A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

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A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

Modern Physics Letters B ◽

10.1142/s0217984909018953 ◽

2009 ◽

Vol 23 (05) ◽

pp. 731-739

Author(s):

YONGQING ZHANG ◽

YAN LI

Keyword(s):

Spectral Problem ◽

Lie Algebras ◽

Hamiltonian Structure ◽

Soliton Equation ◽

Curvature Equation ◽

Loop Algebras ◽

Zero Curvature ◽

Soliton Hierarchy ◽

Integrable Couplings ◽

Time Part

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.

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On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

Advances in Mathematical Physics ◽

10.1155/2020/6838731 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Xuemei Li ◽

Lutong Li

Keyword(s):

Spectral Problem ◽

Spectral Parameter ◽

Trace Identity ◽

Hamiltonian Structures ◽

Curvature Equation ◽

Zero Curvature ◽

Matrix Spectral Problem

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a 3×3 matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

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A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

Advances in Mathematical Physics ◽

10.1155/2016/3589704 ◽

2016 ◽

Vol 2016 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Yuqin Yao ◽

Shoufeng Shen ◽

Wen-Xiu Ma

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Spectral Parameter ◽

Hamiltonian Structure ◽

Trace Identity ◽

Liouville Integrability ◽

Hamiltonian Structures ◽

Zero Curvature ◽

Soliton Hierarchy ◽

Matrix Spectral Problem

Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.

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Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints

Zeitschrift für Naturforschung A ◽

10.1515/zna-2007-7-808 ◽

2007 ◽

Vol 62 (7-8) ◽

pp. 399-405

Author(s):

Lin Luo ◽

Engui Fan

Keyword(s):

Spectral Problem ◽

Hamiltonian Systems ◽

Hamiltonian Structure ◽

Lax Pairs ◽

Curvature Equation ◽

Finite Dimensional ◽

Zero Curvature ◽

Symmetry Constraints

A hierarchy associated with the Li spectral problem is derived with the help of the zero curvature equation. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in the Liouville sense. Moreover, the mono- and binary-nonlinearization theory can be successfully applied in the spectral problem. Under the Bargmann symmetry constraints, Lax pairs and adjoint Lax pairs are nonlineared into finite-dimensional Hamiltonian systems (FDHS) in the Liouville sense. New involutive solutions for the Li hierarchy are obtained.

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Algebro-geometric constructions of the Heisenberg hierarchy

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0197 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Zhu Li

Keyword(s):

Meromorphic Function ◽

Theta Function ◽

Algebraic Curve ◽

Hamiltonian Structure ◽

Asymptotic Properties ◽

Trace Identity ◽

Arithmetic Genus ◽

Geometric Constructions ◽

Curvature Equation ◽

Zero Curvature

AbstractThe Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero-curvature equation and the trace identity. With the help of the Lax matrix, we introduce an algebraic curve ${\mathcal{K}}_{n}$ of arithmetic genus n, from which we define meromorphic function ϕ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of ϕ.

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A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

Discrete Dynamics in Nature and Society ◽

10.1155/2018/4152917 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Xi-Xiang Xu ◽

Meng Xu

Keyword(s):

Exact Solution ◽

Difference Equations ◽

Hamiltonian Structure ◽

Bäcklund Transformation ◽

Lax Pair ◽

Liouville Integrability ◽

Backlund Transformation ◽

Zero Curvature Representation ◽

Zero Curvature ◽

Matrix Spectral Problem

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.

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New hierarchy of integrable nonlinear differential-difference equations

Modern Physics Letters B ◽

10.1142/s0217984915501900 ◽

2015 ◽

Vol 29 (31) ◽

pp. 1550190

Author(s):

Xianguo Geng ◽

Liang Guan ◽

Bo Xue

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Spectral Parameter ◽

Difference Equations ◽

Trace Identity ◽

Hamiltonian Structures ◽

Zero Curvature ◽

Matrix Spectral Problem

A hierarchy of integrable nonlinear differential-difference equations associated with a discrete [Formula: see text] matrix spectral problem is proposed based on the discrete zero-curvature equations. Then, Hamiltonian structures for this hierarchy are constructed with the aid of the trace identity. Infinitely many conservation laws of the hierarchy are derived by means of spectral parameter expansions.

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A New Super Extension of Dirac Hierarchy

Abstract and Applied Analysis ◽

10.1155/2014/472101 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Jiao Zhang ◽

Fucai You ◽

Yan Zhao

Keyword(s):

Spectral Problem ◽

Integrable Hierarchy ◽

Trace Identity ◽

Hamiltonian Structures ◽

Curvature Equation ◽

Zero Curvature

We derive a new super extension of the Dirac hierarchy associated with a3×3matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.

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Constructing a Variable Coefficient Integrable Coupling Equation Hierarchy and its Hamiltonian Structure

Zeitschrift für Naturforschung A ◽

10.1515/zna-2015-0437 ◽

2016 ◽

Vol 71 (3) ◽

pp. 281-287

Author(s):

Fajun Yu ◽

Shuo Feng

Keyword(s):

Variable Coefficient ◽

Hamiltonian Structure ◽

Trace Identity ◽

Coupling Equation ◽

Lax Pairs ◽

Curvature Equation ◽

Zero Curvature ◽

Integrable Coupling

AbstractHow to construct a variable coefficient integrable coupling equation hierarchy is an important problem. In this paper, we present new Lax pairs with some arbitrary functions and generate a variable coefficient integrable coupling of Ablowitz-Kaup-Newell-Segur hierarchy from a zero-curvature equation. Then the Hamiltonian structure of the variable coefficient coupling equation hierarchy is derived from the variational trace identity. It is also indicated that this method is an efficient and straightforward way to construct the variable coefficient integrable coupling equation hierarchy.

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TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

Modern Physics Letters B ◽

10.1142/s021798491002286x ◽

2010 ◽

Vol 24 (08) ◽

pp. 791-805 ◽

Cited By ~ 4

Author(s):

YUNHU WANG ◽

XIANGQIAN LIANG ◽

HUI WANG

Keyword(s):

Lie Algebra ◽

Hamiltonian Structures ◽

Curvature Equation ◽

Zero Curvature

By means of the Lie algebra G1, we construct an extended Lie algebra G2. Two different isospectral problems are designed by the two different Lie algebra G1 and G2. With the help of the variational identity and the zero curvature equation, two families generalization of the AKNS hierarchies and their Hamiltonian structures are obtained, respectively.

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