A new integrable symplectic map by the binary nonlinearization to the super AKNS system

Li, Xinyue; Zhao, Qiulan

doi:10.1016/j.geomphys.2017.07.010

Cited by 64 publications

(38 citation statements)

References 44 publications

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“…In this paper, starting with the orthosymplectic Lie superalgebra

O S P false(2, 2 false)

, we construct the generalized super AKNS equations hierarchy , which can be written as the super bi‐Hamiltonian structures and . When

ε = 0

in Equation , we find that this equations are exactly ones of Li and Zhao . As for the other super integrable equations, how to construct their generalized hierarchies?…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…To this end, we shall use the super‐trace identity:

\frac{δ}{δ u} \int S t r false(V \frac{\partial U}{\partial λ} false) d x = false(λ^{- s} \frac{\partial}{\partial λ} λ^{s} false) S t r false(\frac{\partial U}{\partial u} V false),

where

S t r

is the abbreviation of the super‐trace, and

s

is an undetermined constant. We all knew that the supertrace identity was discussed in Hu and Chowdhury and Swapna, and rigorously proved in Ma et al In previous studies, the supertrace identity has been generalized to the variational identity. After a direct and simple calculation, we have

\begin{matrix} eqnarrayleft center left \\ eqnarray-1 & eqnarray-2 & eqnarray-3 S t r (V \frac{\partial U}{\partial λ}) = - 2 A, S t r (\frac{\partial U}{\partial p} V) = - C - 2 ε q A, S t r (\frac{\partial U}{\partial q} V) = - B - 2 ε p A, S t r (\frac{\partial U}{\partial α_{1}} V) = - 2 δ - 4 ε α_{2} A, \\ eqnarray-1 & eqnarray-2 & eqnarray-3 S t r (\frac{\partial U}{\partial α_{2}} V) = 2 ρ + 4 ε α_{1} A, S t r (\frac{\partial U}{\partial β_{1}} V) = - 2 μ - 4 ε β_{2} A, S t r (\frac{\partial U}{\partial β} \end{matrix}

…”

Section: Super Bi‐hamiltonian Structuresmentioning

confidence: 99%

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A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

Han

2020

Math Methods in App Sciences

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For the orthosymplectic Lie superalgebra OSPfalse(2,2false), we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi‐Hamiltonian structures.

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“…In this paper, starting with the orthosymplectic Lie superalgebra

O S P false(2, 2 false)

, we construct the generalized super AKNS equations hierarchy , which can be written as the super bi‐Hamiltonian structures and . When

ε = 0

in Equation , we find that this equations are exactly ones of Li and Zhao . As for the other super integrable equations, how to construct their generalized hierarchies?…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…To this end, we shall use the super‐trace identity:

\frac{δ}{δ u} \int S t r false(V \frac{\partial U}{\partial λ} false) d x = false(λ^{- s} \frac{\partial}{\partial λ} λ^{s} false) S t r false(\frac{\partial U}{\partial u} V false),

where

S t r

is the abbreviation of the super‐trace, and

s

\begin{matrix} eqnarrayleft center left \\ eqnarray-1 & eqnarray-2 & eqnarray-3 S t r (V \frac{\partial U}{\partial λ}) = - 2 A, S t r (\frac{\partial U}{\partial p} V) = - C - 2 ε q A, S t r (\frac{\partial U}{\partial q} V) = - B - 2 ε p A, S t r (\frac{\partial U}{\partial α_{1}} V) = - 2 δ - 4 ε α_{2} A, \\ eqnarray-1 & eqnarray-2 & eqnarray-3 S t r (\frac{\partial U}{\partial α_{2}} V) = 2 ρ + 4 ε α_{1} A, S t r (\frac{\partial U}{\partial β_{1}} V) = - 2 μ - 4 ε β_{2} A, S t r (\frac{\partial U}{\partial β} \end{matrix}

…”

Section: Super Bi‐hamiltonian Structuresmentioning

confidence: 99%

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

Han

2020

Math Methods in App Sciences

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“…By using (8)- (10) Besov spaces , (R 2 ) for 0 < < 1 and 1 < , < ∞. For other interesting works on this topic we refer the readers to consult [24][25][26][27][28]. We denote by , (R ) the fractional Sobolev spaces defined by the Bessel potentials.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Liu

2018

Journal of Function Spaces

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We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces , (R 2 ) for 0 ≤ ≤ 1 and 1 < < ∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from ℓ 1 (Z 2 ) to BV(Z 2 ). Here BV(Z 2 ) denotes the set of all functions of bounded variation on Z 2 .

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“…As we all know, the generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1][2][3][4][5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6][7][8][9][10][11]. It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12,13], Hamiltonian structures [14][15][16], exact solutions [17], and the transformed rational function method [18].…”

Section: Introductionmentioning

confidence: 99%

Algebro-Geometric Solutions for a Discrete Integrable Equation

Tao

Dong

2017

Discrete Dynamics in Nature and Society

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With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

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A new integrable symplectic map by the binary nonlinearization to the super AKNS system

Cited by 64 publications

References 44 publications

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Algebro-Geometric Solutions for a Discrete Integrable Equation

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