Field theoretical Lie symmetry analysis

The Möbius group, exact solutions of conformal autonomous systems, and predictive model-building

Kyriakos Christodoulides

Research output: Contribution to journalArticle

Abstract

We study single and coupled first-order differential equations (ODEs) that admit symmetries with tangent vector fields, which satisfy the N-dimensional Cauchy-Riemann equations. In the two-dimensional case, classes of first-order ODEs which are invariant under Möbius transformations are explored. In the N dimensional case we outline a symmetry analysis method for constructing exact solutions for conformal autonomous systems. A very important aspect of this work is that we propose to extend the traditional technical usage of Lie groups to one that could provide testable predictions and guidelines for model-building and model-validation. The Lie symmetries in this paper are constrained and classified by field theoretical considerations and their phenomenological implications. Our results indicate that conformal transformations are appropriate for elucidating a variety of linear and nonlinear systems which could be used for, or inspire, future applications. The presentation is pragmatic and it is addressed to a wide audience.

Original languageEnglish
Pages (from-to)2191-2199
Number of pages9
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume19
Issue number7
DOIs
Publication statusPublished - Jul 2014
Externally publishedYes

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Lie Symmetry
Predictive Model
Autonomous Systems
Differential equations
Exact Solution
Cauchy-Riemann Equations
Symmetry
Tangent vector
Lie groups
Conformal Transformation
Model Validation
First order differential equation
Linear systems
Nonlinear systems
Vector Field
Nonlinear Systems
Linear Systems
Differential equation
First-order
Invariant

Cite this

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title = "Field theoretical Lie symmetry analysis: The M{\"o}bius group, exact solutions of conformal autonomous systems, and predictive model-building",
abstract = "We study single and coupled first-order differential equations (ODEs) that admit symmetries with tangent vector fields, which satisfy the N-dimensional Cauchy-Riemann equations. In the two-dimensional case, classes of first-order ODEs which are invariant under M{\"o}bius transformations are explored. In the N dimensional case we outline a symmetry analysis method for constructing exact solutions for conformal autonomous systems. A very important aspect of this work is that we propose to extend the traditional technical usage of Lie groups to one that could provide testable predictions and guidelines for model-building and model-validation. The Lie symmetries in this paper are constrained and classified by field theoretical considerations and their phenomenological implications. Our results indicate that conformal transformations are appropriate for elucidating a variety of linear and nonlinear systems which could be used for, or inspire, future applications. The presentation is pragmatic and it is addressed to a wide audience.",
keywords = "Autonomous systems, Conformal ODEs, Lie symmetry analysis, M{\"o}bius group",
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TY - JOUR

T1 - Field theoretical Lie symmetry analysis

T2 - The Möbius group, exact solutions of conformal autonomous systems, and predictive model-building

AU - Christodoulides, Kyriakos

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PY - 2014/7

Y1 - 2014/7

N2 - We study single and coupled first-order differential equations (ODEs) that admit symmetries with tangent vector fields, which satisfy the N-dimensional Cauchy-Riemann equations. In the two-dimensional case, classes of first-order ODEs which are invariant under Möbius transformations are explored. In the N dimensional case we outline a symmetry analysis method for constructing exact solutions for conformal autonomous systems. A very important aspect of this work is that we propose to extend the traditional technical usage of Lie groups to one that could provide testable predictions and guidelines for model-building and model-validation. The Lie symmetries in this paper are constrained and classified by field theoretical considerations and their phenomenological implications. Our results indicate that conformal transformations are appropriate for elucidating a variety of linear and nonlinear systems which could be used for, or inspire, future applications. The presentation is pragmatic and it is addressed to a wide audience.

AB - We study single and coupled first-order differential equations (ODEs) that admit symmetries with tangent vector fields, which satisfy the N-dimensional Cauchy-Riemann equations. In the two-dimensional case, classes of first-order ODEs which are invariant under Möbius transformations are explored. In the N dimensional case we outline a symmetry analysis method for constructing exact solutions for conformal autonomous systems. A very important aspect of this work is that we propose to extend the traditional technical usage of Lie groups to one that could provide testable predictions and guidelines for model-building and model-validation. The Lie symmetries in this paper are constrained and classified by field theoretical considerations and their phenomenological implications. Our results indicate that conformal transformations are appropriate for elucidating a variety of linear and nonlinear systems which could be used for, or inspire, future applications. The presentation is pragmatic and it is addressed to a wide audience.

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