The Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems

Yu, Jing; He, Jingsong; Ma, Wen‐Xiu; Cheng, Yi

doi:10.1007/s11401-009-0032-6

Cited by 32 publications

(35 citation statements)

References 23 publications

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“…In order to further elaborate the effectiveness of above method, we choose another set of basis matrices for the Lie superalgebra B (0,1) in this section. Specifically, let us first choose another set of basis matrices for the Lie superalgebra B (0,1)()

\begin{matrix} alignleft \\ align-1 f_{1} & align-2 = (\begin{array}{ccc} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}), f_{2} = (\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}), f_{3} = (\begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}), f_{4} = (\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & - 1 & 0 \end{array}), \\ align-1 f_{5} & align-2 = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}), \end{matrix}

where f 1 , f 2 , f 3 are even elements and f 4 , f 5 are odd ones. Under the super Lie bracket , the nonzero (anti)commutation relations of f 1 , f 2 , f 3 , f 4 , f 5 in are given by

\begin{matrix} alignleft \\ align-1 [f_{1}, f_{2}} & align-2 = - 2 f_{2}, [f_{1}, f_{3}} = 2 f_{} \end{matrix}

…”

Section: Gni‐sdmentioning

confidence: 99%

“…We all knew that the variables satisfied commuting relationship in classical integrable hierarchies. In recent years, super (supersymmetry) integrable hierarchies has aroused the interest of many researchers, including nonlinearization of Lax pairs,() Darboux transformation,() construction of new super integrable hierarchies,() and so forth. () Both commuting and anticommuting variables appeared in super (supersymmetry) integrable hierarchies.…”

Section: Introductionmentioning

confidence: 99%

“…() Both commuting and anticommuting variables appeared in super (supersymmetry) integrable hierarchies. If we chose the Lie superalgebra instead of the Lie algebra g in above procedure, then many isospectral super integrable hierarchies and their super Hamiltonian structures would be constructed, for example, super Ablowitz‐Kaup‐Newell‐Segur (AKNS) hierarchy,() super Kaup‐Newell (KN) hierarchy,() super Dirac hierarchy, and so on. () However, few researchers paid attention to the construction of nonisospectral super integrable hierarchies.…”

Section: Introductionmentioning

confidence: 99%

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Generalized nonisospectral super integrable hierarchies

Zhou

Han

et al. 2019

Math Methods in App Sciences

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For the Lie superalgebra B(0,1), we choose a set of basis matrices. Then we consider a linear combination of the basis matrices, which is exactly the spectral matrix of the spatial part for the super Ablowitz‐Kaup‐Newell‐Segur (AKNS) hierarchy. The compatible condition of the spatial and temporal spectral problems leads to the well‐known zero curvature equation. Here, when the spectral parameter is independent (dependent) of temporal parameter, we obtain isospectral (nonisospectral) super AKNS hierarchy. Furthermore, we derive the generalized nonisospectral super AKNS hierarchy (GNI‐SAKNS). As another example, similar method is successfully applied to the super Dirac hierarchy, and we obtain the generalized nonisospectral super Dirac hierarchy (GNI‐SD).

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\begin{matrix} alignleft \\ align-1 f_{1} & align-2 = (\begin{array}{ccc} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}), f_{2} = (\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}), f_{3} = (\begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}), f_{4} = (\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & - 1 & 0 \end{array}), \\ align-1 f_{5} & align-2 = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}), \end{matrix}

where f 1 , f 2 , f 3 are even elements and f 4 , f 5 are odd ones. Under the super Lie bracket , the nonzero (anti)commutation relations of f 1 , f 2 , f 3 , f 4 , f 5 in are given by

\begin{matrix} alignleft \\ align-1 [f_{1}, f_{2}} & align-2 = - 2 f_{2}, [f_{1}, f_{3}} = 2 f_{} \end{matrix}

…”

Section: Gni‐sdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized nonisospectral super integrable hierarchies

Zhou

Han

et al. 2019

Math Methods in App Sciences

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“…The super coupled Burgers hierarchy and its super-Hamiltonian structure were considered [9]. Recently, Yu et al considered the binary nonlinearization of the super AKNS hierarchy under an implicit symmetry constraint [10] and the Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems [11]. Meanwhile, various systematic methods on classical integrable systems have been developed to obtain exact solutions of the super integrable such as the inverse transformations, the B   cklund and Darboux transformations , the bilinear transformation of Hirota and others [19]- [21].…”

Section: Introductionmentioning

confidence: 99%

Self-Consistent Sources and Conservation Laws for Super Coupled Burgers Equation Hierarchy

Tao¹

2013

IJAPM

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“…Studies provide many examples of super symmetry integrable systems, with super dependent variables and/or super independent variables [11]- [15]. Only very recently, nonlinearization were made for the super AKNS hierarchy , the super Dirac hierarchy and their corresponding super finite dimensional hierarchies were generated [16]- [18]. Li and Dong presented the super Hamiltonian structures of the super Guo hierarchy [19].…”

Section: Introductionmentioning

confidence: 99%

Bargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

Tao¹

2013

IJAPM

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Abstract-An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Guo hierarchy. Under the obtained symmetry constraint, the n-th flow of the super Guo hierarchy is decomposed into two super finite -dimensional integrable Hamiltonian systems, which defined over the super symmetry manifold R 4N|2N with the corresponding dynamical variables x and t n . The integrals of motion required for Liouville integrability are explicitly given.Index Terms-Symmetry constraint, binary nonlinearization, super guo hierarchy, super finitee-dimensional integrable hamiltonian systems. I. 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 [19]. In this paper, we would like to consider the binary nonlinearization of the super Guo hierarchy. This paper is organized as follows. In the next section, we will recall the super Guo soliton hierarchy and its super Hamiltonian structure. Then in Section III, we compute a Bargmann symmetry constraint for the potential of the super Guo hierarchy. In Section IV, we apply the binary nonlinearization method to super Guo hierarchy, and then obtain super finite-dimensional integrable Hamiltonian hierarchy on the super symmetry manifold ℝ 4 |2 , whose integrals of motion are explicitly given. II. THE SUPER GUO HIERARCHYwhere is a spectral parameter, q and r are even variables, and and are odd variables(see [20] ). Taking If we setthen (2) Si-Xing TaoBargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

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The Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems

Cited by 32 publications

References 23 publications

Generalized nonisospectral super integrable hierarchies

Generalized nonisospectral super integrable hierarchies

Self-Consistent Sources and Conservation Laws for Super Coupled Burgers Equation Hierarchy

Bargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

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