Binary Nonlinearization of the Super Akns System Under an Implicit Symmetry Constraint

BINAR Y NONLINEARIZA TION OF THE SUPER AKNS SYSTEM UNDER AN IMPLICIT SYMMETR Y CONSTRAINT JING YU a , JINGWEI HAN a , JINGSONG HE b,c ∗ a Sc ho ol of Science, Hangzhou Dianzi Univ ersit y , Hang zhou, Z hejiang, 31 0018, China b Departmen t of Mathematics, Univ ersit y of Science and T ec hnology of China, Hefei, Anhui, 230026, China c Departmen t of Mathematics, Ningb o Univ ersit y , Ningb o, Zhejiang, 3 15211, China Abstract F or the super AKNS system, an implicit symmetry cons traint betw een the p otentials and the e igenfunctions is prop osed. After in tro ducing some new v aria bles to explicitly expre s s po tent ials, the sup er AK NS sy s tem is decomp osed into tw o compa tible finite-dimensio nal sup e r systems (x-part and t n -part). F urthermore, we show that the obtained sup er systems are integrable sup er Hamiltonian systems in sup ersymmetry manifold R 4 N +2 | 2 N +2 . Key w ords: nonlinear ization, super AKNS system, an implicit symmetry constr aint, finite- dimensional integrable super Hamilto nian systems. P A CS co des: 02.9 0.+p, 02.30.IK 1 In tro duction In 1988, Prof. Cao pr op osed the m ono-nonlinearization method of Lax pairs for the classical in tegrable (1+1)-dimensional system [1]. T he k ey of mon o-nonlin earization is to fi nd the con- strain t b et wee n the p oten tials and the eigenfunctions of the Lax system. After choosing some distinct sp ectral p arameters and considering th e constrain t, the Lax system is decomp osed into finite-dimensional systems whose v ariables can b e separated, and furthermore, the obtained finite-dimensional sy s tems are completely integrable Hamiltonian systems in the Liouville sense. Sev eral y ears later, the method wa s extended to classical integ rable (2+1)-dimensional systems [2, 3, 4]. Thereafter, the metho d wa s conti nued to generaliz e in the follo w ing tw o asp ects. One ∗ Corresponding author, E-mail address:jshe@ustc.edu.cn, hejingsong@nbu.edu.cn 1 w as the b inary nonlinearization metho d of the Lax pairs and its adjoint Lax pairs for the clas- sical in tegrable systems, w hic h w as fi r stly p rop osed by Prof. Ma in Ref. [5 ]. An d the other w as the higher-order constraint s (i. e. im p licit constrain ts), whic h were widely s tu died in Refs. [6, 7, 8]. In a word, the m etho d of nonlinearization w as extensiv ely studied by m any r esearc h er s in the past tw ent y y ears. F ollo w ed from this, man y finite-dimensional integrable Hamiltonian systems w ere obtained. In recen t y ears, sev eral integ rable sup er s ystems [9 , 10, 11] and integ rable sup ersymmetry systems [12, 13, 14] h a v e aroused strong in terests in man y mathematicians and ph ysicists. Su c h as Darb oux tran s formation [15, 16], Hamiltonian str u ctures [17, 18, 19], and s o on. In v ery recen t yea rs, nonlin earization of the sup er AKNS system has b een studied in Ref. [20], where w e considered an explicit symmetry constraint of the sup er AKNS sy s tem, and we pro ved that under the explicit co nstr ain t, the su p er AKNS system w as complete ly in tegrable sup er Hamiltonian system in the Liouville sense. Insp ired b y this and implicit constraint of th e classical in tegrable system, a natural question is app eared whether an implicit sy m metry constrain t is av ailable for the sup er AKNS system. I n the presen t pap er, w e shall solv e this prob lem. The pap er is organized as follo ws. In the next section, w e prop ose an imp licit symmetry constrain t b et we en th e p otenti als and the eigenfunctions of the sup er AKNS system. Then in section 3, un der the constraint , th e sup er AKNS system is decomp osed into t wo compatible finite-dimensional sup er systems. And furthermore, w e sh o w that the obtained finite-dimensional sup er sy s tems are completely in tegrable in the Liouville s en se. Finally , some conclusions and discussions are listed in secti on 4. 2 An Implicit Symmetry Constrain t Of The Sup er AKNS Hi- erarc h y In Ref. [20], w e ha v e considered the binary n on lin earizatio n of the sup er AKNS system under an explicit sym m etry constrain t, and ob tained the finite-dimensional in tegrable sup er Hamiltonian system. Here we w ill pr op ose an imp licit s y m metry constraint of the sup er AKNS system. Therefore, th is pap er can b e regarded as a contin uation of Ref. [20]. F or simplicit y , we omit the detailed deriv ation of the su p er AKNS hierarc hy , which can b e referr ed to Ref. [20]. In what follo ws , w e shall p rop ose an implicit symmetry constraint b et w een the p otent ial and th e eigenfunctions. T o this aim, we consider the sup er AKNS sp ectral p roblem φ x = M φ, M =     − λ q α r λ β β − α 0     , φ =     φ 1 φ 2 φ 3     , (1) 2 and its adjoin t sp ectral problem ψ x = − M S t ψ =     λ − r β − q − λ − α − α − β 0     ψ , ψ =     ψ 1 ψ 2 ψ 3     , (2) where St means sup ertransp ose [21]. By a similar wa y of th e coun terpart in the classical system [5 , 22], it is n ot difficult to get the v ariational deriv ativ e of the parameter with resp ect to the p oten tial δ λ δ U 0 =        δλ δr δλ δq δλ δβ δλ δα        =        ψ 2 φ 1 ψ 1 φ 2 ψ 2 φ 3 − ψ 3 φ 1 ψ 1 φ 3 + ψ 3 φ 2        , (3) where U 0 = ( r, q , β , α ) T . Imp osing the zero b oundary conditions lim | x |→∞ φ i = lim | x |→∞ ψ i = 0( i = 1 , 2 , 3), w e can v erify a simple c haracteristic p rop erty L 1 δ λ δ U 0 = λ δ λ δ U 0 , (4) where L 1 =        − 1 2 ∂ x + q ∂ − 1 x r − q ∂ − 1 x q 1 2 α + 1 2 q ∂ − 1 x β − 1 2 q ∂ − 1 x α r ∂ − 1 x r 1 2 ∂ x − r ∂ − 1 x q 1 2 r ∂ − 1 x β − 1 2 β − 1 2 r ∂ − 1 x α 2 β − 2 α∂ − 1 x r 2 α∂ − 1 x q − ∂ x − α∂ − 1 x β − q + α∂ − 1 x α 2 β ∂ − 1 x r 2 α − 2 β ∂ − 1 x q r + β ∂ − 1 x β ∂ x − β ∂ − 1 x α        , (5) with ∂ x = d dx , ∂ x ∂ − 1 x = ∂ − 1 x ∂ x = 1 . Cho osing N distinct sp ectral parameters λ 1 , · · · , λ N , the sup er AKNS s p ectral problem (1 ) and the adjoin t sp ectral problem (2) b ecome the follo wing fi n ite-dimensional sup er systems                        φ 1 j,x = − λ j φ 1 j + qφ 2 j + αφ 3 j , 1 ≤ j ≤ N , φ 2 j,x = r φ 1 j + λ j φ 2 j + β φ 3 j , 1 ≤ j ≤ N , φ 3 j,x = β φ 1 j − αφ 2 j , 1 ≤ j ≤ N , ψ 1 j,x = λ j ψ 1 j − r ψ 2 j + β ψ 3 j , 1 ≤ j ≤ N , ψ 2 j,x = − q ψ 1 j − λ j ψ 2 j − αψ 3 j , 1 ≤ j ≤ N , ψ 3 j,x = − αψ 1 j − β ψ 2 j , 1 ≤ j ≤ N . (6) In what follo ws, let us consider the traditional s y m metry constrain ts        b k +1 c k +1 − 2 ρ k +1 2 δ k +1        = N X j =1 γ j        δλ j δr δλ j δq δλ j δβ δλ j δα        , (7) 3 where γ j (1 ≤ j ≤ N ) are u sual constan ts and k ≥ 0. In Ref. [20], we ha ve chosen k = 0 and γ j = 1(1 ≤ j ≤ N ) in the ab o ve constrain t. Thus we obtained an explicit symmetry constrain t (i. e. the p otentia ls can b e expressed by the eigenfun ctions explicitly). While in this pap er, we will extend our previously w ork and c ho ose k = 1 and γ j = − 1 2 (1 ≤ j ≤ N ) in Eq. (7). T hat is to sa y , w e obtain the follo w in g implicit symmetry constrain t              q x = < Ψ 2 , Φ 1 >, r x = − < Ψ 1 , Φ 2 >, α x = − 1 4 ( < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) , β x = − 1 4 ( < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) , (8) where Φ i = ( φ i 1 , · · · , φ iN ) T , Ψ i = ( ψ i 1 , · · · , ψ iN ) T , i = 1 , 2 , 3, and < ., . > denotes the standard inner pro d uct in Euclidean space R N . Ob viously , the constrain t (8) is an imp licit constrain t. That is to say , the p oten tials of th e fi nite-dimensional sup er sys tems (6) can not b e expr essed b y the eigenfunctions explicitly , which is different f r om the constrain t in Ref. [20]. In order to consider nonlinearization of the su p er AKNS system un der the imp licit symm etry constrain t (8), w e should tak e some measures. 3 Nonlinearization of the sup er AKNS system under an implicit symmetry constrain t No w we are in a p osition to discuss the nonlinearization of th e su p er AKNS system under the implicit symmetry constrain t (8). T o this aim, w e fir stly in tro duce the f ollo wing new v ariables φ N +1 = q , φ N +2 = 2 α, ψ N +1 = r, ψ N +2 = − 2 β . (9) Considering the new v ariables (9) and substituting th e constrain t (8) in to system (6), we obtain the follo win g finite-dimensional sup er system                                              φ 1 j,x = − λ j φ 1 j + φ N +1 φ 2 j + 1 2 φ N +2 φ 3 j , 1 ≤ j ≤ N , φ 2 j,x = ψ N +1 φ 1 j + λ j φ 2 j − 1 2 ψ N +2 φ 3 j , 1 ≤ j ≤ N , φ 3 j,x = − 1 2 ψ N +2 φ 1 j − 1 2 φ N +2 φ 2 j , 1 ≤ j ≤ N , φ N +1 ,x = < Ψ 2 , Φ 1 >, φ N +2 ,x = − 1 2 ( < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) , ψ 1 j,x = λ j ψ 1 j − ψ N +1 ψ 2 j − 1 2 ψ N +2 ψ 3 j , 1 ≤ j ≤ N , ψ 2 j,x = − φ N +1 ψ 1 j − λ j ψ 2 j − 1 2 φ N +2 ψ 3 j , 1 ≤ j ≤ N , ψ 3 j,x = − 1 2 φ N +2 ψ 1 j + 1 2 ψ N +2 ψ 2 j , 1 ≤ j ≤ N , ψ N +1 ,x = − < Ψ 1 , Φ 2 >, ψ N +2 ,x = 1 2 ( < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) . (10) 4 Ob viously , E q. (10) can b e written b y the follo wing su p er Hamiltonian form:    Φ 1 ,x = ∂ H 1 ∂ Ψ 1 , Φ 2 ,x = ∂ H 1 ∂ Ψ 2 , Φ 3 ,x = ∂ H 1 ∂ Ψ 3 , φ N +1 ,x = ∂ H 1 ∂ ψ N +1 , φ N +2 ,x = ∂ H 1 ∂ ψ N +2 , Ψ 1 ,x = − ∂ H 1 ∂ Φ 1 , Ψ 2 ,x = − ∂ H 1 ∂ Φ 2 , Ψ 3 ,x = ∂ H 1 ∂ Φ 3 , ψ N +1 ,x = − ∂ H 1 ∂ φ N +1 , ψ N +2 ,x = ∂ H 1 ∂ φ N +2 , (11) with the sup er Hamilto nian is giv en by H 1 = − < ΛΨ 1 , Φ 1 > + < ΛΨ 2 , Φ 2 > + φ N +1 < Ψ 1 , Φ 2 > + ψ N +1 < Ψ 2 , Φ 1 > + 1 2 φ N +2 ( < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) − 1 2 ψ N +2 ( < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) . That is to sa y , the n onlinearized fin ite-dimensional sup er system (10) is a sup er Hamiltonian system. In what follo ws, let us consider the temp oral part of the su p er AKNS h ierarc h y φ t n = N ( n ) φ = ( λ n N ) + φ, (12) with ( λ n N ) + = n X j =0     a j b j ρ j c j − a j δ j δ j − ρ j 0     λ n − j , where the symb ol ”+” denotes taking the nonn egativ e p ow er of λ . When considered N d is- tinct sp ectral parameter λ 1 , · · · , λ N , the temp oral part of the su p er AKNS s ystem b ecomes the follo win g sup er s y s tem     φ 1 j φ 2 j φ 3 j     t n =         n P i =0 a i λ n − i j n P i =0 b i λ n − i j n P i =0 ρ i λ n − i j n P i =0 c i λ n − i j − n P i =0 a i λ n − i j n P i =0 δ i λ n − i j n P i =0 δ i λ n − i j − n P i =0 ρ i λ n − i j 0             φ 1 j φ 2 j φ 3 j     , 1 ≤ j ≤ N , (13) whose adjoin t sup er system is giv en by     ψ 1 j ψ 2 j ψ 3 j     t n =         − n P i =0 a i λ n − i j − n P i =0 c i λ n − i j n P i =0 δ i λ n − i j − n P i =0 b i λ n − i j n P i =0 a i λ n − i j − n P i =0 ρ i λ n − i j − n P i =0 ρ i λ n − i j − n P i =0 δ i λ n − i j 0             ψ 1 j ψ 2 j ψ 3 j     , 1 ≤ j ≤ N . (14) F or n = 1, systems (13) and (14) are exactly th e sp atial systems (1) and (2), resp ectiv ely . I n particular, as f or t 2 -part, the nonlinearized sup er systems (13) and (14) b ecome the f ollo wing 5 system                        φ 1 j,t 2 = ( − λ 2 j + 1 2 q r + αβ ) φ 1 j + ( qλ j − 1 2 q x ) φ 2 j + ( αλ j − α x ) φ 3 j , 1 ≤ j ≤ N , φ 2 j,t 2 = ( r λ j + 1 2 r x ) φ 1 j + ( λ 2 j − 1 2 q r − αβ ) φ 2 j + ( β λ j + β x ) φ 3 j , 1 ≤ j ≤ N , φ 3 j,t 2 = ( β λ j + β x ) φ 1 j + ( − αλ j + α x ) φ 2 j , 1 ≤ j ≤ N , ψ 1 j,t 2 = ( λ 2 j − 1 2 q r − αβ ) ψ 1 j − ( r λ j + 1 2 r x ) ψ 2 j + ( β λ j + + β x ) ψ 3 j , 1 ≤ j ≤ N , ψ 2 j,t 2 = ( − q λ j + 1 2 q x ) ψ 1 j + ( − λ 2 j + 1 2 q r + αβ ) ψ 2 j + ( − αλ j + α x ) ψ 3 j , 1 ≤ j ≤ N , ψ 3 j,t 2 = ( − αλ j + α x ) ψ 1 j − ( β λ j + β x ) ψ 2 j , 1 ≤ j ≤ N . (15) Considering the n ew v ariables (9) and the implicit constraint (8), the ab o v e fi nite-dimensional sup er system (15) b ecomes the follo wing th e nonlinearized sup er system                                                                  φ 1 j,t 2 = ( − λ 2 j + 1 2 φ N +1 ψ N +1 − 1 4 φ N +2 ψ N +2 ) φ 1 j + ( φ N +1 λ j − 1 2 < Ψ 2 , Φ 1 > ) φ 2 j + 1 4 (2 φ N +2 λ j + < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) φ 3 j , φ 2 j,t 2 = ( ψ N +1 λ j − 1 2 < Ψ 1 , Φ 2 > ) φ 1 j + ( λ 2 j − 1 2 φ N +1 ψ N +1 + 1 4 φ N +2 ψ N +2 ) φ 2 j − 1 4 (2 ψ N +2 λ j + < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) φ 3 j , φ 3 j,t 2 = − 1 4 (2 ψ N +2 λ j + < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) φ 1 j − 1 4 (2 φ N +2 λ j + < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) φ 2 j , φ N +1 ,t 2 = 1 2 φ N +1 ( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > )+ < ΛΨ 2 , Φ 1 > + φ 2 N +1 ψ N +1 − 1 2 φ N +1 φ N +2 ψ N +2 , φ N +2 ,t 2 = 1 4 φ N +2 ( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > ) + 1 2 ( < ΛΨ 3 , Φ 1 > − < ΛΨ 2 , Φ 3 > ) , ψ 1 j,t 2 = ( λ 2 j − 1 2 φ N +1 ψ N +1 + 1 4 φ N +2 ψ N +2 ) ψ 1 j − ( ψ N +1 λ j − 1 2 < Ψ 1 , Φ 2 > ) ψ 2 j − 1 4 (2 ψ N +2 λ j + < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) ψ 3 j , ψ 2 j,t 2 = − ( φ N +1 λ j − 1 2 < Ψ 2 , Φ 1 > ) ψ 1 j + ( − λ 2 j + 1 2 φ N +1 ψ N +1 − 1 4 φ N +2 ψ N +2 ) ψ 2 j − 1 4 (2 φ N +2 λ j + < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) ψ 3 j , ψ 3 j,t 2 = − 1 4 (2 φ N +2 λ j + < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) ψ 1 j + 1 4 (2 ψ N +2 λ j + < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) ψ 2 j , ψ N +1 ,t 2 = − < ΛΨ 1 , Φ 2 > − 1 2 ψ N +1 ( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > ) − φ N +1 ψ 2 N +1 + 1 2 φ N +2 ψ N +1 ψ N +2 , ψ N +2 ,t 2 = 1 2 ( < ΛΨ 1 , Φ 3 > + < ΛΨ 3 , Φ 2 > ) − 1 4 ψ N +2 ( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > ) − 1 2 φ N +1 ψ N +1 ψ N +2 , (16) where 1 ≤ j ≤ N . It is a direct b ut tedious chec k that the nonlin earized sup er system (16) can b e wr itten as the follo win g sup er Hamiltonian form    Φ 1 ,t 2 = ∂ H 2 ∂ Ψ 1 , Φ 2 ,t 2 = ∂ H 2 ∂ Ψ 2 , Φ 3 ,t 2 = ∂ H 2 ∂ Ψ 3 , φ N +1 ,t 2 = ∂ H 2 ∂ ψ N +1 , φ N +2 ,t 2 = ∂ H 2 ∂ ψ N +2 , Ψ 1 ,t 2 = − ∂ H 2 ∂ Φ 1 , Ψ 2 ,t 2 = − ∂ H 2 ∂ Φ 2 , Ψ 3 ,t 2 = ∂ H 2 ∂ Φ 3 , ψ N +1 ,t 2 = − ∂ H 2 ∂ φ N +1 , ψ N +2 ,t 2 = ∂ H 2 ∂ φ N +2 , (17) 6 where the sup er Hamilto nian is giv en by H 2 = − < Λ 2 Ψ 1 , Φ 1 > + < Λ 2 Ψ 2 , Φ 2 > + φ N +1 < ΛΨ 1 , Φ 2 > + ψ N +1 < ΛΨ 2 , Φ 1 > + 1 2 φ N +2 ( < ΛΨ 1 , Φ 3 > + < ΛΨ 3 , Φ 2 > ) − 1 2 ψ N +2 ( < ΛΨ 2 , Φ 3 > − < ΛΨ 3 , Φ 1 > ) + 1 4 (2 φ N +1 ψ N +1 − φ N +2 ψ N +2 )( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > ) − 1 2 < Ψ 2 , Φ 1 >< Ψ 1 , Φ 2 > + 1 4 ( < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > )( < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) − 1 2 φ N +1 φ N +2 ψ N +1 ψ N +2 + 1 2 φ 2 N +1 ψ 2 N +1 . That is to s a y , as for t 2 -part, the n onlinearized sup er system (16 ) is fin ite-dimensional sup er Hamiltonian sys tem. I n what follo w, we w an t to pro ve that for an y n ≥ 2, the sup er system (13) and (14) can b e nonlinearized, and furthermore, the obtained nonlinearized system is finite- dimensional su p er Hamiltonian system. Therefore, making use of (4) and E q. (5), w e obtain the constrained a i , b i , c i , ρ i , δ i (1 ≤ i ≤ N ) in systems (13) and (14). Only for d ifferen tiation, ˜ P ( U ) denotes the new expression generate d from P ( U ) by the nonlinear constraint (8). i. e.                  ˜ a i = − 1 4 < Λ i − 2 Ψ 1 , Φ 1 > − 1 2 < Λ i − 2 Ψ 2 , Φ 2 >, i ≥ 2 , ˜ b i = − 1 2 < Λ i − 2 Ψ 2 , Φ 1 >, i ≥ 2 , ˜ c i = − 1 2 < Λ i − 2 Ψ 1 , Φ 2 >, i ≥ 2 , ˜ ρ i = 1 4 ( < Λ i − 2 Ψ 2 , Φ 3 > − < Λ i − 2 Ψ 3 , Φ 1 > ) , i ≥ 2 , ˜ δ i = − 1 4 ( < Λ i − 2 Ψ 1 , Φ 3 > + < Λ i − 2 Ψ 3 , Φ 2 > ) , i ≥ 2 . (18) Substituting (18) in to the s up er systems (13) and (14), we obtain th e nonlinearized sup er system                                            φ 1 j φ 2 j φ 3 j     t n =         n P i =0 ˜ a i λ n − i j n P i =0 ˜ b i λ n − i j n P i =0 ˜ ρ i λ n − i j n P i =0 ˜ c i λ n − i j − n P i =0 ˜ a i λ n − i j n P i =0 ˜ δ i λ n − i j n P i =0 ˜ δ i λ n − i j − n P i =0 ˜ ρ i λ n − i j 0             φ 1 j φ 2 j φ 3 j     , 1 ≤ j ≤ N ,     ψ 1 j ψ 2 j ψ 3 j     t n =         − n P i =0 ˜ a i λ n − i j − n P i =0 ˜ c i λ n − i j n P i =0 ˜ δ i λ n − i j − n P i =0 ˜ b i λ n − i j n P i =0 ˜ a i λ n − i j − n P i =0 ˜ ρ i λ n − i j − n P i =0 ˜ ρ i λ n − i j − n P i =0 ˜ δ i λ n − i j 0             ψ 1 j ψ 2 j ψ 3 j     , 1 ≤ j ≤ N . (19) In wh at follo ws, w e w an t to see that the nonlinearized sup er system (19) is a fi nite-dimensional sup er Hamiltonian system. 7 F rom Eq. (18), we know that the constrained co-adjoin t repr esen tation equation ˜ N x = [ ˜ M , ˜ N ] is still satisfied, and furtherm ore, the equalit y ( ˜ N 2 ) x = [ ˜ M , ˜ N 2 ] is also satisfied. There- fore, let ˜ F = 1 2 S tr ˜ N 2 = ˜ a 2 + ˜ b ˜ c + 2 ˜ ρ ˜ δ . It is n ot difficu lt to calculate that ˜ F x = 0 , whic h means that ˜ F is a generating fu n ction of in tegrals of motion for the nonlinearized spatial s ystem (10). Let ˜ F = P n ≥ 0 ˜ F n λ − n , in tegrals of motion ˜ F n ( n ≥ 0) is giv en by th e follo wing f ormulas ˜ F 0 = 1 , ˜ F 1 = 0 , ˜ F 2 = 1 2 ( < Ψ 1 , Φ 1 > − < Ψ 2 , Φ 2 > ) + φ N +1 ψ N +1 − 1 2 φ N +2 ψ N +2 , ˜ F 3 = 1 2 ( < ΛΨ 1 , Φ 1 > − < ΛΨ 2 , Φ 2 > ) − 1 2 φ N +1 < Ψ 1 , Φ 2 > − 1 2 ψ N +1 < Ψ 2 , Φ 1 > − 1 4 φ N +2 ( < Ψ 1 , Φ 3 > + < Ψ 3 , Φ 2 > ) + 1 4 ψ N +2 ( < Ψ 2 , Φ 3 > − < Ψ 3 , Φ 1 > ) , ˜ F n = n − 1 X i =2 [ 1 16 ( < Λ i − 2 Ψ 1 , Φ 1 > − < Λ i − 2 Ψ 2 , Φ 2 > )( < Λ n − i − 2 Ψ 1 , Φ 1 > − < Λ n − i − 2 Ψ 2 , Φ 2 > ) − 1 8 ( < Λ i − 2 Ψ 2 , Φ 3 > − < Λ i − 2 Ψ 3 , Φ 1 > )( < Λ n − i − 2 Ψ 1 , Φ 3 > + < Λ n − i − 2 Ψ 3 , Φ 2 > ) + 1 4 < Λ i − 2 Ψ 2 , Φ 1 >< Λ n − i − 2 Ψ 1 , Φ 2 > ] + 1 2 ( < Λ n − 2 Ψ 1 , Φ 1 > − < Λ n − 2 Ψ 2 , Φ 2 > ) − 1 4 φ N +2 ( < Λ n − 3 Ψ 1 , Φ 3 > + < Λ n − 3 Ψ 3 , Φ 2 > ) + 1 4 ψ N +2 ( < Λ n − 3 Ψ 2 , Φ 3 > − < Λ n − 3 Ψ 3 , Φ 1 > ) − 1 2 φ N +1 < Λ n − 3 Ψ 1 , Φ 2 > − 1 2 ψ N +1 < Λ n − 3 Ψ 2 , Φ 1 >, n ≥ 4 . (20) After a direct calc ulation, w e hav e    Φ 1 ,t n = − 2 ∂ F n +2 ∂ Ψ 1 , Φ 2 ,t n = − 2 ∂ F n +2 ∂ Ψ 2 , Φ 3 ,t n = − 2 ∂ F n +2 ∂ Ψ 3 , φ N +1 ,t n = − 2 ∂ F N +2 ∂ ψ N +1 , φ N +2 ,t n = − 2 ∂ F N +2 ∂ ψ N +2 , Ψ 1 ,t n = 2 ∂ F n +2 ∂ Φ 1 , Ψ 2 ,t n = 2 ∂ F n +2 ∂ Φ 2 , Ψ 3 ,t n = − 2 ∂ F n +2 ∂ Φ 3 , ψ N +1 ,t n = 2 ∂ F N +2 ∂ φ N +1 , ψ N +2 ,t n = − 2 ∂ F N +2 ∂ φ N +2 , (21) whic h means that the nonlinearized temp oral system (19) is su p er Hamiltonian system. I n conclusion, for any n ( n ≥ 1), the nonlinearized system (19) is a fi nite-dimensional, sup er Hamil- tonian system. I n what follo w s, we only w an t to p ro v e that th e nonlin earized system (19) is completely in tegrable in the Liouville sense. T o this aim, w e c ho ose the follo wing P oisson brac ke t { F , G } = 3 X i =1 N X j =1 ( ∂ F ∂ φ ij ∂ G ∂ ψ ij − ( − 1) p ( φ ij ) p ( ψ ij ) ∂ F ∂ ψ ij ∂ G ∂ φ ij ) + 2 X j =1 ( ∂ F ∂ φ N + j ∂ G ∂ ψ N + j − ( − 1) p ( φ N + j ) p ( ψ N + j ) ∂ F ∂ ψ N + j ∂ G ∂ φ N + j ) , (22) 8 where p ( u ) is a parit y function of u , namely , p ( u ) = 0 if u is an ev en v ariable and p ( u ) = 1 if u is an o dd v ariable. It is not difficult to see that ˜ F n ( n ≥ 0) are also in tegrals of motion for Eq. (19), i. e. { ˜ F m +1 , ˜ F n +2 } = − 1 2 ∂ ∂ t n ˜ F m +1 = 0 , m, n ≥ 0 , whic h means that { ˜ F n } n ≥ 0 are in in v olution in pair. With the help of the resu lt of nonlinearization for classical in tegrable sys tem [5, 22, 23], it is natural for us to s et f k = ψ 1 k φ 1 k + ψ 2 k φ 2 k + ψ 3 k φ 3 k , 1 ≤ k ≤ N , (23) and verify that they are also in tegrals of motion of the constrained spatial system (10) and temp oral system (19). Making use of (22), it is easy to fi nd that (23) are in inv olution in pair. F or th e nonlinearized spatial system (10) and the nonlinearized temp oral system (19), w e c ho ose 3N+2 in tegrals of motion f 1 , · · · , f N , F 2 , F 3 , · · · , F 2 N + 3 , (24) whose inv olution hav e b een verified. In what follo ws, w e wan t to s h o w th e functional indep en- dence of (24 ). Similar to the Refs. [20, 23, 24], 3N+2 fu nctions (2 4 ) are functionally indep endent at least o ver some region of the sup ersymmetry manifold R 4 N + 2 | 2 N +2 . T aking int o account the p receding program, it is n ot difficult to dra w a b elo w. Theorem 1 The c onstr aine d (6N+4)-dimensio anl systems (10) and (19) ar e sup er Hamilto- nian systems , whose 3N+2 i nte gr als of motio n (24) ar e in involution i n p air and functional ly indep endent over sup ersymmetry manifold R 4 N + 2 | 2 N +2 . Remark 1 The main differ enc es b etwe en the finite-dimensional sup er Hamiltonian systems i n pr esent p ap er and r efe r enc e [20] c an b e summarize d b el low: 1) D ue to the implicit c onstr aint, we have to intr o duc e oth er 4 c o or dinates i n Eq.(9) such that finite-dimensional sup er Hamiltonia n system in pr esent p ap er i s (6N+4)-dimensional. However, the c orr esp onding system in r e fer enc e [20] is 6N-dimensional. 2)Corr e sp ondingly, Poisson br acket in Eq.(22) is differ ent fr om Eq . (36) in R ef.[20]. 4 Conclusions and Discussions In this pap er, we p resen ted a new finite-dimensional in tegrable sup er Hamiltonia n system of the su p er AKNS system. The d ifferen ce of this p ap er an d Ref. [20] lies on the symmetry constrain t b et we en the p otentia ls and the eigenfunctions, wh ic h resu lts in generating different 9 finite-dimensional sup er system. In Ref. [20], we hav e pr op osed an explicit symmetry constrain t. Substituting the explicit constraint in to th e spatial system and the temp oral system of the sup er AKNS system, we h a ve obtained the n onlinearized sup er system, and fur th ermore, w e h av e pro ve d the obtained nonlinearized sys tem are complete in tegrable in the Liouville sense. Ho wev er in this pap er, we prop osed an implicit symmetry constrain t (8), which made the p oten tials can not b e expressed by eigenfunctions explicitly . Therefore, w e refer to the metho d of implicit constrain t for classical inte grable s ystem, and introdu ce four new v ariables (9) to explicitly express p oten tials. After th is, we obtained the sup er nonlinearized spatial system (10) and the temp oral system (19), and p ro v ed that the obtained sup er system (10) and (19) is sup er Hamiltonian sys tem and hav e 3N+2 inte grals of motion, w hic h is in in v olution in pair and functionally indep endent at least o ve r s ome region of su p ersym metry manifold R 4 N + 2 | 2 N +2 . At last, we w ould lik e to str ess that the n onlinearization of the sup er AKNS s y s tem pro vides a new and systematic w a y to co nstr u ct finite-dimensional sup er Hamiltonian system, and there are very few examples [25] of Hamiltonian system with fermionic v ariables in literat ur es. Additionally , to illustrate the p oten tial applications of the fin ite-dimensional sup er Hamiltonian system obtained in previous sections, w e w ould lik e to sh o w follo wing t w o p oints of ph ysics and mathematics, resp ectiv ely . On the one hand, it is p ossible to fi nd these systems in th e finite-dimensional sup er physic al theory in the f uture. F or example, the sup er an alogue of the integrable Roso c hatius deformation b ecause fi nite-dimensional in tegrable Roso c hatus systems, whic h are imp ortant in tegrable structures in string th eory , can b e obtained [26] through nonlinearizati on of the AKNS system. On the other hand, finite-dimensional C. Neumann system [27, 28], w hic h describ es the motion of a particle on S N − 1 with a qu adratic p oten tial in N -dimensional space, is deriv ed again from the AKNS b y non lin earizatio n. So, it is p ossible to establish corresp onding sup er C. Neumann in the sup ersymmetry case through the n onlinearization of the sup er AKNS system. Ho wev er, for the sup er AKNS system, we ha v e n ot y et foun d another kind of imp licit sym- metry constrain t, w hic h will engend er finite-dimensional in tegrable su p er Hamiltonian system. The d iffi cu lt y lies in the selection of new v ariables. Once we find prop er new v ariables, the metho d of nonlinearization for the sup er AKNS system under new implicit sym metry constrain t will b e carried out. In addition, under implicit constraints, n on lin earizatio n of the other sup er systems will b e studied in our fu ture w ork. 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