Symmetries for the Ablowitz-Ladik hierarchy: II. Integrable discrete nonlinear Schr"odinger equation and discrete AKNS hierarchy

Symmetries for the Ablo witz-Ladik hierarc h y: I I. In tegrable discrete nonlinear Sc hr¨ odinger equatio n and discrete AKNS hierarc h y Da-jun Zhang ∗ , Shou-ting Chen Dep artment of Mathematics, Shanghai University, Shanghai 200444 , P.R. China April 23, 2022 Abstract In the pap er we contin ue to consider sy mmetr ie s rela ted to the Ablowitz-Ladik hierarch y . W e derive symmetries for the integrable discrete nonlinear Schr¨ odinger hierarchy and discrete AKNS hierarch y . The integrable discrete nonlinear Sc hr¨ o dinger hiera rc hy ar e in sc alar for m and its tw o sets of symmetries are shown to fo r m a Lie alg ebra. W e also present discr ete AKNS isosp ectral flo ws , non-isosp ectral flows and their recurs ion op erator. In cont inuous limit these flo ws go to the con tinuous AKNS flows and the recursion oper ator go es to the square of the AK N S recursion op e rartor. These discrete AKNS flo ws form a Lie algebra which plays a key role in constructing symmetries and their algebraic structur es for both the int e grable discr ete nonlinear Schr¨ odinger hierarchy and discrete AKNS hierarch y . Structures of the obtained algebr as are different structur e s from those in contin uous cases which usually are cen ter less Kac-Mo o dy-Virasoro type. These algebra defor mations ar e explained through contin uous limit and de gr e e in terms of lattice spa cing parameter h . Key w ords : Ablowitz-Ladik hiera rc hies; symmetries; discrete nonlinear Sc hr ¨ odinger e q ua- tion; discr ete AKNS hierarch y; algebr a deformatio n. P A CS: 02.30.Ik , 05.45 .Yv 1 In tro duction In th e previous pap er [1], to w hic h w e refer as Pa rt I, we ha ve d eriv ed s ymmetries and their algebras for the isosp ectral and non-isosp ectral four-p oten tial Ab lo w it z-Ladik (AL) hierarc hies, and these results including symmetries of hiearac hies were sho wn to b e reduced to the t w o- p oten tial case. Among the equations related to the t wo-potenti al AL sp ec tral problem, one of the most physic ally meaningful systems is the integ rable discrete nonlinear Sc hr¨ odinger(IDNLS) equation [2–4]: iQ n,t = Q n +1 + Q n − 1 − 2 Q n + ε | Q n | 2 ( Q n +1 + Q n − 1 ) , (1.1) ∗ Corresponding auth o r. E-mail: d j zhang@staff.shu.edu.cn 1 where ε = ± 1 and i is the imaginary un it . This equation differs fr om the discrete NLS equation iQ n,t = Q n +1 + Q n − 1 − 2 Q n + 2 εQ n | Q n | 2 (1.2) b y the nonlinear term, wh ich is not in tegrable but arose in man y imp ortan t p h ysical con texts (cf. the int r oduction chapter of [4] and the referen c es therein). With regard to sym met ries, Refs. [5, 6] deriv ed p oin t and generalize d symmetries for the IDNLS equation (1.1). T w o su balg eb r as w ere obtained [5] and then a general structure f or the whole symm etry a lgebra w as described [6 ]. In the present pap er we hop e to go fu rther than Refs. [5, 6]. In fact, Eq.(1.1) consists of p ositi v e as w ell as negativ e ord er AL isosp ec tral flo ws, whic h corresp onds to a cen tral-difference discretization for the second order deriv ativ e in the con tinuous NLS equation. Since in Pa r t I and also in [7] we ha ve deriv ed algebraic r e lations for arbitrary t w o fl o ws among the p ositiv e and negativ e-order, isosp ectral and n on-iso sp ectral, hierarc hies r e lated to the t wo-potentia l AL sp ectral pr o blem, it is p ossible to ge t infinitely man y symmetries with a clear alge braic structure for the IDNLS equation (1.1) . Our plan is the follo wing. W e will start from the isosp ectral and n o n -iso sp ectral flo ws of the t wo- p oten tial AL system, wh ic h serve as basic flo w s . By suitable linear com binations w e can get isosp ectral and non-isosp ectral IDNLS flo ws separately (cf. [5, 8] w here a uniformed hierarch y w as giv en). Since the IDNLS equ a tion (1.1) is in scalar form, we then try to d e riv e algebraic structure of the scalar IDNLS flo ws . By the obtained s tr ucture w e can presen t infinitely many symmetries which form a Lie algebra with clear structur e . Going further we can get similar results for the IDNLS h ie rarc hy . As in the con tinuous AKNS system, the IDNLS flo ws can b e a redu c tion of some flo ws whic h we call th e discrete AKNS (DAKNS) flo ws in the p a p er. W e find these flo ws and their recursion op erato r corresp o nd to the contin uous AKNS flo w s and the square of the AKNS recursion op erator. In our p a p er the algebraic structure of the DAKNS flo ws will play a key role in d e riving symmetries and their algebras for the IDNLS h ierarc h y and the DAKNS hierarc hy . Finally , by comparison one can see the d ifferen ce b et w een the obtained algebras and their con tin uous coun terparts. These algebra deformations will also b e explained b y co nsidering co n tinuous limit and in tro ducing de g r e e of discrete elements. The pap er is organized as follo ws. S ec .2 con tains basic results of the t wo-potent ial AL system, including flows, recursion op erator and alge b raic structur e . In S ec .3 w e present isosp ect ral and non-isosp ectral IDNLS flo ws and their algebraic structures. Sec.4 derives symmetries and Lie algebras for the IDNLS equation and hierarch y . Sec.5 discusses the D AKNS flo w s and con tinuous limits. In Sec.6 alg ebra deformations from discrete case to conti n u ous case are listed out and explanation follo ws. Finally , Sec.7 giv es conclusions. 2 Flo ws related to the AL sp ectral p roble m Let us list out the m a in results of Ref. [7] as th e starting p oin t. W e w il l also follo w the notations used in [7] without an y confusion with Part I. The tw o-p ot en tial AL sp ectral problem (also called discrete Z a k h aro v-Shabat sp ectral problem) is Φ n +1 = M Φ n , M = λ Q n R n 1 λ ! , u n = Q n R n ! , Φ n = φ 1 ,n φ 2 ,n ! . (2.1) 2 The isosp ectral AL hierarc hy and non-isosp ectral AL hierarc hy are resp ectiv ely u n,t = K ( l ) = L l K (0) , K (0) = ( Q n , − R n ) T , l ∈ Z , (2.2a) u n,t = σ ( l ) = L l σ (0) , σ (0) = (2 n + 1)( Q n , − R n ) T , l ∈ Z , (2.2b) where L is the recursion op erato r defined as L =  E 0 0 E − 1  +  − Q n E R n  ( E − 1) − 1 ( R n E , Q n E − 1 ) + γ 2 n  − E Q n R n − 1  ( E − 1) − 1 ( R n , Q n ) 1 γ 2 n , (2 .3) γ n = √ 1 − Q n R n and E is a sh ift op erator defined as E j f ( n ) = f ( n + j ) , ∀ j ∈ Z . L is inv ertible with L − 1 =  E − 1 0 0 E  +  Q n − R n E  ( E − 1) − 1 ( R n E − 1 , Q n E ) + γ 2 n  Q n − 1 − E R n  ( E − 1) − 1 ( R n , Q n ) 1 γ 2 n . The isosp ectral flo ws { K ( l ) } and non-isosp ectral flows { σ ( l ) } form a centrel ess Kac-Mo ody- Virasoro (KMV) algebra, ∀ l, s ∈ Z , [ [ K ( l ) , K ( s ) ] ] = 0 , (2.4a) [ [ K ( l ) , σ ( s ) ] ] = 2 lK ( l + s ) , (2.4b) [ [ σ ( l ) , σ ( s ) ] ] = 2( l − s ) σ ( l + s ) , (2.4c) where the Lie pr oduct [ [ · , · ] ] is defi n ed as in P art I, i.e., [ [ f , g ] ] = f ′ [ g ] − g ′ [ f ] , (2.5) in wh ic h f = ( f 1 ( u n ) , f 2 ( u n )) T , g = ( g 1 ( u n ) , g 2 ( u n )) T , f ′ [ g ] is the Gateaux d eriv ativ e of f w.r.t u n in direction g , i.e., f ′ [ g ] = f ( u n ) ′ [ g ] = d dǫ    ǫ =0 f ( u n + ǫg ) , ǫ ∈ R , (2.6) and vice ve rse for g ′ [ f ]. (2.4) is a generaliz ation of the r e sults in [9]. 3 Isosp ectral and non-isosp ectral IDNLS hierarc hies 3.1 Hierarc hies T o deriv e IDNLS hierarchies, let us int r oduce auxiliary flo ws K (0) [0] = K (0) , (3.1a) K (0) [1] = 1 2 ( L − L − 1 ) K (0) = 1 2  (1 − Q n R n )( Q n +1 − Q n − 1 ) (1 − Q n R n )( R n +1 − R n − 1 )  , (3.1b) 3 σ (0) [0] = 1 2 σ (0) , (3.2a) σ (0) [1] = 1 4 ( L − L − 1 ) σ (0) = 1 4  (1 − Q n R n )[(2 n + 3) Q n +1 − (2 n − 1) Q n − 1 )] (1 − Q n R n )[(2 n + 3) R n +1 − (2 n − 1) R n − 1 )]  − 1 2  Q n − R n  ( E − 1) − 1 ( Q n +1 R n − Q n R n +1 ) , (3.2b) and op erator L = L − 2 I + L − 1 =  E + E − 1 − 2 0 0 E + E − 1 − 2  + γ 2 n  Q n − 1 − Q n +1 E R n − 1 − R n +1 E  ( E − 1) − 1 ( R n , Q n ) 1 γ 2 n +  Q n − R n E  ( E − 1) − 1 ( R n E − 1 , Q n E ) −  Q n E − R n  ( E − 1) − 1 ( R n E , Q n E − 1 ) , (3.3) where I is the 2 × 2 unit matrix. By ∗ we denote complex conjugate. Then we define e K (0) [0] = K (0) [0]    R n = − εQ ∗ n = ( Q n , εQ ∗ n ) T , (3.4a) e K (0) [1] = K (0) [1]    R n = − ǫQ ∗ n = 1 2  (1 + ε | Q n | 2 )( Q n +1 − Q n − 1 ) − ε (1 + ε | Q n | 2 )( Q ∗ n +1 − Q ∗ n − 1 )  , (3.4b) e σ (0) [0] = σ (0) [0]    R n = − εQ ∗ n =( n + 1 2 )( Q n , εQ ∗ n ) T , (3.5a) e σ (0) [1] = σ (0) [1]    R n = − εQ ∗ n = 1 4  (1 + ε | Q n | 2 )[(2 n + 3) Q n +1 − (2 n − 1) Q n − 1 )] − ε (1 + ε | Q n | 2 )[(2 n + 3) Q ∗ n +1 − (2 n − 1) Q ∗ n − 1 )]  + 1 2 ε  Q n εQ ∗ n  ( E − 1) − 1 ( Q n +1 Q ∗ n − Q n Q ∗ n +1 ) , (3.5b) the op erator e L = L| R n = − εQ ∗ n , (3.6) and furth e r define the flo ws e K ( l ) [ j ] = e L l e K (0) [ j ] , j ∈ { 0 , 1 } , (3.7a) e σ ( s ) [ j ] = e L s e σ (0) [ j ] , j ∈ { 0 , 1 } . (3.7b) T o mak e clear the relationship of t w o co mp onen ts in e K ( l ) [ j ] and e σ ( s ) [ j ] , w e in tro duce function sets A [ j ] ( ε ) = { ( f 1 , ( − 1) j εf ∗ 1 ) T } , j ∈ { 0 , 1 } , where f 1 is an arbitrary s c alar function. Then one can fi nd that e K (0) [ j ] , e σ (0) [ j ] ∈ A [ j ] ( ε ) , (3.8) 4 and e L provi des a self-transformation f o r b oth A [0] ( ε ) and A [1] ( ε ), i.e., ∀ α ∈ A [0] ( ε ), e L α ∈ A [0] ( ε ), and ∀ β ∈ A [1] ( ε ), e L β ∈ A [1] ( ε ). That means e K ( l ) [ j ] , e σ ( s ) [ j ] ∈ A [ j ] ( ε ) . (3.9) Noting that e u n = u n | R n = − εQ ∗ n = ( Q n , − εQ ∗ n ) T ∈ A [1] ( ε ), we then define the follo wing isosp ec- tral IDNLS hierarch y i 1 − j e u n,t m,j = e K ( m ) [ j ] = e L m e K (0) [ j ] , m = 0 , 1 , 2 , · · · , (3.10a ) and non-isosp ectral IDNLS hierarc h y i 1 − j e u n,t s,j = e σ ( s ) [ j ] = e L s e σ (0) [ j ] , s = 0 , 1 , 2 , · · · , (3.10b) where j ∈ { 0 , 1 } , e L is the r e cursion op erator, and we add subind exe s m, j and s, j for t as lab els of equ a tions in their hierarch y . In fact, the IDNLS equation (1.1) comes from (3.10a) with j = 0 , m = 1, i.e. , i e u n,t 1 , 0 = i  Q n − εQ ∗ n  t 1 , 0 = e K (1) [0] =  Q n +1 + Q n − 1 − 2 Q n + εQ n Q ∗ n ( Q n +1 + Q n − 1 ) εQ ∗ n +1 + εQ ∗ n − 1 − 2 εQ ∗ n + Q n Q ∗ n ( Q ∗ n +1 + Q ∗ n − 1 )  . (3.11) In non-isosp ectral case, (3.10b) with j = 0 and s = 1 reads i  Q n − εQ ∗ n  t 1 , 0 = e σ (1) [0] = 1 2  (1 + εQ n Q ∗ n )[(2 n + 3) Q n +1 + (2 n − 1) Q n − 1 ] − 2(2 n + 1) Q n ε (1 + εQ n Q ∗ n )[(2 n + 3) Q ∗ n +1 + (2 n − 1) Q ∗ n − 1 ] − 2 ε (2 n + 1) Q ∗ n  + ε  Q n εQ ∗ n  ( E − 1) − 1 ( Q n +1 Q ∗ n + Q n Q ∗ n +1 ) , (3.12) of which th e first r o w pro vides a non-isosp ect r al IDNLS equation iQ n,t 1 , 0 = 1 2 (1 + εQ n Q ∗ n )[(2 n + 3) Q n +1 + (2 n − 1) Q n − 1 ] − (2 n + 1) Q n + εQ n ( E − 1) − 1 ( Q n +1 Q ∗ n + Q n Q ∗ n +1 ) , (3.13) whic h goes to a non-isosp ectral NLS equation in con tinuous limit (see S ec .5). The flo w s { i j − 1 e K ( l ) [ j ] } and { i j − 1 e σ ( s ) [ j ] } defined in (3.7) are called isosp ectral and non-isosp ectral IDNLS flo ws in vecto r form, resp ectiv ely . W e add multiplier i j − 1 so th at they are alwa ys in the set A [1] ( ε ) to which e u n b elongs. The fi rst comp onen ts of { i j − 1 e K ( l ) [ j ] } and { i j − 1 e σ ( s ) [ j ] } , whic h w e resp ectiv ely den ote them by { i j − 1 e K ( l ) [ j ] , 1 } and { i j − 1 e σ ( s ) [ j ] , 1 } , are called isosp ectral and non- isosp ectral IDNLS flo ws in scalar form. Thus the scalar form of isospectral and non-isosp ect r al IDNLS hierarc h ie s can b e written as i 1 − j Q n,t m,j = e K ( m ) [ j ] , 1 , m = 0 , 1 , 2 , · · · , (3.14a ) i 1 − j Q n,t s,j = e σ ( s ) [ j ] , 1 , s = 0 , 1 , 2 , · · · . (3.14b) 5 3.2 Algeb raic struct ures of the IN D LS flows Making use of (2.4 ) one can deriv e the algebraic relatio n s for the IDNLS flows. Theorem 3.1. Supp ose that Q n is the only indep endent variable and the Gate aux derivative i s define d w.r.t. Q n . Then the sc alar i sosp e ctr al and non-isosp e ctr al IDNLS flows { i j − 1 e K ( m ) [ j ] , 1 } and { i j − 1 e σ ( s ) [ j ] , 1 } form a Lie algebr a thr ough the Lie pr o duct [ [ · , · ] ] Q n with the fol lowing structur e [ [ i j − 1 e K ( m ) [ j ] , 1 , i k − 1 e K ( s ) [ k ] , 1 ] ] Q n = 0 , (3.15a ) [ [ − i e K ( m ) [0] , 1 , − i e σ ( s ) [0] , 1 ] ] Q n = − 2 m e K ( m + s − 1) [1] , 1 , (3.15b) [ [ − i e K ( m ) [0] , 1 , e σ ( s ) [1] , 1 ] ] Q n = − 1 2 im ( e K ( m + s +1) [0] , 1 + 4 e K ( m + s ) [0] , 1 ) , (3.15 c) [ [ e K ( m ) [1] , 1 , i j − 1 e σ ( s ) [ j ] , 1 ] ] Q n = 1 2 i j − 1 [( m + 1) e K ( m + s +1) [ j ] , 1 + 2(2 m + 1) e K ( m + s ) [ j ] , 1 ] , (3.15d) [ [ − i e σ ( m ) [0] , 1 , − i e σ ( s ) [0] , 1 ] ] Q n = − 2( m − s ) e σ ( m + s − 1) [1] , 1 , (3.15e ) [ [ i j − 1 e σ ( m ) [ j ] , 1 , e σ ( s ) [1] , 1 ] ] Q n = 1 2 i j − 1 [( m − s − 1 + j ) e σ ( m + s +1) [ j ] , 1 + 2(2 m − 2 s − 1 + j ) e σ ( m + s ) [ j ] , 1 ] , (3.15f ) wher e j, k ∈ { 0 , 1 } , m, s ≥ 0 and we set e K ( − 1) [ j ] , 1 = e σ ( − 1) [ j ] , 1 = 0 onc e they app e ar on the r.h.s. of (3.15) . We note that her e after by [ [ · , · ] ] Q n we denote the pr o duct define d thr ough the Gate aux derivative w.r.t. Q n . W e pro ve the theorem by t wo steps. First, we deriv e algebraic structures for the follo wing v ector flo ws K ( l ) [ j ] = L l K (0) [ j ] , σ ( l ) [ j ] = L l σ (0) [ j ] , j ∈ { 0 , 1 } , l = 0 , 1 , · · · , (3.16) where K (0) [ j ] and σ (0) [ j ] are giv en in (3.1 ) and (3.2 ), resp ectiv ely . { K ( l ) [ j ] } and { σ ( l ) [ j ] } are called (semi-)D AKNS flo ws (see Sec.5.1) 1 . F or these flows we ha ve Lemma 3.1. The flows { K ( m ) [ j ] } and { σ ( s ) [ j ] } form a Lie algebr a, denote d by D , thr ough [ [ · , · ] ] with structur e [ [ K ( m ) [ j ] , K ( s ) [ k ] ] ] = 0 , (3.17a ) [ [ K ( m ) [0] , σ ( s ) [0] ] ] = 2 mK ( m + s − 1) [1] , (3.17b) [ [ K ( m ) [0] , σ ( s ) [1] ] ] = 1 2 m ( K ( m + s +1) [0] + 4 K ( m + s ) [0] ) , (3.17c ) [ [ K ( m ) [1] , σ ( s ) [ j ] ] ] = 1 2 [( m + 1) K ( m + s +1) [ j ] + 2(2 m + 1) K ( m + s ) [ j ] ] , (3.17d) [ [ σ ( m ) [0] , σ ( s ) [0] ] ] = 2( m − s ) σ ( m + s − 1) [1] , (3.17e ) [ [ σ ( m ) [ j ] , σ ( s ) [1] ] ] = 1 2 [( m − s − 1 + j ) σ ( m + s +1) [ j ] + 2(2 m − 2 s − 1 + j ) σ ( m + s ) [ j ] ] , (3.17f ) 1 In contin uous limit one can find K (0) [0] ∼ ( q , − r ) T , K (0) [1] ∼ ( q x , r x ) T , σ (0) [0] ∼ ( xq , − xr ) T , σ (0) [1] ∼ ( q + xq x , r + xr x ) T and L ∼ L 2 AK N S where L AK N S is the recursion op erator of the AKNS system. 6 wher e j, k ∈ { 0 , 1 } , the Gate aux derivative is stil l define d w.r.t. u n , m, s ≥ 0 and we set K ( − 1) [ j ] = σ ( − 1) [ j ] = 0 onc e they app e ar on the r.h.s. of (3.17) . Pr o of. W e only p ro v e (3.17c) . Th e others ca n b e pro ved similarly . Noting that L m = ( L − 2 I + L − 1 ) m = m X r =0 m − r X j =0 C r m C j m − r ( − 2) j L m − 2 r − j , (3.18) and by this we write K ( m ) [0] and σ ( s ) [1] as K ( m ) [0] = m X r =0 m − r X j =0 C r m C j m − r ( − 2) j K ( m − 2 r − j ) , σ ( s ) [1] = 1 4 s X h =0 s − h X k =0 C h s C k s − h ( − 2) k ( σ ( s − 2 h − k +1) − σ ( s − 2 h − k − 1) ) . Substituting them into [ [ K ( m ) [0] , σ ( s ) [1] ] ] and making u s ing of the Lie pro duct relation (2 .4b) yield [ [ K ( m ) [0] , σ ( s ) [1] ] ] = A + B + C, (3.19) where A = 1 2 m ( L − L − 1 ) m X r =0 m − r X j =0 C r m C j m − r ( − 2) j s X h =0 s − h X k =0 C h s C k s − h ( − 2) k K ( m + s − 2 r − 2 h − j − k ) , B = − ( L − L − 1 ) m X r =0 m − r X j =0 r C r m C j m − r ( − 2) j s X h =0 s − h X k =0 C h s C k s − h ( − 2) k K ( m + s − 2 r − 2 h − j − k ) , C = − 1 2 ( L − L − 1 ) m X r =0 m − r X j =0 j C r m C j m − r ( − 2) j s X h =0 s − h X k =0 C h s C k s − h ( − 2) k K ( m + s − 2 r − 2 h − j − k ) . Next, still u sing (3.18), the first term A is n ot h ing but A = 1 2 m ( L − L − 1 ) L m L s K (0) = 1 2 m ( L − L − 1 ) K ( m + s ) [0] . F or the second term B w h ere th e s ummatio n for r essentiall y starts from r = 1, again u s ing (3.18) we hav e B = − ( L − L − 1 ) L s m X r =1 r C r m ( L − 2 I ) m − r L − r K (0) . It then follo w s from the formula r C r m = mC r − 1 m − 1 that B = − ( L − L − 1 ) L s m − 1 X r =0 mC r m − 1 ( L − 2 I ) m − 1 − r L − r − 1 K (0) = − m ( L − L − 1 ) L − 1 L s L m − 1 K (0) = − m ( L − L − 1 ) L − 1 K ( m + s − 1) [0] . 7 Similarly , for the last term C , u s ing j C r m C j m − r = ( m − r ) C r m C j − 1 m − r − 1 = mC r m − 1 C j − 1 m − r − 1 w e ha v e C = m ( L − L − 1 ) K ( m + s − 1) [0] . Then w e ha v e [ [ K ( m ) [0] , σ ( s ) [1] ] ] = A + B + C = 1 2 m ( L − L − 1 )( L − 2 L − 1 + 2 I ) K ( m + s − 1) [0] = 1 2 m ( L − L − 1 ) 2 K ( m + s − 1) [0] = 1 2 m ( L 2 + 4 L ) K ( m + s − 1) [0] = 1 2 m ( K ( m + s +1) [0] + 4 K ( m + s ) [0] ) , whic h is just (3.17c). The other relations in (3.17) can b e pro ved in a similar w ay . It is easy to fi nd that the Lie algebra D is generated by the follo wing eleme n ts { K (0) [0] , K (0) [1] ( or K (1) [0] ) , σ (0) [0] , σ (0) [1] ( or σ (1) [0] ) , σ (1) [1] } . (3.20) The second step consists of a d iscu ssio n for the consistency of the reduction of (3.17) und er R n = − εQ ∗ n . Let us first consider 2-dimens io n a l vecto r fun ct ions : f ( u n ) = ( f 1 , f 2 ) T , g ( u n ) = ( g 1 , g 2 ) T , h ( u n ) = ( h 1 , h 2 ) T , (3.21) whic h are related b y [ [ f , g ] ] = h. (3.22) W e note that if Q n and R n are t wo in depend en t v ariables and there is no complex op eration of Q n and R n in f , g , then the linear relati onship holds on th e complex n u m b er fi eld C , i.e., f ( u n ) ′ [ ag ] = af ( u n ) ′ [ g ] , ∀ a ∈ C . (3.23) Ho we v er, when R n = − εQ ∗ n and Q n is considered to b e the only one indep endent v ariable, in general the ab o ve linear relationship do es not hold any longer unless a is real. 2 F or a reasonable reduction for (3.17) the problem w e ha v e to conquer is the ‘consistency’ of the pro duct (3.22): • Firs t, a co n siste n t reduction requ ir e s tw o comp onen ts are someho w related after reduction, for example, h 2 = − εh ∗ 1 . 2 F or example, f 1 = Q 2 n + Q ∗ n , g 1 = Q n,x , f 1 ( Q n ) ′ [ g 1 ] = 2 Q n Q n,x + Q ∗ n,x but f 1 ( Q n ) ′ [ ig 1 ] = i (2 Q n Q n,x − Q ∗ n,x ) 6 = if 1 ( Q n ) ′ [ g 1 ]. 8 • S ec on d , when Q n b ecomes the only one indep enden t v ariable in stead of ( Q n , R n ), a consisten t reduction for the pr oduct (3.22) | R n = − εQ ∗ n should pr o v id e [ [ f 1 , g 1 ] ] Q n = f 1 ( Q n ) ′ [ g 1 ] − g 1 ( Q n ) ′ [ f 1 ] = h 1 . (3.24) Suc h consistency for the pro duct (3.22) can b e guaran teed b y taking f ( e u n ) , g ( e u n ) and e u n are in the same fun ction set, i.e., f ( e u n ) , g ( e u n ) ∈ A [1] ( ε ) , (3.25) same as e u n . After the ab o ve d isc ussion for the consistency of reduction, firs t, we m ultiply K ( m ) [ j ] and σ ( s ) [ j ] on the l.h.s. of (3.17) by i j − 1 whic h just guaran tees { i j − 1 e K ( l ) [ j ] } and { i j − 1 e σ ( s ) [ j ] } are in the set A [1] ( ε ) to wh ic h e u n b elongs. Then w e take the redu ction R n = − εQ ∗ n and follo wing (3.24) we get th e algebraic relatio n s for those first comp onent s, whic h are listed in Theorem 3.1. T aking (3.17c) as an example, w e fir s t multiply K ( m ) [0] b y − i and then rewrite (3.17c) to [ [ − iK ( m ) [0] , σ ( s ) [1] ] ] = − 1 2 im ( K ( m + s +1) [0] + 4 K ( m + s ) [0] ) . (3.26) This is then r eady f o r a consisten t reduction and after taking R n = − εQ ∗ n w e get (3.1 5c ). 4 Symmetries 4.1 S ymmetries for the ID NLS equation With the algebraic relations (3.15) in hand , w e can constru ct symmetries for the IDNLS equation (1.1), i.e., iQ n,t 1 , 0 = e K (1) [0] , 1 . A scalar fun ct ion τ = τ ( Q n ) is a sym m e try of (1.1), if τ t 1 , 0 = − i e K (1) [0] , 1 ( Q n ) ′ [ τ ] , (4.1) whic h is, equiv alen tly , ˜ ∂ τ ˜ ∂ t 1 , 0 = [ [ − i e K (1) [0] , 1 , τ ] ] Q n , (4.2) where ˜ ∂ τ ˜ ∂ t 1 , 0 sp eciall y denotes the d e riv ativ e of τ w.r.t. t 1 , 0 explicitly included in τ (cf. [1]), and the Gateaux deriv ativ e in (4.1) is d e fined w.r.t. Q n . F rom the algebraic s t ructures in Theorem 3.1 and the definition (4.2 ) w e ha ve th e follo wing symmetries for the IDNLS equatio n (1.1): K -symmetries { i j − 1 e K ( m ) [ j ] , 1 } and τ -symmetries τ (1 ,s ) [0 ,j ] = t 1 , 0 · [ [ − i e K (1) [0] , 1 , i j − 1 e σ ( s ) [ j ] , 1 ] ] Q n + i j − 1 e σ ( s ) [ j ] , 1 , s = 0 , 1 , · · · , (4.3 ) i.e., τ (1 ,s ) [0 , 0] = − 2 t 1 , 0 e K ( s ) [1] , 1 − i e σ ( s ) [0] , 1 , (4.4a) τ (1 ,s ) [0 , 1] = − 1 2 it 1 , 0 ( e K ( s +2) [0] , 1 + 4 e K ( s +1) [0] , 1 ) + e σ ( s ) [1] , 1 . (4.4b) 9 The algebraic relations in (3.15) su g gest an algebra for the symm e tries of the IDNLS equ a tion (1.1). This is concluded by Theorem 4.1. The isosp e ctr al ID NLS e quation (1.1) c an have two sets of symmetries, K - symmetries { i l − 1 e K ( m ) [ l ] , 1 } and τ -symmetries τ (1 ,s ) [0 ,l ] given in (4.4) , which form a Lie algebr a with structur e [ [ i z − 1 e K ( m ) [ z ] , 1 , i l − 1 e K ( s ) [ l ] , 1 ] ] Q n = 0 , (4.5a) [ [ − i e K ( m ) [0] , 1 , τ (1 ,s ) [0 , 0] ] ] Q n = − 2 m e K ( m + s − 1) [1] , 1 , (4.5b) [ [ − i e K ( m ) [0] , 1 , τ (1 ,s ) [0 , 1] ] ] Q n = − 1 2 im ( e K ( m + s +1) [0] , 1 + 4 e K ( m + s ) [0] , 1 ) , (4.5c) [ [ e K ( m ) [1] , 1 , τ (1 ,s ) [0 ,l ] ] ] Q n = 1 2 i l − 1 [( m + 1) e K ( m + s +1) [ l ] , 1 + 2(2 m + 1) e K ( m + s ) [ l ] , 1 ] , (4.5d) [ [ τ (1 ,m ) [0 , 0] , τ (1 ,s ) [0 , 0] ] ] Q n = − 2( m − s ) τ (1 ,m + s − 1) [0 , 1] , (4.5e) [ [ τ (1 ,m ) [0 ,l ] , τ (1 ,s ) [0 , 1] ] ] Q n = 1 2 [( m − s − 1 + l ) τ (1 ,m + s +1) [0 ,l ] + 2(2 m − 2 s − 1 + l ) τ (1 ,m + s ) [0 ,l ] ] , (4.5f ) wher e z , l ∈ { 0 , 1 } , m, s ≥ 0 and we set e K ( − 1) [ l ] , 1 = τ (1 , − 1) [0 ,l ] = 0 onc e they app e ar on the r.h.s. of (4.5) . 4.2 S ymmetries for the isosp ectral ID NLS hierarch y The algebraic structures in (3.15) also enable us to get t wo sets o f symmetries of the isosp ec- tral IDNLS hierarch y (3. 14a). F or this we ha ve the follo wing theorem. Theorem 4.2. Each e quation i 1 − j Q n,t k,j = e K ( k ) [ j ] , 1 in the isosp e ctr al ID N LS hier ar chy (3.14a) has two sets of symmetries. W h en j = 0 these symmetries ar e K -symmetries: { i l − 1 e K ( m ) [ l ] , 1 } , l ∈ { 0 , 1 } , (4.6a) τ - symm e tr i es: τ ( k, s ) [0 , 0] = − 2 k t k , 0 e K ( k + s − 1) [1] , 1 − i e σ ( s ) [0] , 1 , (4.6b) τ ( k, s ) [0 , 1] = − 1 2 ik t k , 0 ( e K ( k + s +1) [0] , 1 + 4 e K ( k + s ) [0] , 1 ) + e σ ( s ) [1] , 1 ; (4.6 c) and when j = 1 the symmetries ar e K -symmetries: { i l − 1 e K ( m ) [ l ] , 1 } , l ∈ { 0 , 1 } , (4.7a) τ - symm e tr i es: τ ( k, s ) [1 , 0] = − 1 2 i ( k + 1) t k , 1 e K ( k + s +1) [0] , 1 − i (2 k + 1) t k , 1 e K ( k + s ) [0] , 1 − i e σ ( s ) [0] , 1 , (4.7b) τ ( k, s ) [1 , 1] = 1 2 ( k + 1) t k , 1 e K ( k + s +1) [1] , 1 + (2 k + 1) t k , 1 e K ( k + s ) [1] , 1 + e σ ( s ) [1] , 1 . (4.7c) 10 Symmetries for e ach e quation c an f orm a Lie algebr a and structur es ar e describ e d as [ [ i z − 1 e K ( m ) [ z ] , 1 , i l − 1 e K ( s ) [ l ] , 1 ] ] Q n = 0 , (4.8a) [ [ − i e K ( m ) [0] , 1 , τ ( k, s ) [ j, 0] ] ] Q n = − 2 m e K ( m + s − 1) [1] , 1 , (4.8b) [ [ − i e K ( m ) [0] , 1 , τ ( k, s ) [ j, 1] ] ] Q n = − 1 2 im ( e K ( m + s +1) [0] , 1 + 4 e K ( m + s ) [0] , 1 ) , (4.8c) [ [ e K ( m ) [1] , 1 , τ ( k, s ) [ j,l ] ] ] Q n = 1 2 i l − 1 [( m + 1) e K ( m + s +1) [ l ] , 1 + 2(2 m + 1) e K ( m + s ) [ l ] , 1 ] , (4.8d) [ [ τ ( k, m ) [ j, 0] , τ ( k, s ) [ j, 0] ] ] Q n = − 2( m − s ) τ ( k, m + s − 1) [ j, 1] , (4.8e) [ [ τ ( k, m ) [ j,l ] , τ ( k, s ) [ j, 1] ] ] Q n = 1 2 [( m − s − 1 + l ) τ ( k, m + s +1) [ j,l ] + 2(2 m − 2 s − 1 + l ) τ ( k, m + s ) [ j,l ] ] , (4.8f ) wher e j, z , l ∈ { 0 , 1 } , k , m, s ≥ 0 and we set e K ( − 1) [ l ] , 1 = τ ( k, − 1) [ j,l ] = 0 onc e they app e ar on the r.h.s. of (4.8) . E sp e cial ly, when j = 0 , k = 1 the ab ove r esults r e duc e to The or em 4.1. 4.3 Rela tions b et ween flo ws and the recursion op erator L Theorem 4.3. The flow s { K ( m ) [ j ] } and { σ ( m ) [ j ] } and their r e cursion op er ator L satisfy L ′ [ K ( m ) [ j ] ] − [ K ( m ) ′ [ j ] , L ] = 0 , j ∈ { 0 , 1 } , (4.9a) L ′ [ σ ( m ) [0] ] − [ σ ( m ) ′ [0] , L ] − L m ( L − L − 1 ) = 0 , (4.9b) L ′ [ σ ( m ) [1] ] − [ σ ( m ) ′ [1] , L ] − 1 2 L m +2 − 2 L m +1 = 0 , (4.9c) wher e m = 0 , 1 , 2 , · · · . Pr o of. W e only pro ve (4.9c). The other t wo can b e pr o ved similarly . W e start from the relation L ′ [ σ ( m ) ] − [ σ ( m ) ′ , L ] − 2 L m +1 = 0 , (4.10a ) ( L − 1 ) ′ [ σ ( m ) ] − [ σ ( m ) ′ , L − 1 ] + 2 L m − 1 = 0 , (4.1 0b ) in wh ic h (4.1 0a ) w as given in Ref. [7 ] and (4.10b) can b e prov ed similarly but we h ere skip the pro of. W e can express L ′ [ σ ( m ) [1] ] − [ σ ( m ) ′ [1] , L ] in terms of L, L − 1 and σ ( s ) and then making use of the ab o ve r e lation we find L ′ [ σ ( m ) [1] ] − [ σ ( m ) ′ [1] , L ] = 1 2 m X h =0 m − h X k =0 C h m C k m − h ( − 2) k ( L m − 2 h − k +2 − L m − 2 h − k − L m − 2 h − k + ( L − 1 ) m − 2 h − k − 2 ) = 1 2 ( L 2 L m − L m − L m + L − 2 L m ) = 1 2 L m +2 + 2 L m +1 . Th us w e complete the pro of. 11 W e note that these relations (4.9) can b e emplo ye d to pro ve the Lemma 3.1 if we use ind uctiv e approac h (cf. [10, 11] for cont inuous cases). 5 Con tin uous limi t 5.1 D AKNS flo ws In Sec.3.2 w e in tro duced fl o w s { K ( l ) [ j ] } and { σ ( l ) [ j ] } giv en by (3.1 6 ), whic h were r eferr ed to as D AKNS flo w s. In fact, in con tinuous limit these flo ws just go to the con tinuous AKNS isosp ectral and non-isosp ectral flo ws . Let us consider th e f ollo wing limit (cf. [4]): • r epla cing Q n and R n with hq n and hr n , wh er e h is the r eal step parameter (the lattice spacing), • n → ∞ , h → 0 suc h that nh fin ite , • introd ucing contin uous v ariable x = x 0 + nh , then for a scala r function, for example, q n , one has q n + j = q ( x + j h ) . F or conv enience we tak e x 0 = 0. F ollo w ing the abov e limit pr ocedure, one can fi nd that K (0) [0] →  q − r  (5.1) and the leading term is of O ( h ) ; K (0) [1] →  q r  x (5.2) and the leading term is of O ( h 2 ). Besides, for an y giv en scalar fun c tions f 1 ,n and f 2 ,n , applying the abov e contin uous limit on the op e rator L (3.3) an d defining integrate op erator ∂ − 1 x ∼ h ( E − 1) − 1 , one can fi n d that th e leading term is of O ( h 2 ), which give s L  f 1 ,n f 2 ,n  →  ∂ 2 x − 4 q r − 2  q x r x  ∂ − 1 x ( r , q ) − 2  q − r  ∂ − 1 x ( − r x , q x )   f 1 ( x ) f 2 ( x )  = L 2 AK N S  f 1 ( x ) f 2 ( x )  , (5.3) where L AK N S is the well kno wn recur sio n op erato r of the AKNS system, defin ed as (cf. [10 , 12]) L AK N S = − σ 3 ∂ x + 2 σ 3  q r  ∂ − 1 x ( r , q ) , (5.4 ) in which σ 3 is the Pauli matrix  − 1 0 0 1  . T his result means in con tinuous limit L go es to th e square of the AK N S r e cursion op erator. Th e n, noting that (5.1) and (5. 2) are n othin g b ut the first t w o flo ws in the AK NS hierarc hy (cf. [10, 13]), no w it is clear that in con tin u o us limit K ( m ) [0] → K (2 m ) AK N S , leading term O ( h 2 m +1 ) , (5.5a) K ( m ) [1] → K (2 m +1) AK N S , leading term O ( h 2 m +2 ) , (5.5b) 12 where { K ( s ) AK N S } are the hierarch y of the AKNS isosp ect ral flows. After similar d iscussions, for the n on-iso sp ectral flo ws { σ ( m ) [ j ] } we h a ve σ (0) [0] →  xq − xr  , leading term O (1) , (5.6a) σ (0) [1] →  xq xr  x , leading term O ( h ) , (5.6b) whic h are the first t wo AKNS non-isosp ectral flows (cf. [14]), and σ ( m ) [0] → σ (2 m ) AK N S , leading term O ( h 2 m ) , (5.7a) σ ( m ) [1] → σ (2 m +1) AK N S , leading term O ( h 2 m +1 ) , (5. 7b) where { σ ( s ) AK N S } are the hierarch y of the AKNS non-isosp ectral flo ws. 5.2 Isospectral AKNS hierarch y and NLS hierarc hy Based on the ab o ve discussion on con tinuous limit, w e can defin e the D AKNS hierarc hy as u n,t m,j = K ( m ) [ j ] , j ∈ { 0 , 1 } , m = 0 , 1 , · · · , (5.8) whic h is an isosp ectral evo lu ti on equation hierarch y . Consider cont inuous limit of the ab o ve hierarc hy . The dominated terms on b ot h sid e s should ha v e same order in terms of h . Since w e ha ve replaced u n b y h · ( q , r ) T , w e still need to replace t m,j b y t 2 m + j · h − (2 m + j ) , i.e., t m,j → t 2 m + j · h − (2 m + j ) , (5.9) so th at the left side of (5.8) is h 2 m +1+ j · U t 2 m + j , i.e., of O ( h 2 m +1+ j ), wh ere U = ( q , r ) T . Thus in con tinuous limit the D AKNS hierarc hy (5.8) go es to the ANKS isosp ectral ev olution equation hierarc hy U t 2 m + j = K (2 m + j ) AK N S , j ∈ { 0 , 1 } , m = 0 , 1 , · · · . ( 5.10) Then we defin e e K (2 m + j ) AK N S = K (2 m + j ) AK N S | r = − εq ∗ , j ∈ { 0 , 1 } , m = 0 , 1 , · · · . (5.11) Its first elemen t e K (2 m + j ) AK N S, 1 generates the NLS h ie r a rc hy i 1 − j q t 2 m + j = e K (2 m + j ) AK N S, 1 , j ∈ { 0 , 1 } , m = 0 , 1 , · · · , (5.12) whic h is just the con tinuous limit of the I DNLS hierarc h y (3.14 a). 13 5.3 S ymmetries Using the algebra D defined b y (3.17) one can construct t wo sets of s y m met ries for any equation u n,t l,j = K ( l ) [ j ] (5.13) in the DAKNS h ie r arc h y (5.8). When j = 0 these sym m e tries are K -symmetries: K ( m ) [ k ] , k ∈ { 0 , 1 } , (5.14a ) τ -sym met ries: T ( l,s ) [0 , 0] = 2 lt l, 0 K ( l + s − 1) [1] + σ ( s ) [0] , (5.14b) T ( l,s ) [0 , 1] = 1 2 lt l, 0 K ( l + s +1) [0] + 2 lt l, 0 K ( l + s ) [0] + σ ( s ) [1] ; (5.14c ) and when j = 1 the symmetries are K -symmetries: K ( m ) [ k ] , k ∈ { 0 , 1 } , (5.15a ) τ -sym met ries: T ( l,s ) [1 , 0] = 1 2 ( l + 1) t l, 1 K ( l + s +1) [0] + (2 l + 1) t l, 1 K ( l + s ) [0] + σ ( s ) [0] , (5.15b) T ( l,s ) [1 , 1] = 1 2 ( l + 1) t l, 1 K ( k + s +1) [1] + (2 l + 1) t l, 1 K ( l + s ) [1] + σ ( s ) [1] . (5.1 5c) The symmetries for (5.1 3 ) form a Lie algebra with s t ructure [ [ K ( m ) [ z ] , K ( s ) [ k ] ] ] = 0 , (5.16a ) [ [ K ( m ) [0] , T ( l,s ) [ j, 0] ] ] = 2 mK ( m + s − 1) [1] , (5.16b) [ [ K ( m ) [0] , T ( l,s ) [ j, 1] ] ] = 1 2 m ( K ( m + s +1) [0] + 4 K ( m + s ) [0] ) , (5.16c) [ [ K ( m ) [1] , T ( l,s ) [ j,k ] ] ] = 1 2 ( m + 1) K ( m + s +1) [ k ] + (2 m + 1) K ( m + s ) [ k ] , (5.16d) [ [ T ( l,m ) [ j, 0] , T ( l,s ) [ j, 0] ] ] = 2( m − s ) T ( l,m + s − 1) [ j, 1] , (5.16e ) [ [ T ( l,m ) [ j,k ] , T ( l,s ) [ j, 1] ] ] = 1 2 ( m − s − 1 + k ) T ( l,m + s +1) [ j,k ] + (2 m − 2 s − 1 + k ) T ( l,m + s ) [ j,k ] , (5.16f ) where j, k , z ∈ { 0 , 1 } , l, m, s ≥ 0 and we set K ( − 1) [ z ] = T ( l, − 1) [ j,k ] = 0 once they app ear on the r.h.s. of (5.16). F or the con tin uou s flo w s { K ( m ) AK N S } and { σ ( s ) AK N S } , it has b een kno wn that (cf. [10, 15 –1 7 ]) they form an algebra as w ell. W e denote th is alg ebr a b y C . Its s t ructure is [ [ K ( m ) AK N S , K ( s ) AK N S ] ] = 0 , (5.17a ) [ [ K ( m ) AK N S , σ ( s ) AK N S ] ] = mK ( m + s − 1) AK N S , (5.17b) [ [ σ ( m ) AK N S , σ ( s ) AK N S ] ] = ( m − s ) σ ( m + s − 1) AK N S , (5.17c ) where m, s ≥ 0 and we set K ( − 1) AK N S = σ ( − 1) AK N S = 0 once they app ear on the r.h .s. of (5.1 7). This algebra can b e generated by { K (1) AK N S , σ (0) AK N S , σ (3) AK N S } . (5.18) 14 F rom C one can ha ve t wo sets of symmetries for eac h AKNS equation U t l = K ( l ) AK N S (5.19) in the hierarch y (5.1 0), the sym met ries are K -symmetries: K ( m ) AK N S , (5.20a ) τ -sym met ries: T ( l,s ) AK N S = lt l K ( l + s − 1) AK N S + σ ( s ) AK N S , (5.20b) whic h form a Lie algebra with (cf. [10, 15–17]) [ [ K ( m ) AK N S , K ( s ) AK N S ] ] = 0 , (5.21a ) [ [ K ( m ) AK N S , T ( l,s ) AK N S ] ] = mK ( m + s − 1) AK N S , (5.21b) [ [ T ( l,m ) AK N S , T ( l,s ) AK N S ] ] = ( m − s ) T ( l,m + s − 1) AK N S , (5.21c ) where l , m, s ≥ 0 and we set K ( − 1) AK N S = T ( l, − 1) AK N S = 0 once they app ear on the r.h.s. of (5.21). Staring from the relations (5.17) and emplo ying similar discussions as w e ha ve done for the IDNLS hierarc hy in Sec.4, one can ha ve t wo sets of sym m e tries f o r the l th equ ation in the NLS hierarc hy (5.12) : q t l = µ l e K ( l ) AK N S, 1 , µ l =  − i, l is ev en , 1 , l is o d d . (5.22) The symmetries are (cf. [10]) K -symmetries: µ m e K ( m ) AK N S, 1 , ( 5.23a) τ -sym met ries: e T ( l,s ) = µ l µ s lt l e K ( l + s − 1) AK N S, 1 + µ s e σ ( s ) AK N S, 1 , (5.23b) whic h compose a Lie algebra with structur e [ [ µ m e K ( m ) AK N S, 1 , µ s e K ( s ) AK N S, 1 ] ] q = 0 , (5.24a ) [ [ µ m e K ( m ) AK N S, 1 , e T ( l,s ) ] ] q = mµ m µ s e K ( m + s − 1) AK N S, 1 , (5.24b) [ [ e T ( l,m ) , e T ( l,s ) ] ] q = ( m − s ) µ m µ s µ m + s − 1 e T ( l,m + s − 1) , (5.24c ) where l, m , s ≥ 0 and w e set e K ( − 1) AK N S, 1 = e T ( l, − 1) = 0 once they app ear on the r.h.s. of (5.24). In the pro duct [ [ · , · ] ] q the Gatea u x deriv ative is d efi ned w.r.t. q . When l = 1, they are reduced the symmetries and alg eb r a f o r the NLS equatio n . The symmetries (5.14), (5.15) and (5.20) are related together by cont in u o us limit. Same relations hold for the symmetries in Theorem 4.2 and (5.2 3 ). W e skip the detailed discussions for these connections. 15 6 Algebra deformations and understanding W e ha v e present ed symmetries and their alg eb r as for the conti n u ous AKNS and NLS hier- arc hies. Comparing them with those for the discrete cases, one can find not only the form of symmetries but also the structures of algebras are differen t. Since the algebras D and C (see (3.17) and (5.17)) pla y key roles for generating symmetries, let u s fo cus on D and C and see the difference b et wee n them in the ligh t of the corresp ondence (5.5) and (5.7). Sev er al of these deformations from D and C are listed in the follo wing: • d iffe ren t structures and different generators; • { K (0) [0] , K (0) [1] , K (1) [0] , σ (0) [0] } , { σ (0) [0] , σ (0) [1] , σ (1) [0] } and { K (0) AK N S , K (1) AK N S , K (2) AK N S , σ (0) AK N S } , { σ (0) AK N S , σ (1) AK N S , σ (2) AK N S } are subalgebras of D and C resp ectiv ely but with differen t str u c- tures; • { K (0) AK N S , K (1) AK N S , σ (0) AK N S , σ (1) AK N S } is a su balg eb ra of C , b ut f or D w e do not fin d any similar sub a lgebras (co n taining at least tw o non-isosp ec tral flo w s and one isosp ectral flow). T o understand these deformations, w e in tro duce de gr e e for fl o ws { K ( m ) [ j ] } and { σ ( m ) [ j ] } , v ariable t m,j and the recursion op erato r L . F or a f u nctio n f ( n, h, t ) (or an op erator) by d e g f we mean the order of h of th e denominate term (or leading term) in con tin uous limit. So w e can defi n e deg K ( m ) [ j ] = 2 m + 1 + j, (6.1a) deg σ ( m ) [ j ] = 2 m + j, (6 .1b) deg L = 2 , (6.1c) deg t m,j = − (2 m + j ) , (6.1d) deg u n = 1 , (6 .1e) where (6.1d) is from (5.9). Thus, after taking con tinuous limit only the terms with the lo w est degree are left while others disapp ear. W e also note that d ue to (6.1e) and th e definition of Gateaux deriv ativ e deg f ( u n ) ′ [ g ( u n )] = deg [ [ f ( u n ) , g ( u n )] ] = deg f ( u n ) + deg g ( u n ) − deg u n . (6.2) No w let us tak e (3.17 c ) and (5.17b) as an example to see the role that the degrees pla y in con tinuous limit and un derstanding those deformations. (3.17c) r eads [ [ K ( m ) [0] , σ ( s ) [1] ] ] = 1 2 m ( K ( m + s +1) [0] + 4 K ( m + s ) [0] ) , (6.3) in whic h deg [ [ K ( m ) [0] , σ ( s ) [1] ] ] = 2( m + s ) + 1 , deg K ( m + s +1) [0] = 2( m + s ) + 3 , deg K ( m + s ) [0] = 2( m + s ) + 1 . (6.4) With the h el p of de gr e e and n o ting that the corresp ondence (5.5) and (5.7), after taking con tin- uous limit, only those terms with the lo we st degree are left and consequently (6.3) go es to [ [ K (2 m ) AK N S , σ (2 s +1) AK N S ] ] = 2 mK (2( m + s )) AK N S , 16 whic h b elongs to relation (5.17b). S imila r to this example we can examine degrees of other form u la s in (3.17) and in this wa y w e can explain wh y the algebra D go es to C in con tin uou s limit although they ha v e differen t s t ructures. Next w e turn to τ -symmetries. Some of th em con tain differen t n um b er of terms in d iscret e case and contin uous case. T o explain the difference w e consider (5 .14b), (5.14c), (5.15b), (5.15c) and (5.20b) as an example. In (5. 14b) deg ( t l, 0 K ( l + s − 1) [1] ) = − 2 l + [2( l + s − 1) + 1 + 1] = 2 s, d eg σ ( s ) [0] = 2 s, whic h are same. Thus in con tin uous limit (5.14b) yields τ (2 l, 2 s ) AK N S = 2 lt 2 l K (2( l + s ) − 1) AK N S + σ (2 s ) AK N S . In (5.14c) the degrees are deg ( t l, 0 K ( l + s +1) [0] ) = 2 s + 3 , deg ( t l, 0 K ( l + s ) [0] ) = deg σ ( s ) [1] = 2 s + 1 . Th us in contin uous limit the term t l, 0 K ( l + s +1) [0] will disapp ear du e to higher degree and th e n (5.14c) go es to τ (2 l, 2 s +1) AK N S = 2 lt 2 l K (2( l + s )) AK N S + σ (2 s +1) AK N S . Similarly , (5.1 5b ) goes to τ (2 l +1 , 2 s ) AK N S = (2 l + 1) t 2 l +1 K (2( l + s )) AK N S + σ (2 s ) AK N S , and (5.15c) go es to τ (2 l +1 , 2 s +1) AK N S = (2 l + 1) t 2 l +1 K (2( l + s )+1) AK N S + σ (2 s +1) AK N S . The ab o v e four con tinuous limit results ju st comp ose the τ -symmetry (5.20b ). No w by means of de gr e e w e can u nderstand the deformation of algebras and s y m met ries app earing in con tin u o us limit. Finally , let us lo ok at the relatio n (4.9) . In cont in u o us limit we ha ve ( L 2 AK N S ) ′ [ K ( m ) AK N S ] − [ K ( m ) ′ AK N S , L 2 AK N S ] = 0 , (6.5a) ( L 2 AK N S ) ′ [ σ ( m ) AK N S ] − [ σ ( m ) ′ AK N S , L 2 AK N S ] − 2 L m +1 AK N S = 0 . (6.5b) This is consistent with the resu lt for the AKNS recursion op erator and flo w s: L ′ AK N S [ K ( m ) AK N S ] − [ K ( m ) ′ AK N S , L AK N S ] = 0 , (6.6a) L ′ AK N S [ σ ( m ) AK N S ] − [ σ ( m ) ′ AK N S , L AK N S ] − L m AK N S = 0 , (6.6b) where we h a v e made u sed of th e r e sult 1 2 ( L − L − 1 ) → L AK N S , leading term : O ( h ) (6.7) 17 in con tin uou s limit, whic h can b e seen through a pro ce dure similar to (5.3 ). App lyin g L AK N S to (6.6) and using the Leibniz rule for tw o op erators F and G : ( F G ) ′ [ h ] = F ′ [ h ] G + F G ′ [ h ], one can d e riv e (6.5) f rom (6.6). How ev er, one ma y w onder that no w that in con tin u ous limit w e ha v e the op erator 1 2 ( L − L − 1 ) → L AK N S but L → L 2 AK N S , w h y w e use L in stead of 1 2 ( L − L − 1 ) to generate D AKNS hierarc hy? I n fact, it is true that differen t d isc rete flo w s ca n go to the same in con tinuous limit. Applying 1 2 ( L − L − 1 ) on K (0) [0] t wice and taking Q n = − εR ∗ n w e hav e a second IDNLS equation, iQ n,t 2 = 1 4 (1 + εQ n Q ∗ n )[ Q n +2 (1 + εQ n +1 Q ∗ n +1 ) + εQ n Q n +1 Q ∗ n − 1 + εQ 2 n +1 Q ∗ n + Q n − 2 (1 + εQ n − 1 Q ∗ n − 1 ) + εQ 2 n − 1 Q ∗ n + εQ n − 1 Q n Q ∗ n +1 ] − 1 2 Q n . (6.8) It lo oks more complicated than the IDNLS equation (1.1) bu t it do es go to the con tin u o us NLS equation. Therefore we prefer to L although it corresp onds to L 2 AK N S . 7 Conclusions One of th e main results of the pap er is that w e got in finitely many symmetries for the IDNLS equ a tion (1.1) and the IDNLS hierarch y (3.14a). T his w as done through constructing the r ec u rsion op erator e L , isospectral and non-isosp ectral IDNLS flo ws in scal ar form and their algebraic structures (3.15). A second result is on the D AKNS flo ws. These fl ows are generated b y th e basic flo ws K (0) [0] , K (0) [1] , σ (0) [0] , σ (0) [1] and th e recursion op erator L . B y con tin u ous limit one can bu ild d irec t corresp ondence b et w een these flows and the conti n u o us AKNS isosp ectral and non-isosp ectral flo ws. Mean wh ile , L goes to the squ a re of the AKNS recursion op erartor L AK N S in the same con tinuous limit pro cedure. Th e se DAKNS flows form a Lie algebra D of w hic h the structur es (3.17) are d e riv ed from the basic algebraic relations (2.4) of th o s e t wo-potentia l AL flo ws. It has b een sh own that this algebra D pla ys a key role in constructing symmetries and their algebraic structures for b oth the IDNLS hierarch y and D AKNS h ierarch y . The final main resu lt is on the algebra deformations and explanatio n s. W e listed out some deformations of algebras when they go to con tinuous case. I n fact, in Ref. [5 ] a contrac tion of subalgebras has b een rep orte d, but in that case the corresp ondence b et w een discrete case and con tinuous case is n o t direct and some lin ear com b in a tions were inv olv ed. As we can see in the present p a p er the co rresp ondence b et w een the discrete and con tin uous AKNS flo ws is direct (see (5.5) and (5.7)); an d b y means of contin uous limit and the lattice sp acing parameter h w e in tro duced de gr e e f o r discrete elemen ts, as listed in (6.1). C a lculating the degree of eac h term one can und e rstand the algebra deform ations b efo re and after taking con tin u ou s limit. Finally , w e note that D is not a cen terless KMV algebra, bu t it is someho w related to this t yp e. On one hand , D is derived from the cen terless KMV algebra (2.4). O n the other h a nd, D and C are a cont in u um in co ntin uous limit a nd C is a cent erless K M V algebra. It is not rare to see the algebraic structure c hanges in discrete cases. F or example, b esides the subalgebra con traction found in [5] and the algebra deformations listed in this pap er, the symmetry algebra of the differen tial-difference KP equation also has a non-centrele s s Kac-Moo dy-Virasoro structure [18]. W e b elie ve con tinuous limit and de gr e e are goo d means to u nderstand these changes. 18 Ac kno wledgemen t This pro jec t is supp orted b y the Nat ional Natural Science F oundation of China (106711 21) and Shanghai Leading Academic Discipline Pro ject (No.J50101). References [1] D.J. Zhang, S.T. Chen, Symmetries for the Ablowitz-Ladik hiera r c hy: I. F our- potential case, arXiv:100 4.0751 [nlin.SI]. [2] M.J . Ablowitz, J.F. La dik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975 ) 598-6 03. [3] M.J . Ablowitz, J.F. Ladik, Nonlinear differential-difference equations and F o ur ier analysis, J. Math. Phys., 1 7 (1976) 1011-8 . [4] M.J . Ablowitz, B. Prinari, A.D. T rubatch, Discr ete and Continuous N o nline ar Schr¨ odinger S ystems , Cambridge Univ. Press, 2004 . [5] R. Hern´ andez Her e dero, D. Levi, P . Win ternitz, Symmetries of the discre te nonlinear Sc hr¨ odinger equation, Theor e. Math. Phys., 127 (2001) 729-3 7. [6] R. Hern´ a ndez Heredero, D. Lev i, The discrete nonlinear Schr¨ o dinger equation and its Lie symmetry reductions, J. Nonl. Math. Phys., 10, Suppl.2 (2 0 03) 77-94. [7] D.J. Zhang, T.K . Ning, J .B. Bi, D.Y. C he n, New symmetries for the Ablo witz-La dik hier a rc hies, Phys. L e tt . A, 359 (2006) 458- 6 6. [8] S.C. Chiu, J.F. La dik, Generating exactly soluble nonlinear discrete evolution equations by a gen- eralized W ronskia n tec hnique, J. Math. Phys., 18 (1977) 690 -700. [9] K .M. T a mizhmani, W.X. Ma, Master symmetr ies fr o m Lax op erators for cer tain la tt ice so liton hierarchies, J. Phys. So c. Jpn., 69 (2000) 35 1-61. [10] Y.S. Li, C.C. Zhu, New set of symmetries of the int egrable eq ua tions, Lie a lgebra and non-is o spectral evolution e quations. II. AKNS system J. Phys. A: Math. Gen., 1 9 (1986) 3713-2 5. [11] C. Tian, Symmetries, in Soliton The ory and its Applic ations , Ed. C.H. Gu, Spring er-V er leg, Berlin, 1996. [12] M.J. Ablowitz, D.J. K aup, A.C. Newell, H. Segur , The inverse scattering transform — F ourier analysis for nonlinea r problems, Stud. Appl. Ma t h., 53 (1974) 249-3 15. [13] D.Y. Chen, Intr o duction to Soliton The ory , Science Press, Beijing, 2006. [14] T.K. Ning, D.Y. Chen, D.J. Zhang, The exac t solutions for the no nisospectra l AKNS hierar c hy through the inv er se scattering transform, Physica A, 339 (20 04) 2 48-66. [15] G.Z. T u, The Lie a lgebraic structure of symmetries ge nerated by hereditary symmetries, J. Phys. A: Math. Gen., 21 (1988) 1951- 7. [16] D.Y. Chen, H.W. Z hang, Lie algebr a ic str ucture for the AKNS system, J. P h ys. A: Ma th. Gen., 24 (1991) 377 - 83. [17] D.Y. Chen, D.J. Z ha ng, Lie algebraic structures of (1+1 )-dimensional Lax integrable systems, J. Math. Phys., 37 (1996 ) 552 4-38. [18] X.L. Sun, D.J. Zhang , X.Y. Zh u, D.Y. Chen, Symmetries a nd Lie a lgebra o f the differen tial-difference Kadomstev-Petviashvili hierar c hy , arXiv:09 08.0382 [nlin.SI], to app ear in Mo d. Phys. Lett. B. 19