逼近与重整化群对称

1LieGroupAnalysisinOutline

1.1Continuouspointtransformationgroups

1.1.1One-parametergroups

1.1.2Infinitesimaltransformations

1.1.3Lieequations

1.1.4Exponentialmap

1.1,5Canonicalvariables

1.1,6Invariantsandinvariantequations

1.2Symmetriesofordinarydifferentialequations

1.2.1Frameofdifferentialequations

1.2.2Extensionofgroupactionstoderivatives

1.2.3Generatorsofprolongedgroups

1.2.4Definitionofasymmetrygroup

1.2.5Mainpropertyofsymmetrygroups

1.2.6Calculationofinfinitesimalsymmetries

1.2.7Anexample

1.2.8Liealgebras

1.3Integrationoffirst-orderequations

1.3.1Liesintegratingfactor

1.3.2Methodofcanonicalvariables

1.4Integrationofsecond-orderequations

1.4.1CanonicalvariablesinLiealgebrasL2

1.4.2Integrationmethod

1.5Symmetriesofpartialdifferentialequations

1.5.1Mainconceptsillustratedbyevolutionequations

1.5.2Invariantsolutions

1.5.3Grouptransformationsofsolutions

1.6Threedefinitionsofsymmetrygroups

1.6.1Frameandextendedframe

1.6.2Firstdefinitionofsymmetrygroup

1.6.3Seconddefinition

1.6.4Thirddefinition

1.7Lie-Backlundtransformationgroups

1.7.1Lie-Backlundoperators

1.7.2Lie-Backlundequationsandtheirintegration

1.7.3Lie-Backlundsymmetries

References

2ApproximateTransformationGroupsandSymmetries

2.1Approximatetransformationgroups

2.1.1Notationanddefinitions

2.1.2ApproximateLieequations

2.1.3Approximateexponentialmap

2.2Approximatesymmetries

2.2.1Definitionofapproximatesymmetries

2.2.2Determiningequations&Stablesymmetries

2.2.3Calculationofapproximatesymmetries

2.2.4Examplesofapproximatesymmetries

2.2.5Integrationusingapproximatesymmetries

2.2.6Integrationusingstablesymmetries

2.2.7Approximatelyinvariantsolutions

2.2.8Approximateconservationlaws(firstintegrals)

References

3SymmetriesofIntegro-DifferentialEquations

3.1Definitionandinfinitesimaltest

3.1.1Definitionofsymmetrygroup

3.1.2Variationalderivativeforfunctionals

3.1.3Infinitesimalcriterion

3.1.4Prolongationonnonlocalvariables

3.2CalculationofsymmetriesillustratedbyVlasov-Maxwellequations

3.2.1One-dimensionalelectrongas

3.2.2Three-dimensionalplasmakineticequations

3.2.3PlasmakineticequationswithLagrangianvelocity

3.2.4Electron-ionplasmaequationsinquasi-neutralapproximation

References

4RenormgroupSymmetries

4.1Introduction

4.2Renormgroupalgorithm

4.2.1Basicmanifold

4.2.2Admittedgroup

4.2.3Restrictionofadmittedgrouponsolutions

4.2.4Renormgroupinvariantsolutions

4.3Examples

4.3.1ModifiedBurgersequation

4.3.2Examplefromgeometricaloptics

4.3.3Methodbasedonembeddingequations

4.3.4Renormgroupanddifferentialconstraints

References

5ApplicationsofRenormgroupSymmetries

5.1Nonlinearoptics

5.1.1Nonlineargeometricaloptics

5.1.2Nonlinearwaveoptics

5.1.3Renormgroupalgorithmusingfunctionals

5.2Plasmaphysics

5.2.1Harmonicsgenerationininhomogeneousplasma

5.2.2Nonlineardielectricpermittivityofplasma

5.2.3Adiabaticexpansionofplasmabunches

References

Index

　　ApproximateandRenormgroupSymmetriesdealswithapproximatetransformationgroups,symmetriesofintegro-differentialequationsandrenormgroupsymmetries.Itincludesaconciseandself-containedintroductiontobasicconceptsandmethodsofLiegroupanalysis,andprovidesaneasy-to-followintroductiontothetheoryofapproximatetransformationgroupsandsymmetriesofintegrodifferentialequations.　　Thebookisdesignedforspecialistsinnonlinearphysics--mathematiciansandnon-mathematicians--interestedinmethodsofappliedgroupanalysisforinvestigatingnonlinearproblemsinphysicalscienceandengineering.