scholarly journals Lie Algebra Classification, Conservation Laws, and Invariant Solutions for a Generalization of the Levinson–Smith Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
G. Loaiza ◽  
Y. Acevedo ◽  
O.M.L. Duque ◽  
Danilo A. García Hernández
Keyword(s):  
Nonlinear Dynamics ◽  
Lie Algebra ◽  
Generalized Equation ◽  
Group Method ◽  
Invariant Solutions ◽  
Lie Group Method ◽  
Points Of View ◽  
Generating Operators ◽  
The Given ◽  
Lie Algebra Classification

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.

scholarly journals Lie algebra classification, conservation laws and invariant solutions for modification of the generalization of the Emden--Fowler equation.

2021 ◽  
Author(s):  
Yeisson Acevedo Agudelo ◽  
Gabriel Loaiza Ossa ◽  
Oscar Londoño Duque ◽  
Danilo García Hernández
Keyword(s):  
Conservation Laws ◽  
Lie Algebra ◽  
Invariant Solutions ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Generating Operators ◽  
Variational Symmetries ◽  
The Given ◽  
Lie Algebra Classification ◽  
Fowler Equation

We obtain the optimal system’s generating operators associated to a modification of the generalization of the Emden–Fowler Equation. equation. Using those operators we characterize all invariant solutions associated to a generalized. Moreover, we present the variational symmetries and the corresponding conservation laws, using Noether’s theorem and Ibragimov’s method. Finally, we classify the Lie algebra associated to the given equation.

scholarly journals Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Ben Gao ◽  
Yanxia Wang
Keyword(s):  
Chaplygin Gas ◽  
Group Method ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
Lie Group Method ◽  
Two Component ◽  
Present Universe ◽  
Group Invariant Solutions ◽  
Similarity Reductions

In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.

scholarly journals Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Na Lv ◽  
Xuegang Yuan ◽  
Jinzhi Wang
Keyword(s):  
Direct Method ◽  
Lax Pair ◽  
Group Method ◽  
Symmetry Groups ◽  
Invariant Solutions ◽  
Lax Pairs ◽  
Group Invariant ◽  
Lie Group Method ◽  
Group Invariant Solutions ◽  
Symmetry Transformations

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

Symmetry and Painlevé analysis for the extended Sakovich equation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Gangwei Wang ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Lie Group ◽  
Design Methodology ◽  
Original Work ◽  
Soliton Solutions ◽  
Symmetry Analysis ◽  
Group Method ◽  
Invariant Solutions ◽  
Content Type ◽  
Lie Group Method ◽  
Practical Implications

Purpose The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation. Design/methodology/approach The newly developed Sakovich equation has been handled by using the Lie symmetries via using the Lie group method. Findings The developed extended Sakovich model exhibit symmetries and invariant solutions. Research limitations/implications The present study is to address the two main motivations: the study of symmetry analysis and the study of soliton solutions of the extended Sakovich equation. Practical implications The work introduces symmetry analysis to the Painlevé-integrable extended Sakovich equation. Social implications The work presents useful symmetry algorithms for handling new integrable equations. Originality/value The paper presents an original work with symmetry analysis and shows useful findings.

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 Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1313
Author(s):  
Hengtai Wang ◽  
Zhiwei Zou ◽  
Xin Shen
Keyword(s):  
Nonlinear Equation ◽  
Lie Group ◽  
Conservation Law ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Group Method ◽  
Lie Symmetry Analysis ◽  
Lie Group Method ◽  
The Given ◽  
Similarity Reductions

In the present paper, we mainly focus on the symmetry of the solutions of a given PDE via Lie group method. Meanwhile we transfer the given PDE to ODEs by making use of similarity reductions. Furthermore, it is shown that the given PDE is self-adjoining, and we also study the conservation law via multiplier approach.

scholarly journals Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa
Keyword(s):  
Lie Group ◽  
Traveling Wave ◽  
Invariant Solution ◽  
Traveling Wave Solutions ◽  
Group Method ◽  
Lie Group Method ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Negative Order ◽  
Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

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