A soliton hierarchy associated with a new spectral problem and its Hamiltonian structure

2015 ◽  
Vol 56 (2) ◽  
pp. 021502 ◽  
Author(s):  
Solomon Manukure ◽  
Wen-Xiu Ma
Keyword(s):  
Spectral Problem ◽  
Hamiltonian Structure ◽  
Soliton Hierarchy

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.

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.

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.

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

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)].

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 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
Sign in / Sign up

Export Citation Format

Share Document