Bargmann Symmetry Constraint and Binary Nonlinearization of Super NLS-MKdV Hierarchy

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried 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 required for Liouville integrability are explicitly given. 4 |2  N N


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 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.The nonlinearization of Lax pairs is classified into mono-nonlinearization [5,6] and binary nonlinearization [7,8].
The technique of nonlinearization has been successfully 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, nonlinearization were made for the super AKNS hierarchy , the super Dirac hierarchy and their corresponding super finite-dimensional hierarchies were generated in Refs.[10][11][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-MKdV 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 dimensional integrable Hamiltonian hierarchy on the super symmetry manifold , whose integrals of motion are explicitly given.

The Super NLS-MKdV Hierarchy
The super NLS-MKdV spectral problem associated with the Lie super algebra is given by (0,1) B , where  is a spectral parameter, q and r are even variables,  and  are odd variables [14].Taking the co-adjoint equation associated with Equation (1) [ , ] ).

 
then Equation ( 2) is equivalent to (4) which results in the recurrence relations , 0. where Upon choosing the initial conditions can be worked out by the recurrence relations Equation (5).The first few sets are: 4 r q r q r q r q r q r q r q r   3 8 8 .
Let us associate the spectral problem Equation (1) with the following auxiliary problem ( ) ( ) where the symbol "  " denotes taking the non-negative part in the power of  .
The compatible conditions of the spectral problem Equation (1) and the auxiliary problem Equation ( 7) are which infer the super NLS-MKdV hierarchy Here in Equation ( 9) is called the n-th NLS -MKdV flow of this hierarchy.
where Str means the super trace [14,15], we have Therefore, the super NLS-MKdV soliton hierarchy Equation ( 9) can be written as the following super Hamiltonian form: where , , is a super symplectic operator, and n H is given by Equation (11).

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The first non-trivial nonlinear equation of the super NLS-MKdV hierarcy ( 9) is given by its second flow which possesses a Lax pair of U defined in Equation ( 1) and defined by (2) V

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: 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).
v is an even variable p v v is an odd variable By Lemma 1, it is not difficult to find that where we can obtain a characteristic property-a recurrence relation for the variational derivative of  : where L and u   are given by Equation ( 4) and Equation (18), respectively.
Let us now discuss the spatial systems: and the temporal systems: where and 1 2 1 j N   , , ,  N    are N distinct pa- rameters.Now for systems Equation (21) and Equation ( 22), we have the following symmetry constraints: The symmetry constraints in the case of 0 k  is called a Bargmann constraint (see [8]).If taking , then it leads to an expression for the potential , i.e.
where we use the following notation and ,   denotes the standard inner product of the Euclidean space  N .

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 Equation (21) and Equation ( 22), and then we obtain the following nonlinearized spatial Lax pairs and adjoint Lax pairs: and where 1 j N   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 independent variables u, but for a given , thepart of the nonlinearized system Equation ( 26) is a system of ordinary differential equations.Obviously, the system Equation (25) can be written as where 1 diag( , , ).
 N     When , the system Equation ( 26) is exactly the system Equation (25) with The system Equation (25) or Equation ( 27) can be written as the following super Hamiltonian form: where where , , ,     q r   , and , , ,     x x x x q r   denote the functions , , , q r   defined by the explicit constraint Equation (24) are given by re compu atial con (27).The system (29) can be represented as the following super Hamiltonian form: which a ted through using the sp strained flow Equation 22 2 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 Equation (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 ) Then the co-adjoint representation equation remains true.Furthermore, we know that also true.Let Then it is easy to find that That is easy to see, F is a generating function of f motion for the system Equation (25) or Equation (27).Due to 0.
we obtain the following formulas of integrals of motion: Copyright © 2013 SciRes.JMP Substituting Equation (32) into the above formulas of motion, we obtain the following expression of ,

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: , , , , This can be checked pretty easily.For example, we can show one equality in the above system as follows: .
At this time, we still have an equality ] and after a similar discussion, we know that 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 { } m m F  ket Equa-  f  are involution in pair.Similar to the methods in [10,16,17], we can verify that the 3N functions

Acknowledgements
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 { } m m F  , under the corresponding Poission bracket are integrals of motion for Equation (25) and Equation (26).It is not difficult to verify that the fun

1 6 )
functionally independent over some region of the super symmetry 4 |2  N N .Now, all of above analysis give manifold s the following theorem.Theorem Both the spatial and temporal flows tion (25) and Equation (26) are Liouville integrable suand Equation (41).ation (3