Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations

Persistent Link:
http://hdl.handle.net/10150/193894
Title:
Painleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equations
Author:
Lu, Yixia
Issue Date:
2005
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations.In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation.After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups.The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.
Type:
text; Electronic Dissertation
Keywords:
Painleve analysis; Lie symmetries; Integrability; ODE
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Ercolani, Nicholas M.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titlePainleve Analysis, Lie Symmetries and Integrability of Nonlinear Ordinary Differential Equationsen_US
dc.creatorLu, Yixiaen_US
dc.contributor.authorLu, Yixiaen_US
dc.date.issued2005en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations.In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation.After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups.The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectPainleve analysisen_US
dc.subjectLie symmetriesen_US
dc.subjectIntegrabilityen_US
dc.subjectODEen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairErcolani, Nicholas M.en_US
dc.contributor.committeememberErcolani, Nicholas M.en_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberTabor, Michaelen_US
dc.identifier.proquest1103en_US
dc.identifier.oclc137353986en_US
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