非线性物理科学：变换群和李代数（英文版）

　　《非线性物理科学：变换群和李代数（英文版）》为作者在俄罗斯、美国、南非和瑞典多年讲述变换群和李群分析课程的讲义。书中所讨论的局部李群方法提供了求解非线性微分方程解析解通用且非常有效的方法，而近似变换群可以提高构造含少量参数的微分方程的技巧。《非线性物理科学：变换群和李代数（英文版）》通俗易懂、叙述清晰，并提供丰富的模型，能帮助读者轻松地逐步深入各种主题。

　　伊布拉基莫夫（Ibragimov，N.H.），教授，瑞士科学家，被公认为是在微分方程对称分析方面世界上最具权威的专家之一。他发起并构建了现代群分析理论，并推动了该理论在多方面的应用。

Preface

PartⅠLocalTransformationGroups

1Preliminaries

1.1Changesofframesofreferenceandpointtransformations

1.1.1Translations

1.1.2Rotations

1.1.3Galileantransformation

1.2Introductionoftransformationgroups

1.2.1Definitionsandexamples

1.2.2Differenttypesofgroups

1.3Someusefulgroups

1.3.1Finitecontinuousgroupsonthestraightline

1.3.2Groupsontheplane

1.3.3GroupsinIRn

ExercisestoChapter1

2One-parametergroupsandtheirinvariants

2.1Localgroupsoftransformations

2.1.1Notationanddefinition

2.1.2Groupswritteninacanonicalparameter

2.1.3Infinitesimaltransformationsandgenerators

2.1.4Lieequations

2.1.5Exponentialmap

2.1.6Determinationofacanonicalparameter

2.2Invariants

2.2.1Definitionandinfinitesimaltest

2.2.2Canonicalvariables

2.2.3Constructionofgroupsusingcanonicalvariables

2.2.4Frequentlyusedgroupsintheplane

2.3Invariantequations

2.3.1Definitionandinfinitesimaltest

2.3.2Invariantrepresentationofinvariantmanifolds

2.3.3ProofofTheorem

2.3.4ExamplesonTheorem

ExercisestoChapter2

3Groupsadnuttedbydifferentialequations

3.1Preliminaries

3.1.1Differentialvariablesandfunctions

3.1.2Pointtransformations

3.1.3Frameofdifferentialequations

3.2Ptolongationofgrouptransformations

3.2.10ne-dimensionalcase

3.2.2Prolongationwithseveraldifferentialvariables

3.2.3Generalcase

3.3Prolongationofgroupgenerators

3.3.10ne-dimensionalcase

3.3.2Severaldifferentialvariables

3.3.3Generalcase

3.4Firstdefinitionofsymmetrygroups

3.4.1Definition

3.4.2Examples

3.5Seconddefinitionofsymmetrygroups

3.5.1Definitionanddeterminingequations

3.5.2Determiningequationforsecond-orderODEs

3.5.3Examplesonsolutionofdeterminingequations

ExercisestoChapter3

4Liealgebrasofoperators

4.1Basicdefinitions

4.1.2Propertiesofthecommutator

4.1.3Propertiesofdeterminingequations

4.2Basicproperties

4.2.1Notation

4.2.2Subalgebraandideal

4.2.3Derivedalgebras

4.2.4SolvableLiealgebras

4.3Isomorphismandsimilarity

4.3.1IsomorphicLieakebras

4.3.2SimilarLiealgebras

4.4Low-dimensionalLiealgebras

4.4.10ne-dimensionalalgebras

4.4.2Two-dimensionalalgebrasintheplane

4.4.3Three-dimensionalalgebrasintheplane

4.4.4Three-dimensionalalgebrasinlR3

4.5Liealgebrasandmulti-parametergroups

4.5.1Definitionofmulti-parametergroups

4.5.2Constructionofmulti-parametergroups

5Galoisgroupsviasymmetries

5.1Preliminaries

5.2Symmetriesofalgebraicequations

5.2.1Determiningequation

5.2.2Firstexample

5.2.3Secondexample

5.2.4Thirdexample

5.3ConstructionofGaloisgroups

5.3.1Firstexample

5.3.2Secondexample

5.3.3Thirdexample

5.3.4Concludingremarks

AssignmenttoPartI

PartIIApproximateTransformationGroups

6.1Motivation

6.2AsketchonLietransformationgroups

6.2.10ne-parametertransformationgroups

6.2.2Canonicalparameter

6.2.3GroupgeneratorandLieequations

6.3ApproximateCauchyproblem

6.3.1Notation

6.3.2DefinitionoftheapproximateCauchyproblem

7Approximatetransformations

7.1Approximatetransformationsdefined

7.2Approximateone-parametergroups

7.2.1Introductoryremark

7.2.2Definitionofone-parameterapproximate

7.2.3Generatorofapproximatetransformationgroup

7.3Infinitesimaldescription

7.3.1ApproximateLieequations

7.3.2Approximateexponentialmap

ExercisestoChapter7

8Approximatesymmetries

8.1Definitionofapproximatesymmetries

8.2Calculationofapproximatesymmetries

8.2.1Determiningequations

8.2.2Stablesymmetries

8.2.3Algorithmforcalculation

8.3.2ApproximatecommutatorandLiealgebras

9.1Integrationofequationswithasmallparameterusingapproximatesymmetries

9.1.1Equationhavingnoexactpointsymmetries

9.1.2Utilizationofstablesymmetries

9.2Approximatelyinvariantsolutions

9.2.1Nonlinearwaveequation

9.2.2ApproximatetravellingwavesofKdVequation

9.3Approximateconservationlaws

ExercisestoChapter9

AssignmenttoPartII

Bibliography

Index