Hamiltonian structure of graded and super evolution equations

1988 ◽  
Vol 16 (3) ◽  
pp. 199-206 ◽  
Author(s):  
Pierre Mathieu
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

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.

On the Hamiltonian structure of evolution equations

1980 ◽  
Vol 88 (1) ◽  
pp. 71-88 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Formal Theory ◽  
Natural Generalization ◽  
Line Integral ◽  
Theory Of Evolution ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Integral Invariants ◽  
Generalized Symmetries

AbstractThe theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.

scholarly journals Higher spin Chern–Simons theory and the super Boussinesq hierarchy

2018 ◽  
Vol 33 (14n15) ◽  
pp. 1850085
Author(s):  
Michael Gutperle ◽  
Yi Li
Keyword(s):  
Time Evolution ◽  
Gauge Transformation ◽  
Poisson Structure ◽  
Higher Spin Gravity ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Simons Theory ◽  
Higher Spin ◽  
Chern Simons ◽  
Chern Simons Theory

In this paper, we construct a map between a solution of supersymmetric Chern–Simons higher spin gravity based on the superalgebra [Formula: see text] with Lifshitz scaling and the [Formula: see text] super Boussinesq hierarchy. We show that under this map the time evolution equations of both theories coincide. In addition, we identify the Poisson structure of the Chern–Simons theory induced by gauge transformation with the second Hamiltonian structure of the super Boussinesq hierarchy.

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.

INTEGRABLE HIERARCHIES FROM BRST COHOMOLOGY

2005 ◽  
Vol 20 (01) ◽  
pp. 51-59
Author(s):  
V. CALIAN ◽  
G. STOENESCU
Keyword(s):  
Field Theory ◽  
Hamiltonian System ◽  
Hamiltonian Structure ◽  
Integrable Hierarchies ◽  
Evolution Equations ◽  
Classical Field Theory ◽  
Classical Field ◽  
Integrable Hierarchy ◽  
Brst Cohomology ◽  
Gauge Fixing

We demonstrate that the Hamiltonian structure and the integrability of a system of evolution equations can be formulated in terms of a classical field theory using BRST and anti-BRST symmetries. We derive the field theory action and explicitly generate the integrable hierarchy associated to a bi-Hamiltonian system based on cohomological arguments and gauge-fixing deformations.

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