scholarly journals A generalized Dirac soliton hierarchy and its bi-Hamiltonian structure

2016 ◽  
Vol 60 ◽  
pp. 67-72 ◽  
Author(s):  
Yujian Ye ◽  
Zhihui Li ◽  
Chunxia Li ◽  
Shoufeng Shen ◽  
Wen-Xiu Ma
Keyword(s):  
Hamiltonian Structure ◽  
Soliton Hierarchy

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

2010 ◽  
Vol 24 (14) ◽  
pp. 1573-1594 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
JIANQIN MEI
Keyword(s):  
Heat Conduction ◽  
Quadratic Form ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Arbitrary Parameter ◽  
Higher Dimensional ◽  
Soliton Hierarchy ◽  
Equation Of Heat Conduction ◽  
Hamiltonian Functions

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

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.

A Super-Soliton Hierarchy and Its Super-Hamiltonian Structure

2009 ◽  
Vol 48 (7) ◽  
pp. 2172-2176 ◽  
Author(s):  
Zhu Li ◽  
Huanhe Dong ◽  
Hongwei Yang
Keyword(s):  
Hamiltonian Structure ◽  
Soliton Hierarchy

SUPER-BURGERS SOLITON HIERARCHY AND ITS SUPER-HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (24) ◽  
pp. 2907-2914 ◽  
Author(s):  
ZHU LI ◽  
HONGWEI YANG ◽  
HUANHE DONG
Keyword(s):  
Hamiltonian Structure ◽  
Trace Identity ◽  
Soliton Hierarchy ◽  
Burgers Hierarchy

A super-Burgers hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and is associated with super-trace identity.

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 NEW SOLITON HIERARCHY AND ITS TWO KINDS OF EXPANDING INTEGRABLE MODELS AS WELL AS HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (14) ◽  
pp. 3059-3072
Author(s):  
YUFENG ZHANG ◽  
HUANHE DONG ◽  
Y. C. HON
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Models ◽  
Loop Algebras ◽  
Soliton Hierarchy

With the help of two different Lie algebras and the corresponding loop algebras, the first and second kind of expanding integrable models of a new soliton hierarchy of evolution equations are obtained, respectively. The Hamiltonian structure of the first one is worked out by the quadratic-form identity. The bi-Hamiltonian structure of the second one is also generated. From the paper, we conclude that various Lie algebras really produce different soliton hierarchies of evolution equations. The approach presented in the paper provides a way for generating different integrable soliton expanding systems of the known soliton hierarchy of equations.

Tri-integrable coupling of the Kaup-Newell soliton hierarchy and Liouville integrability

2016 ◽  
Vol 30 (21) ◽  
pp. 1650277 ◽  
Author(s):  
Shuimeng Yu ◽  
Yujian Ye ◽  
Jun Zhang ◽  
Junquan Song
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Block Matrices ◽  
Text Block ◽  
Soliton Hierarchy ◽  
Integrable Coupling

Based on a matrix Lie algebra consisting of [Formula: see text] block matrices, new tri-integrable coupling of the Kaup–Newell soliton hierarchy is constructed. Then, the bi-Hamiltonian structure which leads to Liouville integrability of this coupling is furnished by the variational identity.

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.

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