2 1 j ) x , (33a) φ 1 j x = φ 2 j , φ 2 j x = − ( λ j + u ) φ 1 j + µ j φ 3 1 j , j = 1 , · · · , N (33b) whic h has the zero-curv ature represen tation (6) with the N (2) giv en b y N (2) =   u x 4 λ − u 2 − λ 2 − u 2 λ + 1 4 u xx + 1 2 u 2 + 1 16 N P j =1 φ 2 1 j − u x 4   + 1 16 N X j =1 1 λ − λ j φ 1 j φ 2 j − φ 2 1 j φ 2 2 j + µ j φ 2 1 j − φ 1 j φ 2 j ! . (34) Remark 2: The fact that the stationary equation of (33) is a FDIHS (26) and (33) has zero-curv ature represen tation (6) implies the integrabilit y o f t he Roso c hatius deformation of the KdV equation with self-consisten t source (33). In general t he Roso c hatius deformation o f the KdV hierarch y with self-consisten t sources is giv en by u t n = ∂ [ δ H n δ u − 2 4 n N X j =1 φ 2 1 j ] , (35a) φ 1 j x = φ 2 j , φ 2 j,x = − ( λ j + u ) φ 1 j + µ j φ 3 1 j , j = 1 , 2 , · · · , N . (35b) whic h has the zero-curv ature represen tation (6) with N ( n ) giv en b y N ( n ) = n X i =1  a i b i c i − a i  λ n − i +   0 0 b n +1 + 1 4 n N P j =1 φ 2 1 j 0   + 1 4 n N X j =1 1 λ − λ j φ 1 j φ 2 j − φ 2 1 j φ 2 2 j + µ j φ 2 1 j − φ 1 j φ 2 j ! . 3 The Roso c hatius deformed AKNS hie rarc h y w ith self-cons istent sources F or AKNS eigen v alue problem [40]  φ 1 φ 2  x = U  φ 1 φ 2  , U =  − λ q r λ  , 9 and evolution equation of eigenfunction  φ 1 φ 2  t n = V ( n )  φ 1 φ 2  , V ( n ) = n X i =1  a i b i c i − a i  λ n − i , the asso ciat ed AKNS hierarc hy reads u t n =  q r  t n = J  c n +1 b n +1  = J δ H n +1 δ u where a 0 = − 1 , b 0 = c 0 = 0 , a 1 = 0 , b 1 = q , c 1 = r, · · · ,  c n +1 b n +1  = L n  r q  , L = 1 2  ∂ − 2 r ∂ − 1 q 2 r∂ − 1 r − 2 q ∂ − 1 q − ∂ + 2 q ∂ − 1 r  , a n,x = q c n − rb n , H n = 2 n a n +1 , J =  0 − 2 2 0  , w e ha v e δ λ δ q = 1 2 φ 2 2 , δ λ δ r = − 1 2 φ 2 1 . The hig her-order constrained flows of t he AKNS hierarc hy is g iv en b y [26, 27] δ H n +1 δ u − 1 2 N X j =1 δ λ j δ u =  c n +1 b n +1  − 1 4  h Φ 2 , Φ 2 i −h Φ 1 , Φ 1 i  = 0 , (36a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j , j = 1 , 2 , · · · , N . (36b) (36) fo r n = 2 gives the first higher-o rder constrained flow − q xx + 2 q 2 r − N X j =1 φ 2 1 j = 0 , r xx − 2 q r 2 − N X j =1 φ 2 2 j = 0 , (37a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j , j = 1 , 2 , · · · , N . (37b) Let q 1 = q , q 2 = r, p 1 = − 1 2 r x , p 2 = − 1 2 q x , Q = ( φ 11 , φ 12 , · · · , φ 1 N , q 1 , q 2 ) T , P = ( φ 21 , φ 22 , · · · , φ 2 N , p 1 , p 2 ) T . (37) b ecomes a FDIHS (19) with H = −h Λ Φ 1 , Φ 2 i + q 1 2 h Φ 2 , Φ 2 i − q 2 2 h Φ 1 , Φ 1 i + 1 2 q 2 1 q 2 2 − 2 p 1 p 2 , and ha s the Lax represen tation (4) with the en tries of Lax ma t r ix N (2) giv en b y A ( λ ) = − 2 λ 2 + q 1 q 2 + 1 2 N X j =1 φ 1 j φ 2 j λ − λ j , B ( λ ) = 2 λq 1 + 2 p 2 − 1 2 N X j =1 φ 2 1 j λ − λ j , C ( λ ) = 2 λ q 2 − 2 p 1 + 1 2 N X j =1 φ 2 2 j λ − λ j . 10 With resp ect to the standard P oisson brac k et it is found that { A ( λ ) , A ( µ ) } = { B ( λ ) , B ( µ ) } = { C ( λ ) , C ( µ ) } = 0 , { A ( λ ) , B ( µ ) } = B ( λ ) − B ( µ ) λ − µ , { A ( λ ) , C ( µ ) } = C ( µ ) − C ( λ ) λ − µ , { B ( λ ) , C ( µ ) } = 2[ A ( λ ) − A ( µ )] λ − µ , (38) whic h giv es rise to (22). No w w e consider the Ro so c hatius deformation ˜ N (2) of the Lax matrix N (2) ˜ A ( λ ) = A ( λ ) , ˜ B ( λ ) = B ( λ ) , ˜ C ( λ ) = C ( λ ) + 1 2 N X j =1 µ j ( λ − λ j ) φ 2 1 j . (39) It is easy to find that the elemen ts in ˜ N (2) still k eep the relations of the P oisson bra ck ets (38) and (22). A direct calculation gives ˜ A 2 ( λ ) + ˜ B ( λ ) ˜ C ( λ ) = 4 λ 4 + P 0 λ + P 1 + N X j =1 P j λ − λ j − 1 4 N X j =1 µ j ( λ − λ j ) 2 (40) where P 0 = − 4 q 1 p 1 + 4 q 2 p 2 − 2 h Φ 1 , Φ 2 i , P 1 = − 4 p 1 p 2 + q 2 1 q 2 2 + q 1 ( h Φ 2 , Φ 2 i + N X j =1 µ j φ 2 1 j ) − q 2 h Φ 1 , Φ 1 i − 2 h ΛΦ 1 , Φ 2 i , P j +1 = ( λ j q 1 + p 2 )( φ 2 2 j + µ j φ 2 1 j ) − ( λ j q 2 − p 1 ) φ 2 1 j + ( q 1 q 2 − 2 λ 2 j ) φ 1 j φ 2 j (41) + 1 4 X k 6 = j 1 λ j − λ k [2 φ 1 j φ 2 j φ 1 k φ 2 k − φ 2 1 j ( φ 2 2 k + µ k φ 2 1 k ) − φ 2 1 k ( φ 2 2 j + µ j φ 2 1 j )] , j = 1 , · · · , N . Cho osing 1 2 P 1 = ˜ H as a Hamiltonian function, we get the following Hamiltonian system q 1 x = − 2 p 2 , q 2 x = − 2 p 1 , (42a) p 1 x = − 1 2 h Φ 2 , Φ 2 i − N X j =1 µ j 2 φ 3 1 j − q 2 2 q 1 , p 2 x = 1 2 h Φ 1 , Φ 1 i − q 2 1 q 2 , (42b) φ 1 j x = − λ j φ 1 j + q 1 φ 2 j , φ 2 j x = λ j φ 2 j + q 2 φ 1 j + µ j q 1 φ 3 1 j , j = 1 , · · · , N (42c) whic h has the La x represen ta t ion (4) with N (2) replaced b y ˜ N (2) . Then (22) a nd (27) implies that P 0 , P 1 , · · · , P N +1 are N + 2 indep enden t first integrals in inv olution, so (42) is a FDIHS [41]. 11 In general, in the similar wa y as in Sec.2, (3 6) b ecomes a FDIHS (19) with H = n X i =1 q i,x p i − H n +1 − h ΛΦ 1 , Φ 2 i + q 1 2 h Φ 2 , Φ 2 i − q 2 2 h Φ 1 , Φ 1 i . Then the integrable Roso c hatius deformat io n of the hig her-order constrained flow ( 3 6) is generated by the follow ing Hamiltonian function ˜ H = n X i =1 q i,x p i − H n +1 − h ΛΦ 1 , Φ 2 i + q 1 2 h Φ 2 , Φ 2 i − q 2 2 h Φ 1 , Φ 1 i + 1 2 N X j =1 µ j q 1 φ 2 1 j . (43) The AK NS hierarch y with self-consisten t sources is [26]  q r  t n = J [ δ H n +1 δ u − 1 2 N X j =1 δ λ j δ u ] = J [  c n +1 b n +1  − 1 4  h Φ 2 , Φ 2 i −h Φ 1 , Φ 1 i  ] , (44a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j , j = 1 , 2 , · · · , N . (44b) When n = 2, the AKNS equation with self-consisten t sources reads [26] 2 q t = − q xx + 2 q 2 r − N X j =1 φ 2 1 j , 2 r t = r xx − 2 q r 2 − N X j =1 φ 2 2 j , (45a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j , j = 1 , · · · , N . (45b) Based on (42), w e obtain the in tegrable Roso chatius defor ma t io n o f the AKNS equation with self-consisten t sources 2 q t = − q xx + 2 q 2 r − N X j =1 φ 2 1 j , 2 r t = r xx − 2 q r 2 − N X j =1 φ 2 2 j − N X j =1 µ j φ 2 1 j , (46a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j + µ j q φ 3 1 j , j = 1 , · · · , N (46b) whic h has the zero-curv ature represen tation (6) with N (2) N (2) =  − 2 λ 2 + q r 2 λq − q x 2 λr + r x 2 λ 2 − q r  + 1 2 N X j =1 1 λ − λ j φ 1 j φ 2 j − φ 2 1 j φ 2 2 j + µ j φ 2 1 j − φ 1 j φ 2 j ! . (47) In general t he integrable Roso c hatius deformation of t he AK NS hierarch y with self- consisten t sources is give n by  q r  t n = J [  c n +1 b n +1  − 1 4   h Φ 2 , Φ 2 i + N P j =1 µ j φ 2 1 j −h Φ 1 , Φ 1 i   ] , (48a) φ 1 j x = − λ j φ 1 j + q φ 2 j , φ 2 j x = λ j φ 2 j + r φ 1 j + µ j q φ 3 1 j , j = 1 , 2 , · · · , N . (48b) 12 whic h has the zero-curv ature represen tation (6) with N ( n ) giv en b y N ( n ) = V ( n ) + 1 2 N X j =1 1 λ − λ j φ 1 j φ 2 j − φ 2 1 j φ 2 2 j + µ j φ 2 1 j − φ 1 j φ 2 j ! . Remark 3. In con trast with the Roso c ha tius deformation of KdV hierar c h y with self- consisten t sources, the Roso chatius deformation of AKNS hierarc h y with self-consisten t sources has the deformed term in b oth (48a) a nd (4 8b). 4 The Roso c hatiu s deformed mKdV hierarc h y w ith self-cons istent sources F or mKdV sp ectral problem [32]  φ 1 φ 2  x = U  φ 1 φ 2  , U =  − u λ λ u  , and evolution equation of eigenfunction  φ 1 φ 2  t n = V ( n )  φ 1 φ 2  , V ( n ) = n − 1 X i =1  a i λ b i c i − a i λ  λ 2 n − 2 i − 3 +  a n 0 0 − a n  , the asso ciat ed mKdV hierarc hy reads u t n = − ∂ a n = ∂ δ H n δ u , where a 0 = 0 , b 0 = c 0 = 1 , a 1 = − u, b 1 = − u 2 2 + u x 2 , c 1 = − u 2 2 − u x 2 , · · · , a n +1 = La n , L = 1 4 ∂ 2 − u∂ − 1 u∂ b n = ∂ − 1 u∂ a n − 1 2 a nx , c n = ∂ − 1 u∂ a n + 1 2 a nx , w e ha v e δ λ δ u = 1 2 φ 1 φ 2 . The hig her-order constrained flows of t he mKdV hierarc hy is δ H n δ u + 2 N X j =1 δ λ j δ u ≡ − a n + N X j =1 φ 1 j φ 2 j = 0 , (49a) φ 1 j,x = − uφ 1 j + λ j φ 2 j , φ 2 j,x = λ j φ 1 j + uφ 2 j , j = 1 , 2 , · · · , N . (49b) 13 When n = 2, (49 ) give s the first higher-order constrained flo w u xx − 2 u 3 = − 4 N X j =1 φ 1 j φ 2 j = − 4 h Φ 1 , Φ 2 i , (50a) φ 1 j x = λ j φ 2 j − uφ 1 j , φ 2 j x = λ j φ 1 j + uφ 2 j , j = 1 , · · · , N . (50b) Let q 1 = u, p 1 = − u x 4 , it b ecomes a FDIHS (1 9) with H = − q 1 h Φ 1 , Φ 2 i + 1 2 h ΛΦ 2 , Φ 2 i − 1 2 h ΛΦ 1 , Φ 1 i − 2 p 2 1 + 1 8 q 4 1 , and ha s the Lax represen tation (4) with the en tries of Lax ma t r ix N (2) giv en b y A ( λ ) = − q 1 λ + N X j =1 λφ 1 j φ 2 j λ 2 − λ 2 j , B ( λ ) = λ 2 − 1 2 q 2 1 − 2 p 1 − N X j =1 λ j φ 2 1 j λ 2 − λ 2 j , (51) C ( λ ) = λ 2 − 1 2 q 2 1 + 2 p 1 + N X j =1 λ j φ 2 2 j λ 2 − λ 2 j . With resp ect to the standard P oisson brac k et, a direct calculation giv es { A ( λ ) , A ( µ ) } = { B ( λ ) , B ( µ ) } = { C ( λ ) , C ( µ ) } = 0 , { A ( λ ) , B ( µ ) } = 2 λ B ( λ ) − B ( µ ) λ 2 − µ 2 , { A ( λ ) , C ( µ ) } = 2 λ C ( λ ) − C ( µ ) µ 2 − λ 2 , { B ( λ ) , C ( µ ) } = 4[ A ( µ ) µ − A ( λ ) λ ] µ 2 − λ 2 (52) whic h leads t o (22). No w w e consider the integrable Roso chatius deformation ˜ N (2) of the Lax matrix N (2) ˜ A ( λ ) = A ( λ ) , ˜ B ( λ ) = B ( λ ) , ˜ C ( λ ) = C ( λ ) + N X j =1 µ j λ j ( λ 2 − λ 2 j ) φ 2 1 j . (53) It is not difficult to find that the elemen ts in ˜ N (2) still k eep the relations of the P oisson brac k ets (5 2) and (22). A direct calculation giv es ˜ A 2 ( λ ) + ˜ B ( λ ) ˜ C ( λ ) = λ 4 + P 0 + N X j =1 P j λ 2 − λ 2 j − N X j =1 λ j µ j ( λ 2 − λ 2 j ) 2 (54) where P 0 = − 2 q 1 h Φ 1 , Φ 2 i + h ΛΦ 2 , Φ 2 i − h ΛΦ 1 , Φ 1 i + N X j =1 λ j µ j φ 2 1 j + 1 4 q 4 1 − 4 p 2 1 P j = − 2 q 1 λ 2 j φ 1 j φ 2 j + λ 3 j ( φ 2 2 j + µ j φ 2 1 j ) − ( 1 2 q 2 1 + 2 p 1 )( λ j φ 2 2 j + λ j µ j φ 2 1 j ) − λ 3 j φ 2 1 j + ( 1 2 q 2 1 − 2 p 1 ) λ j φ 2 1 j (55) 14 − X k 6 = j 1 λ 2 j − λ 2 k [2 λ 2 j φ 1 j φ 2 j φ 1 k φ 2 k + λ j λ k φ 2 1 j ( φ 2 2 k + µ k φ 2 1 k ) + λ j λ k φ 2 1 k ( φ 2 2 j + µ j φ 2 1 j )] . Cho osing 1 2 P 0 = ˜ H as Hamiltonian function, w e get the fo llowing Hamiltonian system q 1 x = − 4 p 1 , p 1 x = h Φ 1 , Φ 2 i − 1 2 q 3 1 , (56a) φ 1 j x = λ j φ 2 j − q 1 φ 1 j , φ 2 j x = λ j φ 1 j + q 1 φ 2 j + λ j µ j φ 3 1 j , j = 1 , · · · , N (56b) whic h has the Lax represen ta t io n (4) with N (2) replaced b y ˜ N (2) and is a FDIHS. Similarly , (49) can b e transformed in to a FDIHS (1 9) with N + 1 indep enden t first in tegrals P 0 , P 1 , · · · , P N in inv olution H = n − 1 X i =1 q i,x p i − H n − q 1 h Φ 1 , Φ 2 i + 1 2 h ΛΦ 2 , Φ 2 i − 1 2 h ΛΦ 1 , Φ 1 i . Then the integrable Roso c hatius deformat io n of the hig her-order constrained flow ( 4 9) is generated by the follow ing Hamiltonian function ˜ H = n − 1 X i =1 q i,x p i − H n − q 1 h Φ 1 , Φ 2 i + 1 2 h ΛΦ 2 , Φ 2 i − 1 2 h ΛΦ 1 , Φ 1 i + 1 2 N X j =1 λ j µ j φ 2 1 j . (57) The mKdV hierarch y with self-consisten t sources is u t n = ∂ [ δ H n δ u + 2 N X j =1 δ λ j δ u ] ≡ ∂ [ − a n + N X j =1 φ 1 j φ 2 j ] , ( 5 8a) φ 1 j,x = − uφ 1 j + λ j φ 2 j , φ 2 j,x = λ j φ 1 j + uφ 2 j , j = 1 , 2 , · · · , N . (58b) When n = 2, (58 ) give s the mKdV equation with self-consisten t sources u t = u xxx 4 − 3 2 u 2 u x + N X j =1 ( φ 1 j φ 2 j ) x , (59a) φ 1 j x = λ j φ 2 j − uφ 1 j , φ 2 j x = λ j φ 1 j + uφ 2 j , j = 1 , · · · , N . (59b) Based o n (56), the integrable Roso c hatius deformation o f the mKdV equation with self- consisten t sources is give n by u t = u xx 4 − 3 2 u 2 u x + N X j =1 ( φ 1 j φ 2 j ) x , ( 6 0a) φ 1 j x = λ j φ 2 j − uφ 1 j , φ 2 j x = λ j φ 1 j + uφ 2 j + λ j µ j φ 3 1 j , j = 1 , · · · , N . (60b) 15 whic h has the zero-curv ature represen tation (6) with N (2) giv en b y N (2) =  − uλ 2 − u xx 4 + u 3 2 λ 3 − ( u 2 2 − u x 2 ) λ λ 3 − ( u 2 2 + u x 2 ) λ u λ 2 + u xx 4 − u 3 2  +     − N P j =1 φ 1 j φ 2 j 0 0 N P j =1 φ 1 j φ 2 j     + N X j =1 λ λ 2 − λ 2 j λφ 1 j φ 2 j − λ j φ 2 1 j λ j ( φ 2 2 j + µ j φ 2 1 j ) − λφ 1 j φ 2 j ! (61) In general the in tegrable Ro so c hatius deformation of the mKdV hierarch y with self- consisten t sources is give n by u t n = ∂ [ δ H n δ u + 2 N X j =1 δ λ j δ u ] = ∂ [ − a n + N X j =1 φ 1 j φ 2 j ] (62a) φ 1 j x = λ j φ 2 j − uφ 1 j , φ 2 j x = λ j φ 1 j + uφ 2 j + λ j µ j φ 3 1 j , j = 1 , · · · , N . (62b) whic h has the zero-curv ature represen tation (6) with N ( n ) giv en b y N ( n ) = n − 1 X i =1  a i λ b i c i − a i λ  λ 2 n − 2 i − 3 +     a n − N P j =1 φ 1 j φ 2 j 0 0 a n + N P j =1 φ 1 j φ 2 j     + N X j =1 λ λ 2 − λ 2 j λφ 1 j φ 2 j − λ j φ 2 1 j λ j ( φ 2 2 j + µ j φ 2 1 j ) − λφ 1 j φ 2 j ! . 5 Conclus ion Roso c hatius-type integrable systems hav e imp ortan t ph ysical a pplications. Ho w ev er, the studies on Ro so c hatius deformation are limited t o few finite dimensional in tegrable Hamil- tonian system(FDIHS) a t presen t. The main purp ose of this pap er is to pro p ose a system- atic metho d to generalize the Roso c hatius deformation for FDIHS to the Roso c hatius de- formation for infinite dimensional integrable equations. W e first construct infinite num b er of in tegrable Ro so c hatius deformations of FDIHSs obtained fro m higher-o rder constrained flo ws o f some soliton hierarchies . Then, based on the Roso c hatius deformed higher-order constrained flo ws, w e establish t he inte gra ble Roso c hatius deformat io ns of some soliton hierarc hies with self-consisten t sources. The Roso chatius deformations of t he KdV hi- erarc h y with self-consisten t sources, of the AKNS hierarc hy with self-consisten t sources and of the mKdV hierarc h y with self-consisten t sources, together with their Lax pairs are obtained. The a ppro ac h presen ted here can b e applied to other cases. 16 Ac kno wledg men t s This w ork is supp o rted by Nationa l Basic Researc h Program of China ( 9 73 Program) (2007CB81480 0) and Nationa l Natura l Science F oundation of China (10 6 71121). References [1] Roso c ha t ius E, 1877 dissertation, Unive rsity of G ¨ o tingen. [2] Neumann C, 185 9 J. Reine Angew. Math. 56 46. [3] W o jciec how ski S, 198 5 Ph ys. Scr. 31 433. [4] T ondo G , 1995 J. Ph ys. A 28 5097. [5] Kub o R, Og ura W, Saito T and Y asui Y, 1999 Ph ys. Lett. A 251 6 . 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