奇非线性波方程：分支和精确解（英文版）

　　浅水波，非线性光学、电磁学、等离子物理、凝聚态物理、生物及化学、通讯等领域均存在非线性波运动.对其数学模型——波方程的解研究有重要价值.上世纪90年代，数学家发现了行波方程的非光滑的孤粒子解（peakon）、有限支集解（compacton）和圈解（loopsolution）等，为理解这些解，特别是非光滑解的出现，导致用动力系统的分支理论及方法对奇行波方程进行研究的新方向，《奇非线性波方程：分支和精确解（英文版）》介绍两类奇行波方程的研究的动力系统方法，及对大量数学物理问题的应用。

Preface

Chapter1SomePhysicalModelsWhichYieldTwoClassesofSingularTravellingWaveSystems

1．1Nonlinearwaveequationshavingthefirstclassofsingularnonlineartravellingwavesystems

1．2Nonlinearwaveequationshavingthesecondclassofsingularnonlineartravellingwaveequations

Chapter2DynamicsofSolutionsofSingularTravellingSystems

2．1Somepreliminaryknowledgeofdynamicalsystems

2．2Phaseportraitsoftravellingwavesystemshavingsingularstraightlines

2．3Dynamicalbehavioroforbitsinneighborhoodsofthesingular

straightline：thecaseof$1，2aresaddlepoints

2．4Dynamicalbehavioroforbitsinneighborhoodsofthesingularstraightline：thecaseof$1，2arenodepoints

2．5Dynamicalbehavioroforbitsdividedbythesingularcurves：

singulartravellingwavesystemsofthesecondclass

Chapter3ExactTravellingWaveSolutionsandTheirBifurcationsfortheKudryashov-SinelshchikovEquation

3．1Bifurcationsofphaseportraitsofsystem(3．0．4)

3．2Exacttravellingwavesolutionsfor/---3，-4

3．3Exacttravellingwavesolutionsfor/---1，2

Chapter4BifurcationsofTravellingWaveSolutionsofGeneralizedCamassa-HolmEquation(I)

4．1Bifurcationsofphaseportraitsof(4．0．2)

4．2Theexactparametricrepresentationsoftravellingwavesolutionsof(4．0．1)

4．3Theexistenceofsmoothsolitarywavesolutionsandperiodicwavesolutions

Chapter5BifurcationsofTravellingWaveSolutionsofHigherOrderKorteweg-DeVriesEquations

5．1TravellingwavesolutionsofthesecondorderKorteweg-DeVriesequationintheparameterconditiongroup(I)

5．2TravellingwavesolutionsofthesecondorderKorteweg-DeVries

equationintheparameterconditiongroup(II)

5．3TravellingwavesolutionsforthegeneralizationformofmodifiedKorteweg-DeVriesequation

Chapter6TheBifurcationsoftheTravellingWaveSolutionsofK(m，n)Equation

6．1Bifurcationsofphaseportraitsofsystem(6．0．2)

6．2Someexactexplicitparametricrepresentationsoftravellingwavesolutions

6．3Existenceofsmoothandnon-smoothsolitarywaveandperiodicwavesolutions

6．4Theexistenceofuncountablyinfinitemanybreakingwave

solutionsandconvergenceofsmoothandnon-smoothtravelling

wavesolutionsasparametersarevaried

Chapter7KinkWaveSolutionsDeterminedbyaParabolaSolutionofPlanarDynamicalSystems

7．1Sixclassesofnonlinearwaveequations

7．2Existenceofparabolasolutionsof(7．1．2)andtheirparametricrepresentations

7．3Kinkwavesolutionsfor6classesofnonlinearwaveequations

Chapter8ExactDarkSoliton，PeriodicSolutionsandChaoticDynamics

inaPerturbedGeneralizedNonlinearSchrhdingerEquation

8．1Theexactsolutionsof(8．0．2)forthecubicNLSequationwithf(q)=aq

8．2Theexactsolutionsof(8．0．2)forthecubic-quinticNLSequationwithf(q)=cqq2

8．3Thepersistenceofdarksolitionfortheperturbedcubic-quinticNLSequation(8．0．12)withoutthetermV(x)u

8．4Chaoticbehaviorofthetravellingwavesolutionsfortheperturbedcubic-quinticNLSequation(8．0．12)

Chapter9BifurcationsandSomeExactTravellingWaveSolutionsofaGeneralizedCamassa-HolmEquation(II)

9．1Bifurcationsofphaseportraitsof(9．0．5)

9．2Someexacttravellingwavesolutionof(9．0．2)inthesymmetrycases．

9．3Theexacttravellingwavesolutionsofequation(9．0．2)inanon-symmetriccase

Chapter10BifurcationsofBreatherSolutionsofSomeNonlinearWaveEquations

10．1Introduction

10．2Bifurcationsoftravellingwavesolutionsofsystem(10．1．7)whenVRp(0，r)givenby(10．1．2)

10．3Travellingwavesolutionsofsystem(10．1．1)withVap(0，r)givenby(10．1．2)

10．4Bifurcationsofsolutionsof(10．1．7)withVRp(0，r)givenby(10．1．3)．

10．5Travellingwavesolutionsof(10．1．1)withVRP(O，r)givenby(10．1．3)

10．6Bifurcationsofbreathersolutionsof(10．1．4)

Chapter11BoundedSolutionsof(n+1)-dimensionalSine-andSinh-GordonEquations

11．1(n+1)-dimensionalSine-andSinh-Gordonequations

11．2Theboundedsolutionsofthesystems(11．1．4)and(11．1．5)

11．3Theboundedtravellingwavesolutionsoftheform(ll．l．2a)of(ll．l．la)

Chapter12ExactLoopSolutionsandTheirDynamicsofSomeNonlinearWaveEquations

12．1Theelasticbeamequation

12．2ThereducedOstrovskyequation

12．3Theshortpulseequation

12．4Morenonlinearwaveequationswhichhavebreakingloop-solutions-．

Chapter13ExactSolitaryWave，PeriodicandQuasi-periodicWaveSolutionsfortheKdV6Equations

13．1Theequilibriumpointsandlinearizedsystemsof(13．0．6)

13．2Exactsolitarywaveandquasi-periodicwavesolutionsoftheCDGequation(13．0．12)

13．3ExactsolutionsoftheKaup-Kupershmidtequation(13．0．10)

13．4ExactsolutionsoftheKdV6equation(13．0．2)

13．5ExactsolutionsoftheKdV6equation(13．0．3)

Chapter14ExactTravellingWaveSolutionsandTheirDynamicsforaClassCoupledNonlinearWaveEquations

14．1Exactexplicitsolutionsy=xl()of(14．0．4a)whenP(t)hasthefactorization(14．1．1)

14．2Somepropertiesofsolutionsv()ofequation(14．0．45)

Chapter15OntheTravellingWaveSolutionsforaNonlinearDiffusion-convectionEquation

15．1Thedynamicsofthetravellingwavesolutionsandtheexistenceofglobalmonotonicwavefrontsolutionsof(15．0．1)

15．2Dynamicalbehaviorofsystem(15．0．3)

15．3Exacttravellingwavesolutionsof(15．0．3)

References