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

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.

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

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.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

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 ◽  
Matrix Spectral Problem

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.

scholarly journals Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yu-Qing Li ◽  
Bao-Shu Yin
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Integrable Lattice ◽  
Self Consistent ◽  
Matrix Spectral Problem

A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

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.

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

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 ◽  
Matrix Spectral Problem

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

2010 ◽  
Vol 24 (19) ◽  
pp. 2077-2090 ◽  
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 ◽  
Matrix Spectral Problem

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.

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.

scholarly journals Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Qianqian Yang ◽  
Qiulan Zhao ◽  
Xinyue Li
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Explicit Solutions ◽  
Liouville Integrability ◽  
Lattice Equation ◽  
Integrable Lattice ◽  
Matrix Spectral Problem

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

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.

A GENERALIZATION OF TODA LATTICES AND THEIR BI-HAMILTONIAN STRUCTURES

2012 ◽  
Vol 26 (13) ◽  
pp. 1250078 ◽  
Author(s):  
XIANGUO GENG ◽  
FANG LI ◽  
BO XUE
Keyword(s):  
Spectral Problem ◽  
Fourth Order ◽  
Difference Equations ◽  
Toda Lattice ◽  
Trace Identity ◽  
Lattice Equation ◽  
Hamiltonian Structures ◽  
Typical Member ◽  
Toda Lattice Equation ◽  
Toda Lattices

A hierarchy of new nonlinear differential-difference equations associated with fourth-order discrete spectral problem is proposed, in which a typical member is a generalization of the Toda lattice equation. The bi-Hamiltonian structures for this hierarchy are obtained with the help of trace identity.

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