奇非线性行波方程研究的动力系统方法

TheStudiesofsolitonsandcompleteintegrabilityofnonlinearwaveequationsandbifurcationsandchaosofdynamicalystemsaretwoveryactivefieldsinnonlinearscience.Becauseahomoclinicorbitofatravelingwavesystem（ODEs）correspondstoasolitarywavesolutionofanonlinearwaveequation（PDE）.Thisfactprovidesanintersectionpointforabovetwostudyfields.Theaimofthisbookistogiveamoresystematicaccotmtforthebifurcationtheorymethodofdynamicalsystemstofindtravelingwavesolutionswithanemphasisonsingularwavesandunderstandtheirdynamicsforsomeclassesofthewellposednessofnonlinearevolutionequations.Readersshallfindhowstandardmethodsofthetheoryofdynamicalsystemsmaybeusedforthestudyoftravelingwavesolutionseventhecaseofsystemswithdiscontinuities.

Anyreadertryingtounderstandthesubjectofthisbookisonlyrequiredtoknowtheelementarytheoryofdynamicalsystemsandelementaryknowledgeofnonlinearwaveequations.Thisbookshouldbeusefulasaresearchreferenceforgraduatestudents，teachersandengineersindifferentstudyfields.

Chapter1TravelingWaveEquationsofSomePhysicalModels
1Themodelofnonlinearoscillationsofhyperelasticrods
2HigherorderwaveequationsofKorteweg-DeVriestype
3Camassa-Holmequationanditsgeneralizationforms
4Moreclassesofequationsofmathematicalphysics

Chapter2BasicMathematicalTheoryoftheSingularTravelingWaveSystems
2.1Somepreliminaryknowledgeofdynamicalsystems
2.2Phaseportraitsoftravelingwaveequationshavingsingularstraightlines.
2.3Maintheoremstoidentifytheprofilesofwavesandsomeexamples

Chapter3BifurcationsofTravelingWaveSolutionsofNonlinearElasticRodSystems
3.1Bifurcationsofphaseportraitsof（3.0.1）andphysicalacceptablesolutions
3.2Fourtypesofsolitarywavesandthreetypesofperiodicwaves
3.3Thenon-periodicbehaviorofaxialmotions

Chapter4BifurcationsofTravelingWaveSolutionsofGeneralizedCamassa-HolmEquation
4.1Bifurcationsofphaseportraitsofsystem（4.0.2）
4.2Exactparametricrepresentationsoftravelingwavesolutionsof（4.0.1）
4.3Theexistenceofsmoothsolitarywavesolutionsandperiodicwavesolutions

Chapter5BifurcationsofTravelingWaveSolutionsofHigherOrderKorteweg-DeVriesEquations
5.1TravelingwavesolutionsofthesecondorderKorteweg-DeVriesequationintheparameterconditiongroup（I）
5.2TravelingwavesolutionsofthesecondorderKorteweg-DeVriesequationintheparameterconditiongroup（II）
5.3TravelingwavesolutionsforthegeneralizationformofmodifiedKorteweg-DeVriesequation

Chapter6TheBifurcationsoftheTravelingWaveSolutionsofK（m，n）Equation
6.1Bifurcationsofphaseportraitsofsystem（6.0.2）
6.2Someexactexplicitparametricrepresentationsoftravelingwavesolutions
6.3Existenceofsmoothandnon-smoothsolitarywaveandperiodicwavesolutions
6.4Theexistenceofuncountablyinfinitemanybreakingwavesolutionsandconvergenceofsmoothandnon-smoothtravelingwavesolutionsasparametersarevaried

Chapter7KinkWaveSolutionDeterminedbyaParabolaSolutionofPlanarDynamicalSystems
7.1Sixclassesofnonlinearwaveequations
7.2Existenceofparabolasolutionsandtheirparametricrepresentations
7.3Kinkwavesolutionsof6classesofnonlinearwaveequations

Chapter8TravelingWaveSolutionsofCoupledNonlinearWaveEquations
8.1TravelingwaveequationoftheKupershmidtsequation
8.2Bifurcationsofphaseportraitsof（8.7）
8.3Existenceofsmoothsolitarywave，kinkwaveandperiodicwavesolutions
8.4Non-smoothperiodicwavesanduncountablyinfinitemanybreakingwavesolutions

Chapter9SolitaryWavesandChaoticBehaviorforaClassofCoupledFieldEquations
9.1Solitarywavesolutionsoftheintegrablecaseof（9.0.1）
9.2Theexistenceofsmallamplitudeperiodicsolutions
9.3Chaoticbehaviorofsolutionsof（9.0.2）
9.4Theexistenceofarbitrarilymanydistinctperiodicorbits

Chapter10BifurcationsofBreatherSolutionsofSomeNonlinearWaveEquations
10.1Introduction
10.2Bifurcationsoftravelingwavesolutionsofsystem（10.7）whenVRP（θ，r）givenby（10.2）
10.3Travelingwavesolutionsofsystem（10.1）withVRP（θ，r）givenby（10.2）
10.4Bifurcationsofsolutionsof（10.7）withVRP（θ，r）givenby（10.3）
10.5Travelingwavesolutionsof（10.1）withVRP（θ，r）givenby（10.3）
10.6Bifurcationsofbreathersolutionsof（10.4）

Chapter11BoundedSolutionsof（n+1）-DimensionalSineandSinh-GordonEquations
11（n+1）-dimensionalSine-andSinh-Gordonequations
12Theboundedsolutionsofthesystems（14）and（15）
13Theboundedtravelingwavesolutionsoftheform（12a）of（11）

Chapter12ExactExplicitTravelingWaveSolutionsforTwoClassesof（n+1）-DimensionalNonlinearWaveEquations
12.1（n+1）-dimensionalKlein-Gordon-Schrodingerequations
12.2（n+1）-dimensionalKlein-Gordon-Zakharovequations
References