scholarly journals Super-Hamiltonian Structures and Conservation Laws of a New Six-Component Super-Ablowitz-Kaup-Newell-Segur Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Fucai You ◽  
Jiao Zhang ◽  
Yan Zhao
Keyword(s):  
Conservation Laws ◽  
Spectral Parameter ◽  
Lie Superalgebras ◽  
Hamiltonian Structures ◽  
Akns Hierarchy ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Pacs 02.30

A six-component super-Ablowitz-Kaup-Newell-Segur (-AKNS) hierarchy is proposed by the zero curvature equation associated with Lie superalgebras. Supertrace identity is used to furnish the super-Hamiltonian structures for the resulting nonlinear superintegrable hierarchy. Furthermore, we derive the infinite conservation laws of the first two nonlinear super-AKNS equations in the hierarchy by utilizing spectral parameter expansions. PACS: 02.30.Ik; 02.30.Jr; 02.20.Sv.

Three-Component Generalisation of Burgers Equation and Its Bi-Hamiltonian Structures

2017 ◽  
Vol 72 (5) ◽  
pp. 469-475
Author(s):  
Wei Liu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Eigenvalue Problem ◽  
Conservation Laws ◽  
Spectral Parameter ◽  
Burgers Equation ◽  
Trace Identity ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Matrix Eigenvalue ◽  
Zero Curvature

AbstractA hierarchy of three-component generalisation of Burgers equation, which is associated with a 3×3 matrix eigenvalue problem, is generated by using the zero-curvature equation. By means of the trace identity, the bi-Hamiltonian structures of this hierarchy are constructed. Moreover, the infinite conservation laws for the hierarchy are obtained with the aid of spectral parameter expansion.

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

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.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
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.

New hierarchy of integrable nonlinear differential-difference equations

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.

TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

2010 ◽  
Vol 24 (08) ◽  
pp. 791-805 ◽  
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.

Some generalized coupled nonlinear Schrödinger equations and conservation laws

2017 ◽  
Vol 31 (32) ◽  
pp. 1750299 ◽  
Author(s):  
Wei Liu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Matrix Problem ◽  
Nonlinear Schrödinger ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Derivative Nonlinear Schrödinger Equation

Based on zero-curvature equation, a series of new four-component nonlinear Schrödinger-type equations related to a [Formula: see text] matrix problem are proposed by using the polynomial expansion of the spectral parameter. As two special reductions, a generalized coupled nonlinear Schrödinger equation and a generalized coupled derivative nonlinear Schrödinger equation are obtained. And then, the infinite conservation laws for each of these four-component nonlinear Schrödinger-type equations are constructed with the aid of the Riccati-type equations.

TWO SUPER-INTEGRABLE SYSTEMS AND ASSOCIATED SUPER-HAMILTONIAN STRUCTURES

2009 ◽  
Vol 23 (27) ◽  
pp. 3253-3264 ◽  
Author(s):  
QIU-LAN ZHAO ◽  
XIN-YUE LI ◽  
BAI-YING HE
Keyword(s):  
Integrable Systems ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lie Superalgebras ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Zero Curvature

The super extensions of g-cKdV and mKdV integrable systems are proposed. Two hierarchies of super-integrable nonlinear evolution equations are found. In addition, making use of the super-trace identity, we construct the super-Hamiltonian structures of zero-curvature equations associated with Lie superalgebras.

scholarly journals Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jian Zhang ◽  
Chiping Zhang ◽  
Yunan Cui
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

2021 ◽  
pp. 2150156
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian System ◽  
Integrable Model ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Extended System ◽  
Akns Hierarchy ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Higher Dimensional

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

SUPER INTEGRABLE HIERARCHY AND SUPER HAMILTONIAN STRUCTURES ASSOCIATED WITH GUO HIERARCHY

2009 ◽  
Vol 23 (29) ◽  
pp. 3491-3496 ◽  
Author(s):  
NING ZHANG ◽  
HUANHE DONG
Keyword(s):  
Integrable Hierarchies ◽  
Lie Superalgebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

A Lie superalgebra is constructed from which establishes an isospectral problems. By solving the zero curvature equation, a resulting super hierarchies of the Guo hierarchy are obtained. By making use of the super identity, the Hamiltonian structures of the above super integrable hierarchies are generated, this method can be used to other superhierarchy.

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