Integrability of systems of ordinary differential equations via Lie point symmetries

2021 ◽  
Author(s):  
Fatma Al‐Kindi ◽  
Fazal M. Mahomed ◽  
Muhammad Ziad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Point Symmetries

scholarly journals Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 884
Author(s):  
Linyu Peng
Keyword(s):  
Partial Differential Equations ◽  
General Theory ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Vries Equation ◽  
Lie Point Symmetries ◽  
Nonlocal Equations ◽  
De Vries ◽  
Nonlocal Nonlinear

In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.

scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

scholarly journals A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Said Waqas Shah ◽  
F. M. Mahomed ◽  
H. Azad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Second Order ◽  
Two Dimensional ◽  
Lie Point Symmetries ◽  
Complete Integration ◽  
Symmetry Algebras

The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries.

Lie point symmetries of delay ordinary differential equations

2019 ◽  
pp. 89-118
Author(s):  
Vladimir A. Dorodnitsyn ◽  
Roman Kozlov ◽  
Sergey V. Meleshko ◽  
Pavel Winternitz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Point Symmetries
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