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.

LanguageEnglish
Pages2191-2199
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
Möbius
Predictive Model
Autonomous Systems
Differential equations
Exact Solution
Cauchy-Riemann Equations
Symmetry
Möbius Transformation
Tangent vector
Lie groups
Conformal Transformation
Model Validation
First order differential equation
Linear systems
Nonlinear systems
Vector Field
Nonlinear Systems
Linear Systems
Differential equation

Cite this

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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.",
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T2 - Communications in Nonlinear Science and Numerical Simulation

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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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