The direct method in soliton theory

213 Pages·2004·1.315 MB·English

by  Ryogo Hirota, Atsushi Nagai, Jon Nimmo, Claire Gilson

TheDirectMethodinSolitonTheory
CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
B.BOLLOBA´S,W.FULTON,A.KATOK,F.KIRWAN,
P.SARNAK,B.SIMON
155 The Direct Method in
Soliton Theory
The Direct Method in Soliton Theory
RYOGO HIROTA
TranslatedfromJapaneseandeditedbyAtsushiNagai,
JonNimmoandClaireGilson
cambridge university press
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© Iwanami Shoten 1992
English translation: Cambridge University Press 2004
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Contents
Foreword pagevii
Preface ix
1 Bilinearizationofsolitonequations 1
1.1 Solitarywavesandsolitons 1
1.2 Nonlinearityanddispersion 2
1.3 Solutionsofnonlineardifferentialequations 9
1.4 Linearizationofnonlineardifferentialequations 12
1.5 Essentialsofthedirectmethod 19
1.6 The D-operator,anewdifferentialoperator 27
1.7 Bilinearizationofnonlineardifferentialequations 37
1.8 Solutionsofbilinearequations 46
1.9 Transformationfrombilineartononlinearform 58
2 Determinantsandpfaffians 60
2.1 Introduction 60
2.2 Pfaffians 61
2.3 Exterioralgebra 64
2.4 Expressionsforgeneraldeterminantsandwronskians 66
2.5 LaplaceexpansionsofdeterminantsandPlu¨ckerrelations 70
2.6 Jacobiidentitiesfordeterminants 77
2.7 Specialdeterminants 84
2.8 Pfaffianidentities 92
2.9 Expansionformulaeforthepfaffian(a ,a ,1,2,..., 2n)97
1 2
2.10 Additionformulaeforpfaffians 99
2.11 Derivativeformulaeforpfaffians 101
v
vi Contents
3 Structureofsolitonequations 110
3.1 Introduction 110
3.2 TheKPequation 111
3.3 TheBKPequation:pfaffiansolutions 127
3.4 ThecoupledKPequation 133
3.5 Thetwo-dimensionalTodalatticeequation 142
3.6 Thetwo-dimensionalTodamoleculeequation 149
4 Ba¨cklundtransformations 157
4.1 WhatisaBa¨cklundtransformation? 157
4.2 Ba¨cklundtransformationsforKdV-typebilinearequations 161
4.3 TheBa¨cklundtransformationfortheKPequation 169
4.4 TheBa¨cklundtransformationfortheBKPequation 175
4.5 ThesolutionofthemodifiedBKPequation 177
4.6 The Ba¨cklund transformation for the two-dimensional Toda
equation 179
4.7 Solutionofthetwo-dimensionalmodifiedTodaequation 184
Afterword 192
References 195
Index 198
Foreword
Thesecondhalfofthetwentiethcenturysawaresurgenceinthestudyofclas-
sicalphysics.Scientistsbeganpayingparticularattentiontotheeffectscaused
by the nonlinearity in dynamical equations. This nonlinearity was found to
have two interesting manifestations of opposite nature: chaos, that is the ap-
parent randomness in the behaviour of perfectly deterministic systems, and
solitons,thatislocalized,stablemovingobjectsthatscatteredelastically.Both
ofthesetopicshavenowbeendevelopedintoparadigms,withsolidmathemat-
icalbackgroundandwithawiderangeofphysicalobservationsandconcrete
applications.
Thisbookisconcernedwithaparticularmethodusedinthestudyofsoli-
tons.Therearemanywaysofstudyingtheintegrablenonlinearevolutionequa-
tionsthathavesolitonsolutions,eachmethodhavingitsownassumptionsand
areasofapplicability.Forexample,theinversescatteringtransform(IST)can
be used to solve initial value problems, but it uses powerful analytical meth-
odsandthereforemakesstrongassumptionsaboutthenonlinearequations.On
theotherhand,onecanfindatravellingwavesolutiontoalmostallequations
byasimplesubstitutionwhichreducestheequationtoanordinarydifferential
equation. Between these two extremes lies Hirota’s direct method. Although
thetransformationwas,atitsheart,inspiredbytheIST,Hirota’smethoddoes
notneedthesamemathematicalassumptionand,asaconsequence,themethod
is applicable to a wider class of equations than the IST. At the same time,
because it does not use such sophisticated techniques, it usually produces a
smaller class of solutions, the multi-soliton solutions. In many problems the
key to further developments is a detailed understanding of soliton scattering,
andinsuchcasesHirota’sbilinearmethodistheoptimaltool.
Overtheyears,manytextbookshavebeenwrittenonvariousaspectsofsoli-
tons.AlthoughsomeofthemhavebrieflymentionedHirota’sbilinearmethod,
therehasnotbeenanyintroductoryEnglishlanguagebookdevotedtoit.When
vii
viii Foreword
some of the western practitioners of Hirota’s method found out about prof.
Hirota’s book, they felt that it should be translated into English for use as
an introduction to the method. Early in 1997, Prof. J. Satsuma recruited his
students A. Nagai for this translation project. With help from his colleagues,
S. Tsujimoto and R. Willox, a first version was made; this was further im-
provedinacollaborationbetweenA.Nagai,J.J.C.NimmoandC.R.Gilson,
leadingtothefinalversionthatispresentedhere.
Inthisbook,Prof.Hirotaexplainshis‘direct’or‘bilinear’method.Thereis
aninterestingintroduction,fromwhichwecanseethemotivationandchainof
thoughtthatledProf.Hirotatoinventhismethod.Therestofthebookisde-
votedtoadetaileddiscussionofvariousapplicationsofthemethod.Littlehas
beenchangedinthetranslationfromtheoriginalbook;oneortwoarguments
havebeenexpanded,someerrorshavebeencorrectedandsomenotationwas
changed to improve consistency. All such changes have been made with the
approvalofProf.Hirota.
JarmoHietarinta
Turku,Finland
JonNimmo
Glasgow,Scotland
Preface
Asolitonisaparticulartypeofsolitarywave,whichisnotdestroyedwhenit
collideswithanotherwaveofthesamekind.Suchbehaviourissuggestedby
numerical simulation, but is it really possible that the soliton completely re-
coversitsoriginalshapeafteracollision?Indetailedanalysisoftheresultsof
suchnumericalsimulations,someripplescanbeobservedafteracollision,and
itthereforeseemsthattheoriginalshapeisnotcompletelyrecovered.There-
fore, in order to clarify whether or not solitons are destroyed through their
collisions,itisnecessarytofindexactsolutionsofsolitonequations.
Generally,itisaveryhardtasktofindexactsolutionsofnonlinearpartial
differentialequations,includingsolitonequations.Moreover,evenifoneman-
ages to find a method for solving one nonlinear equation, in general such a
methodwillnotbeapplicabletootherequations.Doesthereexistanysuccess-
fulanduniversaltoolenablingonetosolvemanytypesofnonlinearequations
whichdoesnotrequireadeepunderstandingofmathematics?Forthispurpose,
adirectmethodhasbeeninvestigated.
In Chapter 1, we discuss in an intuitive way the conditions under which
a solitary wave is formed and we show that a nonlinear solitary wave can-
not be made by the superposition of linear waves. From this observation, we
obtainamethodforfindingsolutionsofanonlinearwaveequationbyreduc-
tive perturbations and derive the fundamental idea for the direct method. We
firstintroducenewdependentvariables F andG toexpressthesolutionofthe
equation as the ratio G/F and then solve the (now bilinear) equations for F
andG.Aspartofthis,anewbinaryoperator,calledtheD-operator,isderived.
General formulae, through which nonlinear partial differential equations are
transformedintobilinear(or,ingeneral,homogeneous)forms,arepresented.
Byvirtueofspecialpropertiesofthe D-operator,solvingthesebilinearforms
by ordinary reductive perturbation methods leads to perturbation expansions
that may sometimes be truncated as finite sums. Such a truncation yields an
ix