Bilinear Discretization of Integrable Quadratic Vector Fields: Algebraic Structure and Algebro-Geometric Solutions [Elektronische Ressource] / Andreas Pfadler. Betreuer: Yuri B. Suris

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• Mathematics

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• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Bilinear Discretization of Integrable Quadratic Vector Fields:
Algebraic Structure and Algebro-Geometric Solutions
vorgelegt von Diplom-Mathematiker
Andreas Pfadler
aus Munc hen
Von der Fakult at II - Mathematik und Naturwissenschaften
der Technischen Universit at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. John Sullivan (TU Berlin)
Berichter/Gutachter: Prof. Dr. Yuri B. Suris (TU Berlin)
Berichter/Gutachter: Prof. Dr. Pantelis A. Damianou (University of Cyprus)
Tag der wissenschaftlichen Aussprache: 14.10.2011
Berlin 2011
D 832
Acknowledgement
First and foremost I would like to thank my family and especially my wife for all
their love and support during my Ph.D. studies. Without them I could not have come
this far.
Secondly I would like to thank my advisor Prof. Dr. Yuri B. Suris for his guidance
and support in the last two years. I am especially grateful for his introducing me to
the fascinating world of integrable systems. The many hours spent together in front
of computers and complicated formulas will not be forgotten.
Also, I would like to thank Dr. Matteo Petrera for many interesting discussions
and all the help he was able to provide in mathematical as well non mathematical
matters. Finally, I thank Prof. Damianou and Prof. Sullivan for agreeing to join the
examination commitee for this thesis.
During the creation of this thesis I was enrolled as a student of the Berlin Mathe-
matical School (BMS), which has provided me with excellent boundary conditions for
this work. Hence, I also wish to express my gratitutde to the whole administrative and
academic sta of BMS.3
Abstract
This thesis discusses the integrability properties of a class of bilinear discretizations
of integrable quadratic vector elds, the so called Hirota-Kimura type discretizations.
This method tends to produce integrable birational mappings. The integrability prop-
erties of these mappings are discussed in detail and - where possible - solved exactly in
terms of elliptic functions or their relatives. Integrability of the mappings under con-
sideration is typically characterized by conserved quantitites, invariant volume forms
and particular invariance relations, formulated in the language of so called HK bases.
After a short introduction into the theory of nite dimensional integrable systems
in the continuous and discrete setting, a general methodology for discovery and proof
of integrability of birational mappings is developed. This methodology is based on the
concept of HK bases. Having recalled the basics of the theory of elliptic functions, the
relations between HK bases and elliptic solutions of integrable birational mappings is
explored. This makes it possible to formulate a general approach to the explicit inte-
gration of integrable birational mappings, provided they are solvable in terms of elliptic
functions. The appealing feature of this approach is that it does not require knowledge
of additional structures typically characterizing integrability (e.g. Lax pairs).
Having discussed the general properties of the HK type discretizations, several ex-
amples are with the help of the previously introduced methods. In particular,
discretizations of the following systems are considered: Euler top, Zhukovsky-Volterra
system, three and four dimensional periodic Volterra systems, Clebsch system, Kirch-
ho System, and Lagrange top. HK bases, conserved quantities and invariant volume
forms are found for all examples. Furthermore, explicit solutions in terms of elliptic
functions or their relatives are obtained for the Volterra systems and the Kirchho
system.
Methodologically this work is based on the concept of experimental mathematics.
This means that discovery and proof of most of the presented results are based on
computer experiments and the usage of specialized symbolic computations.Contents
1 Introduction 6
1.1 Methodological Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Integrability in the Continuous and Discrete Realm 10
2.1 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Complete Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Integrable Discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Detecting and Proving Integrability of Birational Maps . . . . . . . . . . 18
2.4.1 Algebraic Entropy and Diophantine Integrability . . . . . . . . . 20
2.4.2 Hirota-Kimura Bases . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.3 Algorithmic Detection of HK Bases . . . . . . . . . . . . . . . . . 27
2.4.4 HK Bases and Symbolic Computation . . . . . . . . . . . . . . . 30
2.4.5 Invariant Volume Forms for Integrable Birational Maps . . . . . 33
2.4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Elements of the Theory of Elliptic Functions 36
3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Relations Between Elliptic Functions And Addition Theorems . . . . . . 40
3.3 Elliptic Functions, Experimental Mathematics And Discrete Integrability 43
4 The Hirota-Kimura Type Discretizations 46
4.1 First Integrable Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Weierstrass Dierential Equation . . . . . . . . . . . . . . . . . . 51
4.1.2 Euler Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 A More Complicated Example: The Zhukovski-Volterra System . . . . . 57
4.2.1 ZV System with Two Vanishing ’s . . . . . . . . . . . . . . . . 58k
4.2.2 ZV with One V . . . . . . . . . . . . . . . . . 61k
4.2.3 ZV System with All ’s Non-Vanishing . . . . . . . . . . . . . . 61k
4.3 Integrability of the HK type Discretizations . . . . . . . . . . . . . . . . 61
5 3D and 4D Volterra Lattices 64
5.1 Elliptic Solutions of the In nite Volterra Chain . . . . . . . . . . . . . . 64
5.2 Three-periodic Volterra chain: Equations of Motion and Explicit Solution 65
5.3 HK type Discretization of VC . . . . . . . . . . . . . . . . . . . . . . . 673
4Contents 5
5.4 Solution of the Discrete Equations of Motion . . . . . . . . . . . . . . . 68
5.5 Periodic Volterra Chain with N = 4 Particles . . . . . . . . . . . . . . . 72
5.6 HK type Discretization of VC . . . . . . . . . . . . . . . . . . . . . . . 744
5.7 Solution of the Discrete Equations of Motion . . . . . . . . . . . . . . . 75
6 Integrable Cases of the Euler Equations on e(3) 81
6.1 Clebsch System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.1.1 First HK Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.2 Remaining HK Bases . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.3 First Additional HK Basis . . . . . . . . . . . . . . . . . . . . . . 89
6.1.4 Second HK Basis . . . . . . . . . . . . . . . . . . . . 93
6.1.5 Proof for the Bases ; ; . . . . . . . . . . . . . . . . . . . . 941 2 3
6.2 General Flow of the Clebsch System . . . . . . . . . . . . . . . . . . . . 99
6.3 Kirchho System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4 HK type Discretization of the Kirchho System . . . . . . . . . . . . . . 106
6.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4.2 HK Bases and Conserved Quantities . . . . . . . . . . . . . . . . 106
6.5 Solution of the Discrete Kirchho System . . . . . . . . . . . . . . . . . 110
6.6 Lagrange Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Conclusion and Future Perspectives 124
A MAPLE Session Illustrating the Application of the Algorithm (V) i
B The PSLQ Algorithm iii
List of Figures vi
Bibliography vii1
Introduction
The theory of integrable systems is a rich and old eld of mathematics. In a sense
it is as old as the the subject of di erential equations itself. Since Newton’s solution
of the Kepler problem, which might be considered as the rst integrable system in
the history of mathematics, mathematicians and physicists have been trying to nd
di erential equations which could be \integrated", that is solved exactly in terms of
previously known functions.
After Newton, Euler and Lagrange discovered two new integrable systems, which
are now known as the Euler Top and the Lagrange Top. The study of the functions
which characterized their solutions fueled the subsequent development of analysis,
leading to the systematic study of elliptic functions and their higher genus analogs by
Gauss, Abel, Jacobi and their contemporaries.
At this time there was, however, no precise notion of the term integrability. Back
then, integrability of a system of di erential equation, would usually mean, that the
equations of motion could be reduced to simpler equations whose solutions were then
found by inversion of elliptic or hyperelliptic integrals. A precise notion of integrability
was rst formulated by Liouville. He showed that Hamilton’s equations could be
transformed into a simple linear set of di erential equations if the system of equations
possessed enough independent conserved quantities.
Soon, new integrable systems were discovered; among them were the so called
Kirch