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Cover: 9783642887055 | What Is Integrability? | Vladimir E. Zakharov | Taschenbuch | XIV
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What Is Integrability?
Taschenbuch von Vladimir E. Zakharov
Sprache: Englisch

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Beschreibung
The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg­ ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas­ sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields.
The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg­ ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas­ sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields.
Zusammenfassung
This monograph deals with integrable dynamic systems with an infinite number of degrees of freedom. Leading scientists were invited to discuss the notion of integrability with two main points in mind: 1. a presentation of the various recently elaborated methods for determining whether a given system is integrable or not; 2. to understand the increasingly more important role of integrable systems in modern applied mathematics and theoretical physics.
Inhaltsverzeichnis
Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- Addendum.- References.- Painlevé Property and Integrability.- 1. Background.- 2. Integrability.- 3. Riccati Example.- 4. Balances.- 5. Elliptic Example.- 6. Augmented Manifold.- 7. Argument for Integrability.- 8. Separability.- References.- Integrability.- 1. Integrability.- 2. Introduction to the Method.- 3. The Integrable Hénon-Heiles System: A New Result.- 4. A Mikhailov and Shabat Example.- 5. Some Comments on the KdV Hierarchy.- 6. Connection with Symmetries and Algebraic Structure.- 7. Integrating the Nonintegrable.- References.- The Symmetry Approach to Classification of Integrable Equations.- 1. Basic Definitions and Notations.- 2. The Burgers Type Equations.- 3. Canonical Conservation Laws.- 4. Integrable Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 3. Applications to Particular Systems.- Appendix I.- Appendix II.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 2. Integrable Lagrangean Mappings with Discrete Time.- Appendix A.- Appendix B.- References.- The Cauchy Problem for the KdV Equation with Non-Decreasing Initial Data.- 1. Reflectionless Potentials.- 2. Closure of the Sets B(??2).- 3. The Inverse Problem.- References.
Details
Erscheinungsjahr: 2012
Fachbereich: Theoretische Physik
Genre: Physik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 344
Reihe: Springer Series in Nonlinear Dynamics
Inhalt: xiv
321 S.
2 s/w Illustr.
321 p. 2 illus.
ISBN-13: 9783642887055
ISBN-10: 3642887058
Sprache: Englisch
Herstellernummer: 86017866
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Redaktion: Zakharov, Vladimir E.
Herausgeber: Vladimir E Zakharov
Auflage: Softcover reprint of the original 1st ed. 1991
Hersteller: Springer Berlin
Springer Berlin Heidelberg
Springer Series in Nonlinear Dynamics
Maße: 235 x 155 x 19 mm
Von/Mit: Vladimir E. Zakharov
Erscheinungsdatum: 27.04.2012
Gewicht: 0,522 kg
preigu-id: 105278685
Zusammenfassung
This monograph deals with integrable dynamic systems with an infinite number of degrees of freedom. Leading scientists were invited to discuss the notion of integrability with two main points in mind: 1. a presentation of the various recently elaborated methods for determining whether a given system is integrable or not; 2. to understand the increasingly more important role of integrable systems in modern applied mathematics and theoretical physics.
Inhaltsverzeichnis
Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- Addendum.- References.- Painlevé Property and Integrability.- 1. Background.- 2. Integrability.- 3. Riccati Example.- 4. Balances.- 5. Elliptic Example.- 6. Augmented Manifold.- 7. Argument for Integrability.- 8. Separability.- References.- Integrability.- 1. Integrability.- 2. Introduction to the Method.- 3. The Integrable Hénon-Heiles System: A New Result.- 4. A Mikhailov and Shabat Example.- 5. Some Comments on the KdV Hierarchy.- 6. Connection with Symmetries and Algebraic Structure.- 7. Integrating the Nonintegrable.- References.- The Symmetry Approach to Classification of Integrable Equations.- 1. Basic Definitions and Notations.- 2. The Burgers Type Equations.- 3. Canonical Conservation Laws.- 4. Integrable Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 3. Applications to Particular Systems.- Appendix I.- Appendix II.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 2. Integrable Lagrangean Mappings with Discrete Time.- Appendix A.- Appendix B.- References.- The Cauchy Problem for the KdV Equation with Non-Decreasing Initial Data.- 1. Reflectionless Potentials.- 2. Closure of the Sets B(??2).- 3. The Inverse Problem.- References.
Details
Erscheinungsjahr: 2012
Fachbereich: Theoretische Physik
Genre: Physik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 344
Reihe: Springer Series in Nonlinear Dynamics
Inhalt: xiv
321 S.
2 s/w Illustr.
321 p. 2 illus.
ISBN-13: 9783642887055
ISBN-10: 3642887058
Sprache: Englisch
Herstellernummer: 86017866
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Redaktion: Zakharov, Vladimir E.
Herausgeber: Vladimir E Zakharov
Auflage: Softcover reprint of the original 1st ed. 1991
Hersteller: Springer Berlin
Springer Berlin Heidelberg
Springer Series in Nonlinear Dynamics
Maße: 235 x 155 x 19 mm
Von/Mit: Vladimir E. Zakharov
Erscheinungsdatum: 27.04.2012
Gewicht: 0,522 kg
preigu-id: 105278685
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