THE NEGATIVE AND POSITIVE INTEGRABLE COUPLING HIERARCHIES DERIVED FROM A FOUR BY FOUR MATRIX SPECTRAL PROBLEM

2010 ◽  
Vol 24 (30) ◽  
pp. 2955-2970
Author(s):  
XI-XIANG XU
Keyword(s):  
Hamiltonian System ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Form ◽  
Liouville Integrability ◽  
Positive Power ◽  
Integrable Lattice ◽  
Lax Operator ◽  
Matrix Spectral Problem ◽  
Integrable Coupling

Discrete integrable coupling hierarchies of two existing integrable lattice families are derived from a four by four discrete matrix spectral problem. It is shown that the obtained integrable coupling hierarchies respectively corresponds to negative and positive power expansions of the Lax operator with respect to the spectral parameter. Then, the Hamiltonian form of the negative integrable coupling hierarchy is constructed by using the discrete variational identity. Finally, Liouville integrability of each obtained discrete Hamiltonian system is demonstrated.

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 INTEGRABLE COUPLING SYSTEMS AND ITS LIOUVILLE INTEGRABILITY

2009 ◽  
Vol 23 (13) ◽  
pp. 1671-1685
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Liouville Integrability ◽  
Discrete Integrable Systems ◽  
Direct Sum ◽  
The Family ◽  
Matrix Spectral Problem ◽  
Hamiltonian Forms ◽  
Integrable Coupling

A discrete matrix spectral problem and corresponding family of discrete integrable systems are discussed. A semi-direct sum of Lie algebras of four-by-four matrices is introduced, and the related integrable coupling systems of resulting discrete integrable systems are derived. The obtained discrete integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, Liouville integrability of the family of obtained integrable coupling systems is demonstrated.

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.

ADJOINT SYMMETRY CONSTRAINTS AND INTEGRABLE COUPLING SYSTEM OF MULTICOMPONENT KN EQUATIONS

2011 ◽  
Vol 25 (21) ◽  
pp. 2841-2852 ◽  
Author(s):  
FA-JUN YU
Keyword(s):  
Spectral Problem ◽  
Arbitrary Order ◽  
Hamiltonian Formulation ◽  
Soliton Equation ◽  
Order Matrix ◽  
Coupling System ◽  
Soliton Hierarchy ◽  
Symmetry Constraints ◽  
Matrix Spectral Problem ◽  
Integrable Coupling

A soliton hierarchy of multicomponent KN equations is generated from an arbitrary order matrix spectral problem, along with its bi-Hamiltonian formulation. Adjoint symmetry constraints are presented to manipulate binary nonlinearization for the associated arbitrary order matrix spectral problem. Finally, a class of integrable coupling systems of the multicomponent KN soliton equation hierarchy is obtained using Ma's method associated with enlarging spectral problems [W. X. Ma, Phys. Lett. A316, 72–76 (2003)].

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 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.

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.

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