Zeros of polynomials and solvable nonlinear evolution equations

179 Pages·2018·1.4 MB·English

by  Calogero, Francesco

ZEROS OF POLYNOMIALS AND SOLVABLE
NONLINEAR EVOLUTION EQUATIONS
Reporting a novel breakthrough in the identification and investigation of solvable
and integrable systems of nonlinearly coupled evolution equations, this book
includes many examples that illustrate this approach. Beginning with systems of
ODEs,includingsecond-orderODEsofNewtoniantype,itthendiscussessystems
of PDEs, and systems evolving in discrete time. It reports a novel, differential
algorithm to evaluate all the zeros of a generic polynomial of arbitrary degree:
a remarkable development of a fundamental mathematical problem with a long
history. This book will be of interest to applied mathematicians and mathematical
physicists working in the area of integrable and solvable nonlinear evolution
equations; it can also be used as supplementary reading material for general
AppliedMathematicsorMathematicalPhysicscourses.
francesco calogero is Professor of Theoretical Physics (Emeritus) at the
SapienzaUniversityofRome,Italy.Hehaspublishednumerouspapersandbooks
on physics and mathematics as well as on arms control and disarmament topics
and he served as Secretary General of the Pugwash Conferences on Science and
WorldAffairsfrom1989–1997(1995NobelPeacePrize).

ZEROS OF POLYNOMIALS AND
SOLVABLE NONLINEAR
EVOLUTION EQUATIONS
FRANCESCO CALOGERO
PhysicsDepartment,UniversityofRome“LaSapienza”,Rome,Italy
INFN,SezionediRoma1
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108428590
DOI:10.1017/9781108553124
©FrancescoCalogero2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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Firstpublished2018
PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Calogero,Francesco,1955–author.
Title:Zerosofpolynomialsandsolvablenonlinearevolutionequations/
FrancescoCalogero(SapienzaUniversityofRome,retired).
Description:Cambridge:CambridgeUniversityPress,2018.|
Includesbibliographicalreferences.
Identifiers:LCCN2018017272|ISBN9781108428590(hardback)
Subjects:LCSH:Evolutionequations,Nonlinear.|Zero(Thenumber)|
Polynomials.|Differentialequations,Partial.|Differentialequations,Nonlinear.
Classification:LCCQC20.7.E88C2682018|DDC530.1/55353–dc23
LCrecordavailableathttps://lccn.loc.gov/2018017272
ISBN978-1-108-42859-0Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
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Contents
Preface pageix
1 Introduction 1
2 Parameter-DependentMonicPolynomials:Definitions,KeyFormulas
andOtherPreliminaries 4
2.1 Notation 4
2.2 KeyFormulas 6
2.2.1 First-OrderandSecond-OrdertDerivativesoftheZeros
ofaMonict-DependentPolynomial 7
2.2.2 Third-OrderandFourth-Ordert-DerivativesoftheZeros
ofaMonict-DependentPolynomial 7
2.2.3 Proofs 8
2.3 PeriodicityoftheZerosofaTime-DependentPeriodicPolynomial 9
2.4 HowCertainEvolutionEq(cid:2)ua(cid:3)tionsCanBeMadePeriodic 10
2.4.1 TheODEγ(cid:2)(cid:2) = c γ(cid:2) 2γ−1 andItsIsochronousVariant 12
(cid:2) (cid:3)
2.4.2 TheODEγ(cid:2)(cid:2) = (y¯(cid:2))−1(cid:3) γ(cid:2) 2 andItsIsochronousVariant 16
2.4.3 TheODEγ(cid:2)(cid:2) = c γ(cid:2) α andItsIsochronousVariant 17
(cid:2) (cid:3)
2.4.4 TheSolvableODEγ(cid:2)(cid:2) = ρ γ(cid:2) 2γ−1+cγρ andRelated
IsochronousVersions (cid:2) (cid:3) 19
2.4.5 OtherSolvableCasesoftheODEγ(cid:2)(cid:2) = c γ(cid:2) αγβand
RelatedIsochronousVariants 21
2.N NotesonChapter2 25
3 ADifferentialAlgorithmtoComputeAlltheZerosofaGeneric
Polynomial 26
3.1 ADifferentialAlgorithmBasedonaSystemofFirst-Order
ODEsSuitabletoEvaluateAlltheZerosofaGenericPolynomial 26
v
vi Contents
3.2 ADifferentialAlgorithmBasedonaSystemofSecond-Order
ODEsSuitabletoEvaluateAlltheZerosofaGenericPolynomial 29
3.2.1 Introduction 29
3.2.2 TheTask 29
3.2.3 TheAlgorithm 30
3.2.4 TheProof 30
3.2.5 Remarks 31
3.N NotesonChapter3 32
4 SolvableandIntegrableNonlinearDynamicalSystems:Mainly
NewtonianN-BodyProblemsinthePlane 34
4.1 The“Goldfish” 36
4.2 IntegrableHamiltonianN-BodyProblemsofGoldfish
TypeFeaturingN ArbitraryFunctions 39
4.2.1 AVerySimpleIsochronousCase 41
4.2.2 ACaseThatInvolvesEllipticFunctionsintheSolutions 42
4.2.3 ARatherSimpleCaseandItsIsochronousVariant 43
4.2.4 AnotherRatherSimpleCase,ItsIsochronousVariant,
andItsTreatmentintheRealContext 47
4.2.5 TheGoldfish-CMModel 61
4.2.6 AnN-BodyProblemInvolvingN2 ArbitraryCoupling
Constants 64
4.3 SolvableN-BodyProblemswithAdditional
Velocity-DependentForces 65
4.3.1 AVerySimpleExample 66
4.3.2 AnotherSimpleExampleandItsIsochronousVariant 67
4.3.3 YetAnotherSimpleIsochronousModel 70
4.3.4 ASolvableN-BodyProblemofGoldfishTypeFeaturing
ManyArbitraryParameters,andItsIsochronousVariant 72
4.3.5 AnIntegrableHamiltonianN-BodyProblemofGoldfish
Type,andItsIsochronousVariant 76
4.4 AnotherN-BodyProblemWhichIsIsochronousWhenIts2N
ParametersAreArbitraryRationalNumbers 78
4.5 TwoClassesofSolvableN-BodyProblemsofGoldfishType 85
4.6 SolvableN-BodyProblemsIdentifiedviatheFindingsof
Subsection2.2.2 90
4.7 SolvableDynamicalSystems(SetsofFirst-OrderODEs) 99
4.7.1 AnIsochronousDynamicalSystem 99
4.7.2 AnotherIsochronousDynamicalSystem 102
Contents vii
4.8 HowtoManufactureManyMoreSolvableDynamicalSystems 105
4.N NotesonChapter4 107
5 SolvableSystemsofNonlinearPartialDifferentialEquations(PDEs) 110
5.1 AC-IntegrableSystemofNonlinearPDEs 112
5.2 AS-IntegrableSystemofNonlinearPDEs 113
5.3 Proofs 115
5.4 Outlook 116
5.N NotesonChapter5 118
6 GenerationsofMonicPolynomials 119
6.1 HierarchiesofSolvableN-BodyProblemsRelatedto
GenerationsofPolynomials 125
6.2 ThePeculiarMonicPolynomialsTheZerosofWhichCoincide
withTheirCoefficients 131
6.N NotesonChapter6 141
7 DiscreteTime 143
7.1 SolvableDiscrete-TimeDynamicalSystems 145
7.2 Examples 153
7.3 Outlook 158
7.N NotesonChapter7 159
8 Outlook 160
8.N NotesonChapter8 161
Appendix ComplexNumbersandReal2-Vectors 162
References 164

Preface
This short book is based on the algebraic relationship among the N zeros and
the N coefficients of a (monic) polynomial of arbitrary degree N. This is a core
mathematical topic since time immemorial; indeed the important mathematical
subdiscipline called “algebraic geometry” originated from investigations of this
relationshipandisfocusedonitsramifications,whicharequitedeep.Theapproach
used in this book is instead quite elementary; it is essentially based on the simple
idea to consider (monic) polynomials of arbitrary degree N in their argument—
say,thecomplexvariablez—whichmoreoveralsodependonaparameter—saythe
realvariablet—generallyconvenientlyinterpretedas“time”(thedependenceona
secondparameterplayingtheroleofadditional“space”variableisalsoconsidered,
see Chapter 5). This implies that both the N coefficients and the N zeros of such a
time-dependent monic polynomial depend on time, and this simple notion allows
a certain number of interesting developments relevant to the topics mentioned in
the title of this book. Some of these developments emerged quite recently; indeed
this book is a compilation of results obtained and published by its author—alone
orwithcollaborators—overthelastthreeyears.Thisisreflectedinitsbibliography
and in the fact that occasionally the text below is drawn—even verbatim—from
thesepublications.Butunpublishedfindingsarealsoincluded.
Theauthorisverygratefultohisco-authors—OksanaBihun,MarioBruschi,and
Franc¸ois Leyvraz—who have been instrumental in obtaining some of the findings
reported in this book (as specified below), and also to his co-authors of previous
papersonrelatedtopics—severalofwhichareidentifiedinthebibliographicnotes
at the end of each chapter, and/or are referred to in the papers quoted there—
including Fabio Briscese, Antonio Degasperis, Silvana De Lillo, Luca Di Cerbo,
Riccardo Droghei, Marianna Euler, Norbert Euler, Jean Pierre Franc¸oise, David
Go´mez-Ullate, Sandro Graffi, Vladimir I. Inozemtsev, Sabino Iona, Edwin Lang-
mann, Mauro Mariani, Paolo Maria Santini, Matteo Sommacal, Ji Xiaoda, and
Ge Yi. Interactions with these colleagues and with others I met in Rome and at
ix