scholarly journals Tri-Integrable Couplings of the Giachetti-Johnson Soliton Hierarchy as well as Their Hamiltonian Structure

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lei Wang ◽  
Ya-Ning Tang
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Soliton Equations ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings

Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.

A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (05) ◽  
pp. 731-739
Author(s):  
YONGQING ZHANG ◽  
YAN LI
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Curvature Equation ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Time Part

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.

ON INTEGRABLE COUPLINGS OF THE DISPERSIVE LONG WAVE HIERARCHY AND THEIR HAMILTONIAN STRUCTURE

2007 ◽  
Vol 21 (01) ◽  
pp. 37-44 ◽  
Author(s):  
YUFENG ZHANG
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Compatibility Condition ◽  
Hamiltonian Structure ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Long Wave ◽  
Integrable Couplings

A new subalgebra of the loop algebra Ã3 is directly constructed and used to build a pair of Lax matrix isospectral problems. The resulting compatibility condition, i.e., zero curvature equation, gives rise to integrable couplings of the dispersive long wave hierarchy, as an application example. Through using a proper isomorphic map between two Lie algebras, two equivalent zero curvature equations are presented from which the Hamiltonian structure of the integrable couplings is obtained by the quadratic-form identity. The proposed method can be applied to the construction of integrable couplings and the corresponding Hamiltonian structures of other existing soliton hierarchies.

Non-Semisimple Lie Algebras of Block Matrices and Applications to Bi-Integrable Couplings

2013 ◽  
Vol 5 (05) ◽  
pp. 652-670
Author(s):  
Jinghan Meng ◽  
Wen-Xiu Ma
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structures ◽  
Semisimple Lie Algebras ◽  
Block Matrices ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings

AbstractWe propose a class of non-semisimple matrix loop algebras consisting of 3 × 3 block matrices, and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings. Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.

scholarly journals Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

2017 ◽  
Vol 15 (1) ◽  
pp. 203-217
Author(s):  
Jian Zhang ◽  
Chiping Zhang ◽  
Yunan Cui
Keyword(s):  
Lie Algebra ◽  
Discrete Case ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

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 Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jian Zhang ◽  
Chiping Zhang ◽  
Yunan Cui
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

SUBALGEBRAS OF THE LIE ALGEBRA R6 AND THEIR APPLICATIONS

2007 ◽  
Vol 21 (22) ◽  
pp. 3809-3824 ◽  
Author(s):  
YU-FENG ZHANG ◽  
EN-GUI FAN
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Soliton Equations ◽  
Integrable Couplings ◽  
Potential Component ◽  
Lagrange Mechanics ◽  
Important Significance

As we all know, the Hamiltonian systems are the same describing forms as Newton mechanics and Lagrange mechanics. Therefore, researching for a new Hamiltonian structure of the soliton equations has important significance. In the paper, firstly, with the help of the Lie algebra R6, a few types of subalgebras are constructed, from which the corresponding equivalent tensor systems are given. For their applications, two integrable couplings hierarchies along with the multi-potential component functions generated from the soliton theory and the Virasoro symmetric algebra are obtained. Secondly, the Hamiltonian structures of the above integrable couplings are worked out, which may become another describing expression for the Newton and Lagrange mechanics. In particular, one of the integrable couplings presented above reduces to the famous AKNS hierarchy of soliton equations.

scholarly journals A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Hanyu Wei ◽  
Tiecheng Xia ◽  
Guoliang He
Keyword(s):  
Conservation Laws ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Soliton Equations ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Orthogonal Lie Algebra

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

scholarly journals A Complex Integrable Hierarchy and Its Hamiltonian Structure for Integrable Couplings of WKI Soliton Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fajun Yu ◽  
Shuo Feng ◽  
Yanyu Zhao
Keyword(s):  
Hamiltonian Structure ◽  
Direct Application ◽  
Integrable Hierarchy ◽  
Soliton Equation ◽  
Zero Curvature ◽  
Coupling System ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Nonlinear Composite ◽  
Integrable Coupling

We generate complex integrable couplings from zero curvature equations associated with matrix spectral problems in this paper. A direct application to the WKI spectral problem leads to a novel soliton equation hierarchy of integrable coupling system; then we consider the Hamiltonian structure of the integrable coupling system. We select theU¯,V¯and generate the nonlinear composite parts, which generate new extended WKI integrable couplings. It is also indicated that the method of block matrix is an efficient and straightforward way to construct the integrable coupling system.

A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations

2013 ◽  
Vol 3 (3) ◽  
pp. 171-189 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Huiqun Zhang ◽  
Jinghan Meng
Keyword(s):  
Block Matrix ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Dirac Equations ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Matrix Loop Algebra

AbstractA non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

Sign in / Sign up

Export Citation Format

Share Document