SUPER INTEGRABLE HIERARCHY AND SUPER HAMILTONIAN STRUCTURES ASSOCIATED WITH GUO HIERARCHY

2009 ◽  
Vol 23 (29) ◽  
pp. 3491-3496 ◽  
Author(s):  
NING ZHANG ◽  
HUANHE DONG
Keyword(s):  
Integrable Hierarchies ◽  
Lie Superalgebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

A Lie superalgebra is constructed from which establishes an isospectral problems. By solving the zero curvature equation, a resulting super hierarchies of the Guo hierarchy are obtained. By making use of the super identity, the Hamiltonian structures of the above super integrable hierarchies are generated, this method can be used to other superhierarchy.

scholarly journals A New Super Extension of Dirac Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jiao Zhang ◽  
Fucai You ◽  
Yan Zhao
Keyword(s):  
Spectral Problem ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

We derive a new super extension of the Dirac hierarchy associated with a3×3matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.

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.

scholarly journals Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Mats Vermeeren
Keyword(s):  
Differential Equations ◽  
Integrable Hierarchies ◽  
Kepler Problem ◽  
Lagrange Function ◽  
Lagrangian Formulation ◽  
Integrable Hierarchy ◽  
Poisson Brackets ◽  
Exterior Derivative ◽  
Hamiltonian Structures ◽  
Fundamental Object

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

scholarly journals On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xuemei Li ◽  
Lutong Li
Keyword(s):  
Spectral Problem ◽  
Spectral Parameter ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Matrix Spectral Problem

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a 3×3 matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

scholarly journals Algebro-Geometric Solutions for a Discrete Integrable Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Mengshuang Tao ◽  
Huanhe Dong
Keyword(s):  
Integrable Systems ◽  
Theta Functions ◽  
Discrete Systems ◽  
Integrable Equation ◽  
Integrable Hierarchy ◽  
Discrete Integrable Systems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Riemann Theta Functions ◽  
Geometric Solutions

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

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.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

2021 ◽  
pp. 2150156
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian System ◽  
Integrable Model ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Extended System ◽  
Akns Hierarchy ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Higher Dimensional

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

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.

TWO (2+1)-DIMENSIONAL INTEGRABLE COUPLINGS OF THE KN HIERARCHY AND CORRESPONDING HAMILTONIAN STRUCTURES

2009 ◽  
Vol 23 (30) ◽  
pp. 3643-3658
Author(s):  
CHAO YUE ◽  
ZHAOJUN LIU ◽  
JIADONG YU
Keyword(s):  
Quadratic Form ◽  
Burgers Equation ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Curvature Equation ◽  
Reduced Equations ◽  
Zero Curvature ◽  
Integrable Couplings

A (2+1) zero curvature equation is generated from one of the reduced equations of the self-dual Yang–Mills equations. As its applications, two (2+1)-dimensional integrable couplings of the famous KN hierarchy are obtained with the help of a subalgebra of the Lie subalgebra R9, which can be reduced to the Burgers equation. Furthermore, their Hamiltonian structures are worked out by taking use of the quadratic-form identity and the variational identity, respectively.

The Zero Curvature Form of Integrable Hierarchies in the ℤ × ℤ-Matrices

2012 ◽  
Vol 19 (02) ◽  
pp. 237-262
Author(s):  
G. F. Helminck ◽  
A. V. Opimakh
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchies ◽  
Evolution Equations ◽  
Integrable Hierarchy ◽  
Connection Form ◽  
Minimal Realization ◽  
Band Matrices ◽  
Zero Curvature ◽  
Lax Equations ◽  
Finite Band

In this paper it is shown how one can associate to a finite number of commuting directions in the Lie algebra of upper triangular ℤ × ℤ-matrices an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions inside the mentioned Lie algebra. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices, that are the components of a formal connection form. One concludes with the linearization of the hierarchies and the notion of wave matrices at zero, which is the algebraic substitute for a basis of the horizontal sections of the formal connection corresponding to this connection form.

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