scholarly journals Lie Algebra of Infinitesimal Generators of the Symmetry Group of the Heat Equation

2018 ◽  
Vol 06 (02) ◽  
pp. 373-381
Author(s):  
Narmanov Otabek Abdigapparovich
Keyword(s):  
Heat Equation ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Infinitesimal Generators

scholarly journals Lie algebra classification for the Chazy equation and further topics related with this algebra

2021 ◽  
Vol 17 (34) ◽  
pp. 101-109
Author(s):  
Yeisson Alexis Acevedo-Agudelo ◽  
Danilo Andrés García-Hernández ◽  
Oscar Mario Londoño-Duque ◽  
Gabriel Ignacio Loaiza-Ossa
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Classical Problem ◽  
Lie Symmetry ◽  
Infinitesimal Generators ◽  
Chazy Equation ◽  
The One ◽  
Lie Algebra Classification

It is known that the classification of the Lie algebras is a classical problem. Due to Levi’s Theorem the question can be reduced to the classification of semi-simple and solvable Lie algebras. This paper is devoted to classify the Lie algebra generated by the Lie symmetry group of the Chazy equation. We also present explicitly the one parame-ter subgroup related to the infinitesimal generators of the Chazy symmetry group. Moreover the classification of the Lie algebra associated to the optimal system is investigated. La clasificación de las álgebras de Lie es un problema clásico. Acorde al teorema de Levi la cuestión puede reducirse a la clasificación de álgebras de Lie semi-simples y solubles. Este artículo está dedicado a clasificar el álgebra de Lie generada por el grupo de simetría de Lie para la ecuación de Chazy. También presentamos explícitamente los subgrupos a un parámetro  relacionados con los generadores de las simetrías del grupo de Chazy. Además, la clasificación de la álgebra de Lie asociada al sistema optimo es investigada.

scholarly journals A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Do Ngoc Diep
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Vector Bundles ◽  
Sigma Model ◽  
Fock Space ◽  
Loop Algebra ◽  
Target Space ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

scholarly journals SYMMETRY ANALYSIS OF TELEGRAPH EQUATION

2011 ◽  
Vol 04 (01) ◽  
pp. 117-126
Author(s):  
Mehdi Nadjafikhah ◽  
Seyed-Reza Hejazi
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Telegraph Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Group Method ◽  
Parameter Group ◽  
One Dimensional

Lie symmetry group method is applied to study the telegraph equation. The symmetry group and one-parameter group associated to the symmetries with the structure of the Lie algebra symmetries are determined. The reduced version of equation and its one-dimensional optimal system are given.

Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850383 ◽  
Author(s):  
Xuan Zhou ◽  
Wenrui Shan ◽  
Zhilei Niu ◽  
Pengcheng Xiao ◽  
Ying Wang
Keyword(s):  
Exact Solutions ◽  
Detailed Analysis ◽  
Symmetry Group ◽  
Nonlinear Waves ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

REPRESENTATION OF THE HEISENBERG ALGEBRA h4 BY THE LOWEST LANDAU LEVELS AND THEIR COHERENT STATES

2005 ◽  
Vol 20 (30) ◽  
pp. 2295-2303
Author(s):  
H. FAKHRI ◽  
Z. SHADMAN
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Coherent States ◽  
Rotational Symmetry ◽  
Dynamical Symmetry ◽  
Heisenberg Algebra ◽  
Landau Levels ◽  
Shape Invariance ◽  
Half Axis

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H4 with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h4 not only by the lowest Landau levels, but also by their corresponding standard coherent states.

scholarly journals Generalized Symmetries and mCK Method Analysis of the (2+1)-Dimensional Coupled Burgers Equations

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1473 ◽  
Author(s):  
Gangwei Wang ◽  
Yixing Liu ◽  
Shuxin Han ◽  
Hua Wang ◽  
Xing Su
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Transformation Group ◽  
Symmetry Transformation ◽  
Burgers Equations ◽  
Generalized Symmetries ◽  
Method Analysis

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation group theorem is presented. From symmetry transformation group theorem, a great many of new solutions can be derived. Lastly, Lie algebra for given symmetry group are considered.

scholarly journals Symmetry Group for Solving Elliptic Euler-Poisson-Darboux Equation

2021 ◽  
pp. 228-232
Author(s):  
Zainab John
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Partial Differential ◽  
Darboux Equation

The aim of this article is to study the solution of  Elliptic Euler-Poisson-Darboux equation, by using the symmetry of Lie Algebra of orders two and three, as a contribution in partial differential equations and their solutions.

scholarly journals Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

2021 ◽  
Vol 39 (2) ◽  
Author(s):  
Danilo García Hernández ◽  
Oscar Mario Londoño Duque ◽  
Yeisson Acevedo ◽  
Gabriel Loaiza
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Complete Classification ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Generating Operators ◽  
Alternative Solutions ◽  
Schwarz Equation

We obtain the complete classification of the Lie symmetry group and the optimal system’s generating operators associated with a particular case of the generalized Kummer - Schwarz equation. Using those operators we characterize all invariant solutions, alternative solutions were found for the equation studied and the Lie algebra associated with the symmetry group is classified.

scholarly journals New Solutions of the Heat Equation and Application to Thin Plate Heat Conduction

2020 ◽  
Vol Volume 14 ◽  
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Heat Equation ◽  
Ordinary Differential Equations ◽  
Symmetry Group ◽  
Thin Plate ◽  
Lie Symmetry ◽  
Second Order ◽  
Group Generator ◽  
Plate Heat

Using a Lie symmetry group generator and a generalised form of Euler’s formula for solving second order ordinary differential equations, we determine new symmetries for the heat equation, leading to new solutions. As an application, we test a formula resulting from this approach on thin plate heat conduction

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