Journal of Modern Physics, 2013, 4, 5-11
doi:10.4236/jmp.2013.45B002 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Bargmann Symmetry Constraint and Binary
Nonlinearization of Super NLS-MKdV Hierarchy
Sixing Tao1*, Hui Shi2
1School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China
2School of Physics and Electronic Information, Shangqiu Normal University, Shangqiu, China
Email: *taosixing@163.com
Received 2013
ABSTRACT
An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is car-
ried out for the super NLS-MKdV hierarchy. Under the obtained symmetry constraint, the n-th flow of the super
NLS-MKdV hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over
the super-symmetry manifold with the corresponding dynamical variables x and tn. The integrals of motion re-
quired for Liouville integrability are explicitly given.
4|2
NN
Keywords: Symmetry Constraints; Binary Nonlinearization; Super NLS-MKdV Hierarchy; Super Finite Dimensional
Integrable Hamiltonian Systems
1. Introduction
For 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] or (1 +
1)-dimensional soliton equations [3,4] can be decom-
posed into compatible finite-dimensional integrable sys-
tems. It is known that a crucial idea in carrying out
symmetry constraints is the no nlinearization of Lax pairs
for soliton hierarchies. The nonlinearization of Lax pairs
is classified into mono-nonlinearization [5,6] and binary
nonlinea r i zation [7,8] .
The technique of nonlinearization has been success-
fully applied to many well-known (1+1)-dimensional
soliton equations, such as the AKNS hierarchy [3], the
KdV hierarchy [4] and the Dirac hierarchy [9]. But there
are few results on nonlinearization of super integrable
systems, existing in the literature. Very recently, nonlin-
earization were made for the super AKNS hierarchy , the
super Dirac hierarchy and their corresponding super fi-
nite-dimensional hierarchies were generated in Refs.
[10-12]. Dong presented the super Hamiltonian structures
of the super NLS-MKdV hierarchy [13]. In this paper,
we would like to consider the binary nonlinearization of
the super NLS- M Kd V hierarchy.
This paper is organized as follows. In the next section,
we will recall the super NLS-MKdV soliton hierarchy
and its super Hamiltonian structure. Then in Section 3,
we compute a Bargmann symmetry constraint for the
potential of the super NLS-MKdV hierarchy. In Section
4, we apply binary nonlinearization method to the super
NLS-MKdV hierarchy, and then obtain super finite di-
mensional integrable Hamiltonian hierarchy on the super
symmetry manifold , whose integrals of motion
are explicitly given.
4|2
NN
2. The Super NLS-MKdV Hierarchy
The super NLS-MKdV spectral problem associated with
the Lie super algebra is given by
(0,1)B
11
22
11
22
1
2
3
()
,()
0
,,
x
qr
UU qr
q
r
u
,









 
 



 
 

(1)
where
is a spectral parameter, q and r are even vari-
ables,
and
are odd var i ables[14]. Taking
1
2
1
2,
0
AB
VC A







the co-adjoint equation associated with Equation (1)
[,]
x
VUV
gives
*Corresponding author.
Copyright © 2013 SciRes. JMP
S. X. TAO, H. SHI.
6
1
2
1
2
()()22
()2,
()2,
22(
22(
x
x
x
x
x
AqrBqrC
BBqrA
CCqrA
ABqr
ACqr
,
),
).



 
 
 
 
 

 
,
i
i
(2)
If we set
000
00
,,
,,
ii
ii
iii
ii
ii
ii
AABBCC






 



(3)
then Equation (2) is equivalent to
1
1,
2
1
1,
2
1,
1,
1,111 1
() 2,
() 2,
22(),
2()2,
()()2 2,
iiixi
iiixi
iiiix i
iii iix
ixiii i
BqrAB
CqrAC
AB qr
ACqr
AqrBqrC i


  
 
 

 
 
 
 
 
0.
(4)
which results in the recurrence relations
T
11111 1
T
1
(,,4,4)
(,,4,4),
(()() 22),0.
iiiii i
iiiii i
iiiiiii
BCCB
LBC CB
AqCBrBC i


 
 
 

  
(5)
where
Upon choosing the initial conditions
0000 0
0, 1,BC A
 
all other ,,,,( 1)
iiiii
ABC i
can be worked out by the
recurre n c e r e l ations Equation (5). The fir s t few sets a re :
11
11 1
22
22
11
11 2
22
11 11
22
22 22
22
32 2
1111 1
32244 4
3
1
4
32 2
1111 1
32244 4
0,( ),( ),
,, 4,
,,
2, 2,
4
22,
4
xx xx
xx
xx xxx
xx xxx
ABqrCqr
Aqr
BqrC qr
Bq rqqrqr
rq r
Cqrqqrqr
 


 

 
 
 


 
 
3
1
4
22
11
322
22
11
322
22,
422
422
xx xxxx
xx xxxx
rq r
qr qrqr
qr qrqr
 
388.
x
xx
Aqrqr x

 
Let us associate the spectral problem Equation (1) with
the following aux iliary problem
() ()
n
nn
tVV,
 
 (7)
with
1
2
() 1
2
0,
0
iii
n
nn
iii
i
ii
AB
VCA








i
where the symbol “
” denotes taking the non-negative
part in the power of
.
The compatible conditions of the spectral problem
Equation (1) and th e auxiliary problem Equation (7) are
() ()
[,]0,
n
nn
tx
UV UV
 (8)
which infer the super NLS-MKdV hierarchy
T
11
11111 1
22
(,,,
0.
n
tnnnnnn n
uK BCBC
n

 
 
),
(9)
Here in Equation (9) is called the n-th NLS
–MKdV flow of this hierarchy.
n
n
t
uK
Using the super trace identity
StrdStr,
UU
Vx V
uu


 

 
 

 
(10)
where Str means the super trace [14,15], we have
11
11 2
1
1
,d,
41
4
ii
ii i
ii
i
i
BC
CB A
HH xi
ui




 0.
 



(11)
Therefore, the super NLS-MKdV soliton hierarchy
Equation (9) can be written as the following super Ham-
iltonian for m:
,
n
n
t
H
uJ
u
(12)
where
,
,
 
 

 

1
8
1
8
0100
10 00
.
000
00 0
J







is a super symplectic operator, and n
H
is given by
Equation (11).
111
11
22
1111
11
22
11 11
111
()(
()()
.
44 44222
44 44222
qrqq qq
rr rqrr
Lrq
rqqr
 
 
 
 

 
 
 

 

 


 


 

1
1
)
qr
(6)
Copyright © 2013 SciRes. JMP
S. X. TAO, H. SHI 7
The first non-trivial nonlinear equation of the super
NLS-MKdV hierarcy (9) is given by its second flow
2
2
2
2
23
11
22
32
11
22
22
11 11
22 44
22
11 11
22 44
44 4,
44 4,
2,
2.
txxx x
txxx x
txxxx xx
txxxxxx
qrqrr r
rqq qr q
qrqrqr
qrqrqr
 
 


 

 

(13)
which possesses a Lax pair of U defined in Equation (1)
and defined by
(2)
V
3. Bargmann Symmetry Constraint of Super
NLS-MKdV Hierarchy
In order to compute a Bargmann symmetry constraint,
we consider the following adjoint spectral problem of the
spectral problem:
11 1
22
11 2
22
3
()
() ,
0
St
x
qr
Uqr

 
 



 


 
,
(15)
where means the super transposition. The following
result is a general formula for the variational derivative
with respect to the potential u (see[3] for the classical
case).
St
Lemma 1 [10-12] Let (, )Uu
be an even matrix of
order depending on and a parame-
ter mn,,uu ,,
xxx
u
. Suppose tha t T
(,)
eo

and
satisfy the spectral problem and the adjoint spectral
problem
T
(,
eo

)
(, ),,
St
xx
Uu U


(16)
where

1,,
em

and
1,,
em

,,
om mn
 
om

are even
eigenfunctions, and and

,,mn
are odd eigenfunctions. Then the variational derivative
of the parameter
with respect to the potential u is
given by


()
,( 1),
d
pu U
eo
u
TU
ux



(17)
where we denote
0, ,
() 1, .
v is aneven variable
pv v isan oddvariable
(18)
By Lemma 1, it is not difficult to find that
11
12 21
22
11
12 21
22
13 32
23 31
1.
uE
 
 

 
 





(19)
where 1112 2
2(E
 
 
)d.x
If we consider zero
boundary conditions || ||
lim lim0,
xx

 
 we can
obtain a characteristic property- a recurrence relation for
the variational derivative of
:
,Luu

(20)
where L and u
are given by Equation (4) and Equation
(18), respectively.
Let us now discuss the spatial systems:
11
11
22
11
22
22
33
11
11
22
11
22
22
33
()
() ,
0
()
()
0
jj
jj
jj
x
jj
jj
jj
x
qr
qr
qr
qr
 


 


 

 

 
 


 

 
 


 

 
 


 

,


 
(21)
and the temporal systems:
222
11111
24422
(2)22 2
11 111
22 244
2()() 2
() ()22
22
xx x
xx x
xx
qrqrqr
Vqrqrqr


 

 

 




.
0
(14)
1
2
000
1 1
1
2 2
2
000
3 3
00
1
2
0
1
2
3
,
0
n
n
nnn
ni ni ni
ij ij ij
iii
j j
nnn
nini ni
jij ijij
iii
nn
ni ni
j j
tij ij
ii
nni n
ij ij
i
j
j
jt
AB
CA
AC




 








 

 


 

 

 










j
00
1
12
2
00 0
3
00
,
0
nn
ini
ij
iij
nn n
nini ni
ijijij j
ii i
nn
ni nij
ij ij
ii
BA


 



 












 




(22)
Copyright © 2013 SciRes. JMP
S. X. TAO, H. SHI.
8
where and 12
1jN ,,,N

are N distinct pa-
rameters. Now for systems Equation (21) and Equation
(22), we have the following symm e try constraints:
1,0.
(23)
Nj
kj
j
Hk
uu



The symmetry constraints in the case of 0k
is
called a Bargmann constraint (see[8]). If taking ij
E
1112 2
2(
jjj j
)d
x
 
 
, then it leads to an expression
for the potential , i.e.
u



112 21
2
121 12
2
123 31
4
113 32
4
,,
,,
,,
,,
q
r


 

,
,
,
,
(24)
where we use the following notation

TT
11
,,,,,, 1,2,3.
iiiNiiiN i 
and , denotes the standard inner product of the
Euclidean space
N
.
4. Binary Nonlinearization of Super NLS
–MKdV Hierarchy
In order to perform binary nonlineqrization to the super
NLS-MKdV hierarchy. To this end, let us substituting
Equation (24) into the Lax pairs and adjoint Lax pairs Equa-
tion (21) and Equation (22), and then we obtain the follow-
ing nonlinearized spatial Lax pairs and adjoint Lax pairs:
11
11
22
11
22
22
33
11
11
22
11
22
22
33
()
(),
0
()
()
0




jj
jj
jj
x
jj
jj
jj
x
qr
qr
qr
qr
 





 

 

 
 


 

 

 


 
 

 

 

 

,

(25)
and
where 1jN
and means an expression of
under the explicit constraint Equation (24). Note that the
spatial part of the nonlinearized system Equation (25) is a
system of ordinary differential equations with an inde-
pendent variables u, but for a given , the -
part of the nonlinearized system Equation (26) is a sys-
tem of ordinary differential equations. Obviously, the
system Equation (25) can be written as
P()Pu
n
t(2nn)
where 1
diag( ,,).N
When , the system
Equation (26) is exactly the system Equation (25) with
1
1n
tx
. The system Equation (25) or Equation (27) can be
written as the following super Hamiltonian form:
111
1, 2,3,
12
11
1, 2,3,
12
,,
,,
xxx
xxx
HHH
3
1
3
,
.
H
HH


 

 

(28)
where

11 1
11122233
22 4
13 32
,,,
,,.
H 

1
,
1
2
000
1 1
1
2 2
2
000
3 3
00
1
2
1
2
3
,
0

n
n
nnn
ni ni ni
ij ij ij
iii
j j
nnn
nini ni
jij ijij
iii
nn
ni ni
j j
tij ij
ii
ni
ij
i
j
j
jt
AB
CA
A




 







 

 


 

 

 









j
000
1
12
2
00 0
3
00
,
0

nnn
ni ni
ij ij
iij
nn n
nini ni
ijijij j
iii
nn
ni nij
ij ij
ii
C
BA


 



 












 




(26)




111
1,12 12233 13
22 4
111
2,1 21213323
224
11
3,133 2123312
44
111
1,11 221 3323
224
111
2,211223313
224
1
3,2 3
4
,,,
,,,
,, ,,
,,,
,,
,
x
x
x
x
x
x


 
  

 
,
,
,
,
,,

1
31 113322
4
,,,
,
 
(27)
Copyright © 2013 SciRes. JMP
S. X. TAO, H. SHI 9
When , the system Equation (26) is 2n







2
2
2
2
222
1111 1
1,1 2
2442 2
3
222
11 111
2, 12
22 244
3
3,1 2
2
1

1
1, 24
2()()
2,
() ()2
2,
22,




 
txx
x
txx
x
tx x
t
qrqrqr
qrq rqr




 
 
 
 
 










2
2
22
111
12
422
3
222
11 111
2,1 2
22 244
3
3,1 2
2()()
2,
() ()2
2,
22,
 



 
xx
x
txx
x
tx x
qrqrqr
qrq rqr




 

 
 
 
  
(29)
where ,, ,

qr
, and ,, ,

x
xxx
qr
denote the functions ,, ,qr
defined by the explicit constraint Equation (24)
are given by





111
211212211122
224
111
21121221 1122
224
111
2331112223 31
8816
11 1
133211 22 13 32
8816
,,,,,,
,,,,,,
,,,,,,
,,,,,,
x
x
x
x
q
r
 
  
  
  
,
,
,
.
(30)
re compuatial con
(27).
The system Equation (29) can be represented as the
following super Hamiltonian form:
which ated through using the spstrained
flow Equation
222
222
22
1, 2,3,
12
22
1, 2, 3,
12
2
3
2
,,,
,,.
ttt
ttt
HHH
3
H
HH

 
 

 
 
(31)
where




22
11
211 22
22
1
,,
,,,

 
21 12 1122
4
123311332
8
112 2
11
21 122112
22
123311332
4
123 3
4
,
,,,,
,,
+,,,,
,, ,,
,,
H



  
 


11332
,,.
In what follows, we want to prove that Equation (25)
is a completely integrable Hamiltonian system in the
Liouville sense. Furthermore, we shall prove that Equa-
tion (26) is also completely integrable under the contron
of system Equation (25).
quation (20
and the recurrence relations Equation (5) ensure that
In addition, the characteristic property E)


1
11111
2
1
121
2
1
112
2
1
12331
4
1
11332
4
,,,
,,
0,
,, 0,
,,, 0,
,,, 0.
ii
i
i
i
i
i
ii
i
ii
i
Ai
Bi
Ci
i
i
 

0,
 

 
(32)
Then the co-adjoint representation equation
remains true. Furthermore, we know that
also true. Let
[,]

x
VUV
22
[, ]

VUV is
x
2
Str.
F
V (33)
Then it is easy to find that That is easy to see,
F is a generating function off motion for the
system Equation (25) or Equation (27). Due to
0.
x
F
integrals o
0,
n
n
n
FF
we obtain the following formulas of integrals of motion:
2
1
00101
2
11
,,

 
n
FAFAA

02
124
,2.

nn iniiniini
i
FA
AAABCi



(34)
Copyright © 2013 SciRes. JMP
S. X. TAO, H. SHI.
10
Substituting Equation (32) into the above formulas of
motion, we obtain the following expression of )
(0
m
Fm:



11
01 1122
22
1
21122
2
12112
2
123311332
4
,,,,
,,
,,
,,,,,
FF
F



 
(35)




11
1
11
111
8
1122
2
111
121 12
2
1
1
,,
nin
i
i
n



2 2
,
1
11
11
23 3
4
1
,,
,
,,
nn
n
ii
i
F



2 2
111
1
,,
i
ni ni
nii
 

  

1
113
,
ni
 

132
,,
ni
 
2.n
On the other hand, let us consider the temporal pa
the nonlinearized system Equation (26). Making use of
Equation (32) and Equation (36), the system Equation
(2
(36)
rt of
6) can be written as the following super Hamiltonian
form:
11
1, 2, 3,
12
11
1, 2, 3,
12
,,
,,
nnn
nnn
nn
ttt
nn
ttt
FFF
FF



 
 

 
 
1
3
1
3
,
.
n
n
F
(37)
This can be checked pretty easily. For example, we
can show one equality in the above system as follows:


1
1,1 23
2
000
11
11
11122
24
1
1
121 2
21
11
12331 3
41
1
1
,,
,
,,
=.
n
nnn
ninini
ti ii
iii
n
ni i
i
nini
i
niini
i
n
AB
F





 
 




1
ni
(38)
In order to show the Liouville integrability for the
constrained flows Equation (25) and Equation (26), we
need to prove the commutative propertity of motion
0
{}
mm
F, under the corresponding Poission bracket
3()( )
11
{,}(1) .
ij ij
Npp
ij ijijij ij
F
GF
FG

G



 






(39)
At this time, we still have an equality ]
and after a similar discussion, we know that
()
[,

n
n
t
VVV,
F
is also a
he
hich implies
ge
system Equatior Equation (37), w
nerating function of integrals of motion for Equation
(26). Hence 0
{}
mm
F are integrals of motion for t
n (26) o
11 1
{,}0,,0
mn m
n
FFF mn
t
 
.

(40)
The above equality Equation (40) shows that
are in involution in pair under the Poission brac
tion (39).
In addition, similar to the method in [16], we know
that
0
{}
mm
F
ket Equa-
112 23 3
,1 .
kkkkkkk
f
kN
 
  (41)
are integrals of motion for Equation (25) and Equation
(26). It is not difficult to verify that the fun
3Nctions
2
{}
1
N
mm
F and 1
{}
N
kk
f
are involution in pair. Similar to
the methods in [10,16,17], we can verify that the
3Nfunctions 21
{}
N
mm
F
and 1
{}
N
kk
f are functionally
independent over some region of the super symmetry
4|2
NN
. Now, all of above analysis givemanifold s the
following theorem.
Theorem 1 Both the spatial and temporal flows
tion (25) and Equation (26) are Liouville integrable su-
peuper symmetr
Equa-
r Hamiltonian systems defined on the sy
manifold 4|2
N
N, which possess 3N functionally in-
dependent and involutive integrals 21
{}
N
mm
F and 1
{}
N
kk
f
defined by Equ6) and Equation (41). ation (3
5. Acknowledgements
This work was supported by the Natural Science Founda-
tiod Tech
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