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

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.

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 ◽

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

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 ◽

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.

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0493 ◽

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.

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.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

10.1142/s0217984918500161 ◽

2018 ◽

Vol 32 (02) ◽

pp. 1850016

Author(s):

Siqi Xu ◽

Xianguo Geng ◽

Bo Xue

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Compatibility Condition ◽

Evolution Equations ◽

Nonlinear Evolution ◽

Nonlinear Evolution Equations ◽

Trace Identity ◽

Hamiltonian Structures ◽

Nonlinear Schrödinger ◽

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2021/9912387 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Ning Zhang ◽

Xi-Xiang Xu

Keyword(s):

Spectral Problem ◽

Lie Algebra ◽

Difference Equations ◽

Vector Fields ◽

Hamiltonian Structure ◽

Trace Identity ◽

Zero Curvature Representation ◽

Curvature Equation ◽

Zero Curvature ◽

Liouville Integrable

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.

A HIERARCHY OF LIOUVILLE INTEGRABLE LATTICE EQUATION ASSOCIATED WITH A THREE-BY-THREE DISCRETE SPECTRAL PROBLEM AND ITS INFINITELY MANY CONSERVATION LAWS

International Journal of Modern Physics B ◽

10.1142/s0217979211100321 ◽

2011 ◽

Vol 25 (18) ◽

pp. 2481-2492

Author(s):

YU-QING LI ◽

XI-XIANG XU

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Trace Identity ◽

Lattice Equation ◽

Hamiltonian Structures ◽

Discrete Soliton ◽

Soliton Equations ◽

Integrable Lattice ◽

Matrix Spectral Problem ◽

Liouville Integrable

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

A FAMILY OF DISCRETE HAMILTONIAN EQUATIONS ASSOCIATED WITH A DISCRETE THREE-BY-THREE MATRIX SPECTRAL PROBLEM

International Journal of Modern Physics B ◽

10.1142/s021797920905273x ◽

2009 ◽

Vol 23 (19) ◽

pp. 3859-3869

Author(s):

LIN-LIN MA ◽

XI-XIANG XU

Keyword(s):

Spectral Problem ◽

Trace Identity ◽

Hamiltonian Structures ◽

Hamiltonian Equations ◽

Integrable Lattice ◽

The Family ◽

A family of integrable lattice equations with four potentials is constructed from a new discrete three-by-three matrix spectral problem. The Hamiltonian structures of the integrable lattice equations in the family are derived by applying the discrete trace identity. Finally, infinitely many common commuting conserved functionals of the resulting integrable lattice equations are given.

SOME NEW INTEGRABLE NONLINEAR EVOLUTION EQUATIONS AND THEIR INFINITELY MANY CONSERVATION LAWS

10.1142/s021798491002447x ◽

2010 ◽

Vol 24 (19) ◽

pp. 2077-2090 ◽

Cited By ~ 4

Author(s):

XIANGUO GENG ◽

BO XUE

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Evolution Equations ◽

Nonlinear Evolution ◽

Infinite Sequence ◽

Nonlinear Evolution Equations ◽

Conserved Quantities ◽

Trace Identity ◽

Hamiltonian Structures ◽

A hierarchy of new nonlinear evolution equations associated with a 3×3 matrix spectral problem with two potentials is derived and its Hamiltonian structures are established with the aid of trace identity. The negative flow of the hierarchy is then discussed. A reduction of this hierarchy and its Hamiltonian structures are constructed. An infinite sequence of conserved quantities of several new soliton equations is obtained.

An integrable soliton hierarchy associated with the Boiti–Pempinelli–Tu spectral problem

10.1142/s0217984921502821 ◽

2021 ◽

pp. 2150282

Author(s):

Emmanuel A. Appiah ◽

Solomon Manukure

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Lie Algebra ◽

Mkdv Equation ◽

Trace Identity ◽

Hamiltonian Structures ◽

Soliton Equations ◽

Soliton Hierarchy ◽

Matrix Spectral Problem ◽

Math Phys

Based on the Tu scheme [G.-Z. Tu, J. Math. Phys. 30 (1989) 330], we construct a counterpart of the Boiti–Pempinelli–Tu soliton hierarchy from a matrix spectral problem associated with the Lie algebra [Formula: see text], and formulate Hamiltonian structures for the resulting soliton equations by means of the trace identity. We then show that the newly presented equations possess infinitely many commuting symmetries and conservation laws. Finally, we derive the well-known combined KdV-mKdV equation from the new hierarchy.