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.

INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

2010 ◽  
Vol 24 (07) ◽  
pp. 681-694
Author(s):  
LI-LI ZHU ◽  
JUN DU ◽  
XIAO-YAN MA ◽  
SHENG-JU SANG
Keyword(s):  
Eigenvalue Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Direct Sums ◽  
Direct Sum ◽  
Hamiltonian Lattice ◽  
Integrable Couplings ◽  
Type Lattice ◽  
Integrable Coupling

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

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 The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xia Dong ◽  
Tiecheng Xia ◽  
Desheng Li
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Integrable Systems ◽  
The Other ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Kdv Hierarchy ◽  
Integrable Coupling

By use of the loop algebraG-~, integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.

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.

Split Lie algebras of order 3

2019 ◽  
Vol 19 (04) ◽  
pp. 2050070
Author(s):  
Antonio J. Calderón ◽  
Rosa M. Navarro ◽  
José M. Sánchez
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Subspace ◽  
Type Theorem ◽  
Natural Generalization ◽  
Lie Superalgebras ◽  
Direct Sum ◽  
The Family ◽  
Split Lie Algebra

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

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 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.

scholarly journals A super Degasperis–Procesi equation and related integrable systems

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200780
Author(s):  
Binfang Gao ◽  
Kai Tian ◽  
Qing Ping Liu
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Reciprocal Transformation ◽  
Matrix Spectral Problem

Based on a 4 × 4 matrix spectral problem, a super Degasperis–Procesi (DP) equation is proposed. We show that under a reciprocal transformation, the super DP equation is related to the first negative flow of a super Kaup–Kupershmidt (KK) hierarchy, which turns out to be a particular reduction of a super Boussinesq hierarchy. The bi-Hamiltonian structure of the super Boussinesq hierarchy is established and subsequently produces a Hamiltonian structure, as well as a conjectured symplectic formulation of the super KK hierarchy via suitable reductions. With the help of the reciprocal transformation, the bi-Hamiltonian representation of the super DP equation is constructed from that of the super KK hierarchy. We also calculate a positive flow of the super DP hierarchy and explain its relations with the super KK equation. Infinitely many conservation laws are derived for the super DP equation, as well as its positive flow.

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