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

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.

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A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

Modern Physics Letters B ◽

10.1142/s0217984910022226 ◽

2010 ◽

Vol 24 (02) ◽

pp. 183-193

Author(s):

HAI-YONG DING ◽

HONG-XIANG YANG ◽

YE-PENG SUN ◽

LI-LI ZHU

Keyword(s):

Eigenvalue Problem ◽

Evolution Equations ◽

Trace Identity ◽

Liouville Integrability ◽

Matrix Eigenvalue Problem ◽

Hamiltonian Structures ◽

Hamiltonian Equations ◽

Matrix Eigenvalue ◽

Integrable Evolution

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

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A FOUR-BY-FOUR MATRIX EIGENVALUE PROBLEM, RELATED HIERARCHY OF LAX INTEGRABLE EQUATIONS AND THEIR HAMILTONIAN STRUCTURES

International Journal of Modern Physics B ◽

10.1142/s0217979208048735 ◽

2008 ◽

Vol 22 (23) ◽

pp. 4027-4040 ◽

Cited By ~ 3

Author(s):

XI-XIANG XU ◽

HONG-XIANG YANG ◽

WEI-LI CAO

Keyword(s):

Eigenvalue Problem ◽

Evolution Equations ◽

Trace Identity ◽

Liouville Integrability ◽

Matrix Eigenvalue Problem ◽

Hamiltonian Structures ◽

Hamiltonian Equations ◽

Integrable Equations ◽

Matrix Eigenvalue ◽

Integrable Evolution

Starting from a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is proved.

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

Abstract and Applied Analysis ◽

10.1155/2014/214709 ◽

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.

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TWO (2+1)-DIMENSIONAL INTEGRABLE COUPLINGS OF THE KN HIERARCHY AND CORRESPONDING HAMILTONIAN STRUCTURES

Modern Physics Letters B ◽

10.1142/s0217984909021685 ◽

2009 ◽

Vol 23 (30) ◽

pp. 3643-3658

Author(s):

CHAO YUE ◽

ZHAOJUN LIU ◽

JIADONG YU

Keyword(s):

Quadratic Form ◽

Burgers Equation ◽

The Self ◽

Hamiltonian Structures ◽

Yang Mills ◽

Curvature Equation ◽

Reduced Equations ◽

Zero Curvature ◽

Integrable Couplings

A (2+1) zero curvature equation is generated from one of the reduced equations of the self-dual Yang–Mills equations. As its applications, two (2+1)-dimensional integrable couplings of the famous KN hierarchy are obtained with the help of a subalgebra of the Lie subalgebra R9, which can be reduced to the Burgers equation. Furthermore, their Hamiltonian structures are worked out by taking use of the quadratic-form identity and the variational identity, respectively.

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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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