Fractional Zero Curvature Equation and Generalized Hamiltonian Structure of Soliton Equation Hierarchy

2007 ◽  
Vol 46 (12) ◽  
pp. 3182-3192 ◽  
Author(s):  
Fa-Jun Yu ◽  
Hong-Qing Zhang
Keyword(s):  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Curvature Equation ◽  
Zero Curvature

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 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.

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.

Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints

2007 ◽  
Vol 62 (7-8) ◽  
pp. 399-405
Author(s):  
Lin Luo ◽  
Engui Fan
Keyword(s):  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Lax Pairs ◽  
Curvature Equation ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Symmetry Constraints

A hierarchy associated with the Li spectral problem is derived with the help of the zero curvature equation. It is shown that the hierarchy possesses bi-Hamiltonian structure and is integrable in the Liouville sense. Moreover, the mono- and binary-nonlinearization theory can be successfully applied in the spectral problem. Under the Bargmann symmetry constraints, Lax pairs and adjoint Lax pairs are nonlineared into finite-dimensional Hamiltonian systems (FDHS) in the Liouville sense. New involutive solutions for the Li hierarchy are obtained.

Algebro-geometric constructions of the Heisenberg hierarchy

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhu Li
Keyword(s):  
Meromorphic Function ◽  
Theta Function ◽  
Algebraic Curve ◽  
Hamiltonian Structure ◽  
Asymptotic Properties ◽  
Trace Identity ◽  
Arithmetic Genus ◽  
Geometric Constructions ◽  
Curvature Equation ◽  
Zero Curvature

AbstractThe Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero-curvature equation and the trace identity. With the help of the Lax matrix, we introduce an algebraic curve ${\mathcal{K}}_{n}$ of arithmetic genus n, from which we define meromorphic function ϕ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of ϕ.

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.

Constructing a Variable Coefficient Integrable Coupling Equation Hierarchy and its Hamiltonian Structure

2016 ◽  
Vol 71 (3) ◽  
pp. 281-287
Author(s):  
Fajun Yu ◽  
Shuo Feng
Keyword(s):  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Coupling Equation ◽  
Lax Pairs ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Integrable Coupling

AbstractHow to construct a variable coefficient integrable coupling equation hierarchy is an important problem. In this paper, we present new Lax pairs with some arbitrary functions and generate a variable coefficient integrable coupling of Ablowitz-Kaup-Newell-Segur hierarchy from a zero-curvature equation. Then the Hamiltonian structure of the variable coefficient coupling equation hierarchy is derived from the variational trace identity. It is also indicated that this method is an efficient and straightforward way to construct the variable coefficient integrable coupling equation hierarchy.

ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY

1992 ◽  
Vol 07 (18) ◽  
pp. 4293-4311 ◽  
Author(s):  
ASHOK DAS ◽  
WEN-JUI HUANG ◽  
SHIBAJI ROY
Keyword(s):  
Current Algebra ◽  
Hamiltonian Structure ◽  
Operator Product ◽  
Curvature Condition ◽  
Supergravity Theory ◽  
Kdv Equations ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Graded Lie Algebra ◽  
The One

The N=2 fermionic extensions of the KdV equations are derived from the zero curvature condition associated with the graded Lie algebra of OSp(2/2). These equations lead to two bi-Hamiltonian systems, one of which is supersymmetric. We also derive the one-parameter family of N=2 supersymmetric KdV equations without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for the corresponding 2D supergravity theory. This current algebra is then used to obtain the operator product expansions of various fields of this theory.

TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

2010 ◽  
Vol 24 (08) ◽  
pp. 791-805 ◽  
Author(s):  
YUNHU WANG ◽  
XIANGQIAN LIANG ◽  
HUI WANG
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

By means of the Lie algebra G1, we construct an extended Lie algebra G2. Two different isospectral problems are designed by the two different Lie algebra G1 and G2. With the help of the variational identity and the zero curvature equation, two families generalization of the AKNS hierarchies and their Hamiltonian structures are obtained, respectively.

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