TheBargmannsymmetryconstraintandbinarynonlinearizationofthesuperDiracsystems.doc

the bargmann symmetry constraint and binary nonlinearization of the super dirac systemsjingsong hea , jing yua , wen-xiu mab , yi chengaa department of mathematics, university of science and technology of china, hefei, anhei, 230026, p. r. chinab department of mathematics and statistics, university of south florida,tampa, fl 33620-5700, usaabstractan explicit bargmann symmetry constraint is computed and its associated binary non- linearization of lax pairs is carried out for the super dirac systems. under the obtained symmetry constraint, the n-th flow of the super dirac hierarchy is decomposed into two super finite-dimensional integrable hamiltonian systems, defined over the supersymmetry manifold r4n |2n with the corresponding dynamical variables x and tn . the integrals of motion required for liouville integrability are explicitly given.key words: symmetry constraints, binary nonlinearization, super dirac systems, super finite-dimensional integrable hamiltonian systems.pacs codes(2008): 02.30.ik, 02.90.+p,1 introductionfor almost twenty years, much attention has been paid to the construction of finite-dimensional integrable systems from soliton equations by using symmetry constraints. either (2+1)-dimensional soliton equations 1, 2, 3 or (1+1)-dimensional soliton equations 4, 5 can be decomposed into compatible finite-dimensional integrable systems. it is known that a crucial idea in carrying out symmetry constraints is the nonlinearization of lax pairs for soliton hierarchies, and symmetry corresponding author, e-mail address: , tel:86-551-3600217, fax:86-551-36010053constraints give relations of potentials with eigenfunctions and adjoint eigenfunctions of lax pairs so that solutions to soliton equations can be obtained by solving jacobi inversion problems 6. the nonlinearization of lax pairs is classified into mono-nonlinearization 7, 8, 9 and binary nonlinearization 4, 5 10, 11, 12.the technique of nonlinearization has been successfully applied to many well-known (1+1)- dimensional soliton equations, such as the akns system 48, the kdv equation 5 and the dirac system 13. moreover, studies provide many examples of supersymmetry integrable sys- tems, with super dependent variables and/or super independent variables 14, 15, 16, 17, 18, 19. but there exist few results on nonlinearization of super integrable systems in the literature. very recently, nonlinearization was made for the super akns system 20 and the corresponding super finite-dimensional hamiltonian systems were generated. in this paper, we would like to analyze binary nonlinearization for the super dirac systems under a bargmann symmetry constraint.the paper is organized as follows. in the next section, we will recall the super dirac soliton hierarchy and its super hamiltonian structure. then in section 3, we compute a bargmann symmetry constraint for the potential of the super dirac hierarchy. in section 4, we apply binary nonlinearization to the super dirac hierarchy, and then obtain super finite-dimensionalintegrable hamiltonian systems on the supersymmetry manifold r4n |2n , whose integrals ofmotion are explicitly given. some conclusions and remarks are given in section 5.2 the super dirac hierarchythe super dirac spectral problem associated with the lie super-algebra b(0, 1) is given by 21r + s r s 1 x = u , u = + s r , u = , = 2 , (1) 0 3where is a spectral parameter, r and s are even variables, and and are odd variables.takingc a + b ,v = a b c 0the co-adjoint equation associated with (1) vx = u, v givesx a = 2c + 2rb + , bx = 2ra 2sc ,cx = 2a 2sb + + ,(2)if we set x = (a + b) c + ( + s) + r, x = ( + s) r (a b) + c.a = x ai i , b = x bi i , c = x ci i , = x i i , = x i i , (3)i0equation (2) is equivalent toi0i0i0i0 a0 = c0 = 0 = 0 = 0,11 a = 1 c+ sb , i 0,i+12 i,xi2 i2 ici+1 = 1 ai,x + rbi 1 i + 1 i , i 0,2 2 2(4) i+1 = i,x ri + si (ai bi ) + ci , i 0, i+1 = i,x ri si + (ai + bi ) + ci , i 0, bi+1,x = 2rai+1 2sci+1 i+1 i+1 ,i 0,which results in a recursion relation (ci+1 , ai+1 , i+1 , i+1 )t = l(ci , ai , i , i )t , i 0, bi = 1(2rai 2sci i i ), i 0,(5)where2r1s 1 + 2r1r1 r11 + r12 2 2 1 2s1s 2s1 r 1 s1 1 + s1 l.= 2 2 2 21s + 21 r s 1 + r + 1 + 21 s 21r + r + 1 s 1upon choosing the initial conditionsa0 = c0 = 0 = 0 = 0, b0 = 1,all other ai , bi , ci , i , i , i 1, can be worked out uniquely by the recursion relation (5). the first few results are as followsa1 = s, b1 = 0, c1 = r, 1 = , 1 = ,11 2 2 17a2=2 rx, b2 = 2 (r+ s ) + , c2 = 2 sx, 2 = x, 2 = x,11 2 2 11a3 = 4 sxx + 2 (r1+ s )s + s 2 x + 2 x,b3 = 2 (rsx rxs) + x + x,11 2 2 11c3 = 4 rxx + 2 (r+ s )r + r + 2 x 2 x,1 2 2 113 = xx + 2 (r+ s ) 2 rx 2 sx rx sx,n1 2 2 113= xx + 2 (r+ s ) + 2 rx 2 sx sx + rx.let us associate the spectral problem (1) with the following auxiliary spectral problemtn = v(n) = (nv )+ , (6)withciai + bi i v (n) = x ai bici i i=0 ii0 ni ,where the plus symbol ”+” denotes taking the non-negative part in the power of .the compatible conditions of the spectral problem (1) and the auxiliary spectral problem(6) areut(n)n vx+ u, vwhich infer the super dirac soliton hierarchy(n) = 0, n 0, (7)tutn = kn = (2an+1 , 2cn+1 , n+1 , n+1 ), n 0. (8)here utn = kn in (8) is called the n-th dirac flow of the hierarchy.using the super trace identity 21, 22 z str(v u )dx = ( )str( u v ), (9)u uwhere str means the super trace, we can have ci+1 ai+1=hi , hi =z bi+2dx, i 0. (10) i+1ui+1i + 1therefore, the super dirac soliton hierarchy (8) can be written as the following super hamil-tonian form:whereutn = jhn , (11)u 02 0 0 j = 2 0 0 0 001000022122is a supersymplectic operator, and hn is given by (10).the first non-trivial nonlinear equation of hierarchy (11) is given by the second dirac flow12 2 t2r= 2 sxx + (r+ s )s + 2s x + x,12 22 t2 = xx +1 (r2+ s2 ) + rx sx +1 rx 1 sx,2t2 = xx 1 (r2+ s2) + rx + sx +1 rx +1 sx,which possesses a lax pair of u defined in (1) and v (2) defined byr 1 sx 2 + s + 1 rx + 1 (r2 + s2) + x 2 2 2v (2) = 2 + s + 1 rx 1 (r2 + s2) r + 1 sx + x .2 2 2 + x + x 03 the bargmann symmetry constraintin order to compute a bargmann symmetry constraint, we consider the following adjoint spectral problem of the spectral problem (1):r s 1 x = u st = s r , = 2 , (13) 03where st means the super transposition. the following result is a general formula for the variational derivative with respect to the potential u (see 423 for the classical case).lemma 1 let u (u, ) be an even matrix of order m + n depending on u, ux, uxx, and a parameter . suppose that = (e , o )t and = (e , o )t satisfy the spectral problem and the adjoint spectral problemx = u (u, ), x = u st (u, ),where e = (1, , m ) and e = (1, , m ) are even eigenfunctions, and o = (m+1 , , m+n )and o = (m+1 , , m+n ) are odd eigenfunctions. then, the variational derivative of the spec- tral parameter with respect to the potential u is given by = (e , (1)p(u)o )( u )u , (14)where we denoteu r t ( u )dxp(v) = 0, v is an even variable, 1, v is an odd variable.(15)by lemma 1, it is not difficult to find that 11 22= 12 + 21 . (16)uwhen zero boundary conditions lim 13 + 3223 31 = lim = 0 are imposed, we can obtain a|x|x|characteristic property a recursion relation for the variational derivative of : l u = u , (17)uwhere l and are given by (5) and (16), respectively.let us now discuss two spatial and temporal systems: 1j 1j r j + s 1j 2j = u (u, j ) 2j = j + s r 2j , 3jx3j 0 3j 1j 1j r j s 1j (18) 2j = u st (u, j ) 2j = j s r 2j ; 3jx3j 0 3j111j 1j 2j = v (n) (u, j ) 2j 3jtnnpi=0 n3jjci ninpi=0j(ai + bi )ni nnpi=0nji ni 1j = p(ai bi )ni p ci nip i ni , i=0jnp i nii=0npji nij i=002j 3ji=0jji=0 1j 1j 2j = (v (n)st (u, j ) 2j 3jtn nj p ci ni3jnp (ai bi )ninp i ni ni=0j i=0njj i=0n1j= pi=0(ai + bi )nipji=0ci ni pji=0i ni ; 2j npi ninp i ni03ji=0jji=0(19)where 1 j n and 1, , n are n distinct spectral parameters. now for the systems (18)and (19), we have the following symmetry constraintsn jk h = x , k 0. (20)u uj=1the symmetry constraint in the case of k = 0 is called a bargmann symmetry constraint 12.it leads to an explicit expression for the potential u, i.e.,1 1 2 2 r = , s = + , = + , = + ,(21)where we use the following notation,i = (i1 , , in )t , i = (i1, , in )t , i = 1, 2, 3,and denotes the standard inner product of the euclidian space rn .4 binary nonlinearizationin this section, we want to perform binary nonlinearization for the lax pairs and adjoint lax pairs of the super dirac hierarchy (11). to this end, let us substitute (21) into the lax pairs and adjoint lax pairs (18) and (19), and then we obtain the following nonlinearized lax pairs and adjoint lax pairs 1j 1j rj + s 1j 2j = u (u, j ) 2j = j + sr 2j , 3jx3j 03j 1j 1j rj s 1j (22) 2j = u st (u, j ) 2j = j sr 2j ; 3jx3j 03j1j 1j 2j = v (n) (u, j ) 2j 3jntnpi=03ji jc ninpi=0i(aj+ bi )ninpi=0ji ni 1j n= p(a b )nin p c nin,p ni iij i=0ni j i=0ni j i=02j p ni p3jii=0j ji=0i ni 0 1j 1j 2j = (v (n)st (u, j ) 2j 3jtn n p c ni3jnp (ai bi )ninip ni ini=0j i=0njj i=0n1jp iijp jp j = i=0(a+ b )nii=0ci nii=0i ni 2j ,npi nini p ni03ji=0 jj i=0(23)where 1 j n and p means an expression of p (u) under the explicit constraint (21). note that the spatial part of the nonlinearized system (22) is a system of ordinary differential equations with an independent variable x, but for a given n (n 2), the tn -part of the nonlinearized system(23) is a system of ordinary differential equations. obviously, the system (22) can be written as 1,x = ( )1 + (+ + )2 + ( + )3, 2,x = (+ + )1 ( )2 + (2+ )3, 3,x = ( + )1 ( + )2, 1,x = ( )1 + ( )1 + (+ )3 , 2,x = (+ + )1 + ( )2 ( 1+ )3 ,3,x = ( + )1 ( + )2,(24)where = diag(1 , , n ).when n = 1, the system (23) is exactly the system (22) with t1 = x. when n = 2, the system(23) is12112 2 = (r 2 s ) + (1,t2 x 1+ s + 2 rx + 2 (r+ s ) + )2 + ( x)3 ,1 2 2 = (2 + s + 1 r (r + s ) ) (r 1 s )+ ( + ) ,2,t22 x 21 2 x 2 x 33,t2 = ( + x)1 ( ) ,x212 112 2 (25) 1,t2 = (r 2 sx)1 + ( s 2 rx +2 (r+ s ) + )2 + ( + x)3,2 112 2 1 2,t2 = (+ s + 2 rx + 2 (r+ s ) + )1 + (r 2 sx)2 ( x)3,x3,t2 = ( )1 ( + x)2 ,wherer, s, , denotes the functions r, s, , defined by the explicit constraint (21), andxrx, sx, x, are given by2 2x 1 2 2 1 1 2r = 2 +2 +2 2 , sx = 2 +2 +2( )( ), x = + +( )( + ),x = ( )( + ),which are computed through using the spatial constrained flow (24).in what follows, we want to prove that the system (22) is a completely integrable hamiltonian system in the liouville sense. furthermore, we shall prove that the system (23) is also completely integrable under the control of the system (22).on the one hand, the system (22) or (24) can be represented as the following super hamil-tonian form 1,x =h11, 2,x =h12, 3,x =h1 ,3(26)where 1,x = h11, 2,x = h121, 3,x =h1 ,32h1 = + 2 ( )1+ ( + )22+( + )( + ).in addition, the characteristic property (17) and the recursion relation (5) ensure that a= + ,i 0,i+1 1 2 2 1 bi+1 = ,i 0,ci+1= ,i 0,(27) i+1 = + ,i 0, i+1 = + ,i 0.xthen the co-adjoint representation equation vxthat the equality v 2 = u , v 2 is also true. let= u , v remains true. furthermore, we knowf = 1 strv42. (28)then it is easy to find that fx = 0, that is to say, f is a generating function of integrals ofnm