Lie algebra of coupled higher-dimensional forced Burgers’ equation

2021 ◽  
Author(s):  
Amlan K. Halder ◽  
Kyriakos Charalambous ◽  
R. Sinuvasan ◽  
P. G. L. Leach
Keyword(s):  
Lie Algebra ◽  
Burgers Equation ◽  
Higher Dimensional

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.

HAMILTONIAN STRUCTURES OF TWO INTEGRABLE COUPLINGS OF THE MODIFIED AKNS HIERARCHY

2007 ◽  
Vol 21 (30) ◽  
pp. 2063-2074 ◽  
Author(s):  
YUFENG ZHANG ◽  
Y. C. HON
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Akns Hierarchy ◽  
Higher Dimensional ◽  
Integrable Couplings

The extension of a three-dimensional Lie algebra into two higher-dimensional ones is used to deduce two new integrable couplings of the m-AKNS hierarchy. The Hamiltonian structures of the two integrable couplings are obtained, respectively. Specially, the complex Hamiltonian structure of the second integrable couplings is given.

A higher-dimensional Lie algebra and its decomposed subalgebras

2006 ◽  
Vol 360 (1) ◽  
pp. 92-98 ◽  
Author(s):  
Yufeng Zhang ◽  
Wang Yan
Keyword(s):  
Lie Algebra ◽  
Higher Dimensional

scholarly journals Multipartitions, generalized Durfee squares and affine Lie algebra characters

2002 ◽  
Vol 72 (3) ◽  
pp. 395-408 ◽  
Author(s):  
Peter Bouwknegt
Keyword(s):  
Lie Algebra ◽  
Partition Functions ◽  
Affine Lie Algebra ◽  
Higher Dimensional ◽  
Durfee Square ◽  
Durfee Squares

AbstractWe give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral Partition Functions.

The g-Based Jordan Algebra and Lie Algebra Formulations of the Maxwell Equations

2004 ◽  
Vol 20 (4) ◽  
pp. 285-296
Author(s):  
Chein-Shan Liu
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Algebraic Property ◽  
Gauge Condition ◽  
Linear Matrix ◽  
Lorentz Gauge ◽  
Higher Dimensional ◽  
Necessary And Sufficient

AbstractWhen it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

scholarly journals Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 465
Author(s):  
Javier de Lucas ◽  
Daniel Wysocki
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Matrix Representations ◽  
Lie Bialgebras ◽  
Higher Dimensional ◽  
Trivial Center ◽  
Darboux Polynomial

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

Generating a New Higher-Dimensional Ultra-Short Pulse System: Lie-Algebra Valued Connection and Hidden Structural Symmetries

2012 ◽  
Vol 29 (2) ◽  
pp. 020501 ◽  
Author(s):  
Hermann T. Tchokouansi ◽  
Victor K. Kuetche ◽  
Abbagari Souleymanou ◽  
Thomas B. Bouetou ◽  
Timoleon C. Kofane
Keyword(s):  
Lie Algebra ◽  
Short Pulse ◽  
Pulse System ◽  
Higher Dimensional ◽  
Ultra Short Pulse

scholarly journals Higher-Dimensional WZW Model on Kähler Manifold and Toroidal Lie Algebra

1997 ◽  
Vol 12 (36) ◽  
pp. 2757-2764 ◽  
Author(s):  
Takeo Inami ◽  
Tatsuya Ueno ◽  
Hiroaki Kanno
Keyword(s):  
Lie Algebra ◽  
Sigma Model ◽  
Kähler Manifold ◽  
Classical Equation ◽  
Nonlinear Sigma Model ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Kahler Manifold ◽  
Higher Dimensional ◽  
Toroidal Lie Algebra

We construct a generalization of the two-dimensional Wess–Zumino–Witten model on a 2n-dimensional Kähler manifold as a group-valued nonlinear sigma model with an anomaly term containing the Kähler form. The model is shown to have an infinite-dimensional symmetry which generates an n-toroidal Lie algebra. The classical equation of motion turns out to be the Donaldson–Uhlenbeck–Yau equation, which is a 2n-dimensional generalization of the self-dual Yang–Mills equation.

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