New Integrable Decomposition of Super AKNS Equation

Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 803 808 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 5, November 15, 2010 New Integrable Decomposition of Super AKNS Equation JI Jie ( ) and DONG Ya-Juan (þ ) College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China (Received July 9, 2010; revised manuscript received August 23, 2010) Abstract In this paper, a new 7 7 matrix spectral problem, which is associated with the super AKNS equation is constructed. With the use of the binary nonlinearization method, a new integrable decomposition of the super AKNS equation is presented. PACS numbers: 02.30.Ik Key words: spectral problem, integrable decomposition, super AKNS equation 1 Introduction The nonlinearization method presented a way to obtain integrable Hamiltonian systems and solutions of a partial differential equation by decomposing it into two ordinary differential equations, which is an important and interesting topic in soliton theory. The crucial technique is to obtain constraints between the potentials and the eigenfunctions of the spectral problem associated with the soliton equation. With different constraints of the corresponding soliton equation, we can deduce new integrable Hamiltonian systems and obtain general solutions of soliton equation. Hence, how to find new constraints of the soliton equation has aroused strong interests in soliton theory. [1 12] As we all know, supersymmetry (susy) integrable systems are very interesting topic in soliton theory. [13 20] In recently, the corresponding binary nonlinearization of the super AKNS spectral problem was presented, a supertrace identity on Lie superalgebras for the super spectral problem was established for the super spectral also, the corresponding bi-super-hamiltonian structures of a super- AKNS soliton hierarchy and a super-dirac soliton hierarchy were presented respectively. [20 22] He et al. established the binary nonlinearization approach of the spectral problem of the super AKNS system and obtained the super finite-dimensional integrable Hamiltonian system. According to the existent works, it is nature for us to search different constraints associated new integrable decompositions by constructing different higher-spectral problems. [21 22] Super spectral problem is different from the usual soliton spectral problem, in this discussions, we can find the way to construct higher super spectral problem from a 3 3 super spectral problem is interesting. In this paper, we discuss an example to construct a 7 7 super spectral problem which associated with the super AKNS equation from a 3 3 super AKNS spectral problem. In Sec. 2, we recall some results of hierarchy of super AKNS equation obtained from super AKNS spectral problem in 3 3 matrix representation as a preparatory knowledge for further consideration. In Sec. 3, we formulize a new 7 7 spectral problem from the 3 3 super AKNS spectral problem. It is found that the isospectral evolution equations of the new 7 7 spectral problem turns out to be the super AKNS equation hierarchy. The binary restricted super AKNS flow and its Lax representation are presented, the decompositions of the AKNS equation obtained which is integrable in the sense of Liouville which is different from the existing results. [21 22] 2 Hierarchy of Super AKNS Equation Consider the following super AKNS matrix spectral problem [21] Φ x = UΦ, q λ q u U = U(û) = r λ v r, û = u, (1) v u 0 v where λ, q, and r are even elements such that p(λ) = p(q) = p(r) = 0, but u and v are odd elements such that p(u) = p(v) = 1. λ is an eigenparameter of this system. q, r, u, and v are functions of the usual time-space variables x and t. Here small p(f) denotes the parity of the arbitrary function f. Take a b d a i b i d i V = c a e c i a i e i λ i. (2) e d 0 i=0 e i d i 0 Then, the adjoint representation equation V x = Supported by the National Natural Science Foundation of China under Grant No. 10926036, the Education Department of Zhejiang Province under Grant No. Y200906909 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. Y6090172 Corresponding author, E-mail: jixjiecn@yahoo.com.cn

804 JI Jei and DONG Ya-Juan Vol. 54 [U, V ] = UV V U yields a 1 = 0, b 1 = q, c 1 = r, d 1 = u, e 1 = v, Choose a 0 = 1, then b 0 = c 0 = d 0 = e 0 = 0, a i,x = qc i rb i + ue i + vd i, b i,x = 2(b i+1 + qa i + ud i ), c i,x = 2(c i+1 + ra i + ve i ), i 1, d i,x = d i+1 + qe i ua i vb i, e i,x = e i+1 + rd i uc i + va i. (3) L = where a 2 = 1 2 qr + uv, b 2 = 1 2 q x, c 2 = 1 2 r x, d 2 = u x, e 2 = v x, c n+1 b n+1 e n+1 d n+1 L c n b n e n d n 1 2 r 1 q r 1 r v r 1 u r 1 v q 1 q 1 + q 1 r q 1 u u q 1 v u v 1 q v 1 r v 1 u r v 1 v u 1 q v + u 1 r q u 1 u u v, (4) Consider the auxiliary spectral problem Φ tn = V (n) Φ, where n a i b i d i V (n) = (λ n V ) + = c i a i e i λ n i, (5) i=0 e i d i 0 and zero curvature equation U tn V x (n) + [U, V (n) ] = 0 which gives q r r û tn = = πl n q u v πδh n δû, (6) v u where [15] 0 2 0 0 2 0 0 0 π = 0 0 0 1, H n = 0 0 1 0 For n = 2, we obtain the famous AKNS equation t n 2an+2 n + 1 dx, b n+1 c n+1 2d n+1 2e n+1. q t2 = 1 2 q xx + q 2 r + 2quv 2uu x, r t2 = 1 2 r xx qr 2 2ruv 2vv x, δh n δû, n 0. (7) u t2 = u xx qv x + 1 2 qru 1 2 q xv, v t2 = v xx + ru x + 1 2 r xu 1 qrv, (8) 2 and the corresponding Lax operators are λ q u V (1) = r λ v U, V (2) = v u 0 λ 2 + 1 2 qr + uv qλ 1 2 q x uλ u x rλ + 1 2 r x λ 2 1 2 qr uv vλ + v x. (9) vλ + v x uλ + u x 0 3 A New 7 7 Spectral Problem and One New Integrable Decomposition 3.1 A New 7 7 Spectral Problem Assume ( φ 1, φ 2, φ 3, φ 4, φ 5, φ 6, φ 7 ) = (φ 3 1, φ3 2, φ2 1 φ 2, φ 1 φ 2 2, φ2 1 φ 3, φ 1 φ 2 φ 3, φ 2 2 φ 3), we can get a new 7 7 spectral problem is obtained as follows 3λ 0 3q 0 3u 0 0 φ 1 0 3λ 0 3r 0 0 3v φ 2 r 0 λ 2q v 2u 0 φ Φ x = Ū(û, λ) Φ, 3 Ū(û, λ) = 0 q 2r λ 0 2v u, Φ = φ 4, (10) v u 0 0 2λ 2q 0 φ 5 0 0 v u r 0 q φ 6 0 u 0 v 0 2r 2λ φ 7

No. 5 New Integrable Decomposition of Super AKNS Equation 805 where q, r are still potentials and λ is a spectral parameter, the parameters and variants in this article are all constant. We will show that the isospectral evolution equations of the new 7 7 spectral problem (10) is nothing but the famous super AKNS equation hierarchy. Take 3a 0 3b 0 3d 0 0 0 3a 0 3c 0 0 3e c 0 a 2q e 2d 0 V = 0 b 2c a 0 2e d e d 0 0 2a 2b 0 0 0 e d c 0 b 0 d 0 e 0 2c 2a = i 0 3a i 0 3b i 0 3d i 0 0 0 3a i 0 3c i 0 0 3e i c i 0 a i 2q i e i 2d i 0 0 b i 2c i a i 0 2e i d i e i d i 0 0 2a i 2b i 0 0 0 e i d c i 0 b i 0 d i 0 e i 0 2c i 2a i λ i, with simply calculation and the adjoint representation equation V x = [Ū, V ] which leads to (3) and (4). Take the auxiliary spectral problem as φ 1 3a i 0 3b i 0 3d i 0 0 φ 2 0 3a i 0 3c i 0 0 3e i φ 3 n c i 0 a i 2q i e i 2d i 0 (n) Φ tn = V φ 4, V (n) = 0 b i 2c i a i 0 2e i d i λ n i, (11) φ i=0 5 e i d i 0 0 2a i 2b i 0 φ 6 0 0 e i d c i 0 b i φ 7 0 d i 0 e i 0 2c i 2a i and zero curvature equation Ūt n 3.2 Binary Restricted Super AKNS Flows V (n) x + [Ū, V (n) ] = 0 which just lead to the super AKNS equation hierarchies (6). Consider N distinct constant λ 1,...,λ N and the corresponding spectral problem and the adjoint spectral problem ( φ 1j, φ 2j, φ 3j, φ 4j, φ 5j, φ 6j, φ 7j ) x = Ū(û, λ j)( φ 1j, φ 2j, φ 3j, φ 4j, φ 5j, φ 6j, φ 7j ), 1 j N, ( ψ 1j, ψ 2j, ψ 3j, ψ 4j, ψ 5j, ψ 6j, ψ 7j ) x = ŪT (û, λ j )( ψ 1j, ψ 2j, ψ 3j, ψ 4j, ψ 5j, ψ 6j, ψ 7j ), 1 j N. (12) From Ref. [20], we have δh k δu = b k+1 c k+1 2d k+1 2e k+1 δû = N j=1 δq δr δu δv δr δq δv δu 3 φ 3j ψ1j + 2 φ 4j ψ3j + φ 2j ψ4j 2 φ 6j ψ5j φ 7j ψ6j 3 φ 4j ψ2j + φ 1j ψ3j + 2 φ 3j ψ4j φ 5j ψ6j + 2 φ 6j ψ7j 3 φ 5j ψ1j + 2 φ 6j ψ3j + φ 1j ψ4j + φ 2j ψ5j + φ 2j ψ7j 3 φ 7j ψ2j + φ 5j ψ3j + 2 φ 6j ψ4j φ 1j ψ5j φ 3j ψ6j φ 4j ψ7j As the usual approach of binary nonlinearization of Lax pairs, we consider the constraints 3 Q 4, P 2 + Q 1, P 3 + 2 Q 3, P 4 Q 5, P 6 + 2 Q 6, P 7 3 Q 3, P 1 + 2 Q 4, P 3 + Q 2, P 4 2 Q 6, P 5 Q 7, P 6 which lead to finite-dimensional integrable systems, where δh 0 δû = 3 Q 7, P 2 + Q 5, P 3 + 2 Q 6, P 4 Q 1, P 5 Q 3, P 6 Q 4, P 7 3 Q 5, P 1 + 2 Q 6, P 3 + Q 1, P 4 + Q 2, P 5 + Q 4, P 6 + Q 2, P 7 Λ = diag(λ 1, λ 2,...,λ N ), Qi = ( φ ii,..., φ in ) T, Pi = ( ψ i1,..., ψ in ) T, i = 1, 2, 3, 4, 5, 6, 7. For k = 0, the constraint is q r b 1 c 1 2d 1 2e 1 Under this constraint and 2u 2v 3 Q 4, P 2 + Q 1, P 3 + 2 Q 3, P 4 Q 5, P 6 + 2 Q 6, P 7 3 Q 3, P 1 + 2 Q 4, P 3 + Q 2, P 4 2 Q 6, P 5 Q 7, P 6 3 Q 7, P 2 + Q 5, P 3 + 2 Q 6, P 4 Q 1, P 5 Q 3, P 6 Q 4, P 7 3 Q 5, P 1 + 2 Q 6, P 3 + Q 1, P 4 + Q 2, P 5 + Q 4, P 6 + Q 2, P 7 ( φ 1j, φ 2j, φ 3j, φ 4j, φ 5j, φ 6j, φ 7j ) tn = V (n) (û, λ j )( φ 1j, φ 2j, φ 3j, φ 4j, φ 5j, φ 6j, φ 7j ),. (13), (14). (15) ( ψ 1j, ψ 2j, ψ 3j, ψ 4j, ψ 5j, ψ 6j, ψ 7j ) tn = ( V (n) ) T (û, λ j )( ψ 1j, ψ 2j, ψ 3j, ψ 4j, ψ 5j, ψ 6j, ψ 7j ), (16) we can obtain a hierarchy of finite-dimensional Hamiltonian systems.

806 JI Jei and DONG Ya-Juan Vol. 54 For n = 1, the spectral problem (16) are nonlinearized to the following Hamiltonian systems, x-flows where Q ix = H 1 P i, Pix = H 1 Q i, i = 1, 2, 3, 4, 5, 6, 7, (17) H 1 = 3 Λ Q 1, P 1 + 3 Λ Q 2, P 2 Λ Q 3, P 3 + Λ Q 4, P 4 2 Λ Q 5, P 5 + 2 Λ Q 7, P 7 + (3 Q 4, P 2 + Q 1, P 3 + 2 Q 3, P 4 Q 5, P 6 + 2 Q 6, P 7 ) (3 Q 3, P 1 + 2 Q 4, P 3 + Q 2, P 4 2 Q 6, P 5 Q 7, P 6 ) 1 2 (3 Q 7, P 2 + Q 5, P 3 + 2 Q 6, P 4 Q 1, P 5 Q 3, P 6 Q 4, P 7 ) (3 Q 5, P 1 + 2 Q 6, P 3 + Q 1, P 4 + Q 2, P 5 + Q 4, P 6 + Q 2, P 7 ). For n = 2, the spectral problem (16) are nonlinearized to the following Hamiltonian systems, t-flows where Q it2 = H 2 P i, Pit2 = H 2 Q i, i = 1, 2, 3, 4, 5, 6, 7, (18) H 2 = 3 Λ 2 Q1, P 1 + 3 Λ 2 Q2, P 2 Λ 2 Q3, P 3 + Λ 2 Q4, P 4 2 Λ 2 Q5, P 5 + 2 Λ 2 Q7, P 7 + (3 Λ Q 4, P 2 + Λ Q 1, P 3 + 2 Λ Q 3, P 4 Λ Q 5, P 6 + 2 Λ Q 6, P 7 ) (3 Q 3, P 1 + 2 Q 4, P 3 + Q 2, P 4 2 Q 6, P 5 Q 7, P 6 ) + (3 Q 4, P 2 + Q 1, P 3 + 2 Q 3, P 4 Q 5, P 6 + 2 Q 6, P 7 ) (3 Λ Q 3, P 1 + 2 Λ Q 4, P 3 + Λ Q 2, P 4 2 Λ Q 6, P 5 Λ Q 7, P 6 ) 1 2 (3 Λ Q 7, P 2 + Λ Q 5, P 3 + 2 Λ Q 6, P 4 Λ Q 1, P 5 Λ Q 3, P 6 Λ Q 4, P 7 ) (3 Q 5, P 1 + 2 Q 6, P 3 + Q 1, P 4 + Q 2, P 5 + Q 4, P 6 + Q 2, P 7 ) 1 2 (3 Q 7, P 2 + Q 5, P 3 + 2 Q 6, P 4 Q 1, P 5 Q 3, P 6 Q 4, P 7 ) (3 Λ Q 5, P 1 + 2 Λ Q 6, P 3 + Λ Q 1, P 4 + Λ Q 2, P 5 + Λ Q 4, P 6 + Λ Q 2, P 7 ) + 1 2 ( 3 Q 1, P 1 + 3 Q 2, P 2 Q 3, P 3 + Q 4, P 4 2 Q 5, P 5 + 2 Q 7, P 7 ) (3 Q 4, P 2 + Q 1, P 3 + 2 Q 3, P 4 Q 5, P 6 + 2 Q 6, P 7 ) (3 Q 3, P 1 + 2 Q 4, P 3 + Q 2, P 4 2 Q 6, P 5 Q 7, P 6 ) 1 4 ( 3 Q 1, P 1 + 3 Q 2, P 2 Q 3, P 3 + Q 4, P 4 2 Q 5, P 5 + 2 Q 7, P 7 ) (3 Q 7, P 2 + Q 5, P 3 + 2 Q 6, P 4 Q 1, P 5 Q 3, P 6 Q 4, P 7 ) (3 Q 5, P 1 + 2 Q 6, P 3 + Q 1, P 4 + Q 2, P 5 + Q 4, P 6 + Q 2, P 7 ). We can prove that if P, Q satisfy (17) and (18) then q, r defined by constraint (15) solves the super AKNS equation (8). A new decomposition of the super AKNS equation (8) can be presented. From the spectral problems (12) and the zero boundary conditions, we can find the following result We can obtain a n = 1 2 ( 3 Λn 1 Q1, P 1 + 3 Λ n 1 Q2, P 2 Λ n 1 Q3, P 3 + Λ n 1 Q4, P 4 2 Λ n 1 Q5, P 5 + 2 Λ n 1 Q7, P 7 ), n 1. L δû = λ δû. (19) b n = 3 Λ n 1 Q4, P 2 + Λ n 1 Q1, P 3 + 2 Λ n 1 Q3, P 4 Λ n 1 Q5, P 6 + 2 Λ n 1 Q6, P 7, c n = 3 Λ n 1 Q3, P 1 + 2 Λ n 1 Q4, P 3 + Λ n 1 Q2, P 4 2 Λ n 1 Q6, P 5 Λ n 1 Q7, P 6, d n = 1 2 (3 Λn 1 Q7, P 2 + Λ n 1 Q5, P 3 + 2 Λ n 1 Q6, P 4 Λ n 1 Q1, P 5 Λ n 1 Q3, P 6 Λ n 1 Q4, P 7 ), e n = 1 2 (3 Λn 1 Q5, P 1 + 2 Λ n 1 Q6, P 3 + Λ n 1 Q1, P 4 + Λ n 1 Q2, P 5 + Λ n 1 Q4, P 6 + Λ n 1 Q2, P 7 ). (20)

No. 5 New Integrable Decomposition of Super AKNS Equation 807 From the adjoint representation equation V x = [Ū, V ], V 2 x = [Ū, V 2 ] is also satisfied. Therefore F x = 1 12 (StrV 2 ) x = d dx (a2 + bc + uv) = 0. The identity indicates that F is a generating function of integrable of motion for the nonlinearized spatial systems (12). Assume F = F i λ i, i 0 we can obtain the following formulas F 1 = 2a 1 = 3 Q 1, P 1 + 3 Q 2, P 2 Q 3, P 3 + Q 4, P 4 2 Q 5, P 5 + 2 Q 7, P 7, n 1 F n = (a i a n i + b i c n i + 2d i e n i ) + 2a 0 a n i=1 n 1 [ 1 = 4 ( 3 Λi 1 Q1, P 1 + 3 Λ i 1 Q2, P 2 Λ i 1 Q3, P 3 + Λ i 1 Q4, P 4 i=1 2 Λ i 1 Q5, P 5 + 2 Λ i 1 Q7, P 7 )( 3 Λ n i 1 Q1, P 1 + 3 Λ n i 1 Q2, P 2 Λ n i 1 Q3, P 3 + Λ n i 1 Q4, P 4 2 Λ n i 1 Q5, P 5 + 2 Λ n i 1 Q7, P 7 ) + (3 Λ i 1 Q4, P 2 + Λ i 1 Q1, P 3 + 2 Λ i 1 Q3, P 4 Λ i 1 Q5, P 6 + 2 Λ i 1 Q6, P 7 )(3 Λ n i 1 Q3, P 1 + 2 Λ n i 1 Q4, P 3 + Λ n i 1 Q2, P 4 2 Λ n i 1 Q6, P 5 Λ n i 1 Q7, P 6 ) 1 2 (3 Λi 1 Q7, P 2 + Λ i 1 Q5, P 3 + 2 Λ i 1 Q6, P 4 Λ i 1 Q1, P 5 Λ i 1 Q3, P 6 Λ i 1 Q4, P 7 )(3 Λ n i 1 Q5, P 1 + 2 Λ n i 1 Q6, P 3 + Λ n i 1 Q1, P 4 + Λ n i 1 Q2, P 5 + Λ n i 1 Q4, P 6 + Λ n i 1 Q2, P ] 7 ) 3 Λ n 1 Q1, P 1 + 3 Λ n 1 Q2, P 2 Λ n 1 Q3, P 3 + Λ n 1 Q4, P 4 2 Λ n 1 Q5, P 5 + 2 Λ n 1 Q7, P 7. As usual process, it is not difficult to see that F n (n 0) and f k = φ 1k ψ1k + φ 2k ψ2k + φ 3k ψ3k + φ 4k ψ4k + φ 5k ψ5k + φ 6k ψ6k + φ 7k ψ7k (1 k N) are also integrals of motion for Eq. (16) {F m+1, F n+1 } = t n F m+1 = 0, {F m, f k } = 0, {f m, f n } = 0. We can also obtain the following result (Refs. [6], [21], [22]). Proposition 1 The integrals of motion {f k } N k=1, {F n} 4N n=1 are functionally independent over some region of the supersymmetry manifold R 14N 6N. Here the definition of R M N is given by (x 1, x 2,...,x M, ξ 1, ξ 2,...,ξ N ) with x i R while x i are odd variables. Hence, we get the following theorem Theorem 1 The spatial constrainted flows (17) and the temporal constrained flows (18) of the super AKNS equation (8) are completely integrable in the sense of Liouville, the decomposition of the super AKNS equation (8) to (17) and (18) is an integrable decomposition. 4 Conclusions In this paper, we constructed a properly new 7 7 matrix spectral problem which is associated with the super AKNS equation from the super AKNS spectral problem in 3 3 matrix representation, a new integrable decomposition of the super AKNS equation is presented with the use of the binary nonlinearization method. In fact, if we assume ( φ 1, φ 2, φ 3, φ 4, φ 5 ) = (φ 2 1, φ2 2, φ 1 φ 2, φ 1 φ 3, φ 2 φ 3 ), we can get a new 5 5 super spectral problem is obtained as follows 2λ 0 2q 2u 0 0 2λ 2r 0 2v Φ x = Ū(û, λ) Φ, Ū(û, λ) = r q 0 v u, v 0 u λ q 0 u v r λ φ 1 φ 2 Φ = φ 3, (21) φ 4 φ 5 which is also associated with the super AKNS equation. We can find the odd function φ 3 should be bottom is the key of obtaining higher super spectral problem, which is deduced from 3 3 super spectral problem. Acknowledgments The authors express their appreciations to the referee and Doctor Jing Yu for for their valuable suggestions.

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