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Rogue Waves: Mathematical Theory and Applications in Physics PDF

213 Pages·2017·2.734 MB·English
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BolingGuo,LixinTian,ZhenyaYan,LimingLing,Yu-FengWang RogueWaves Also of interest VanishingViscosityMethod.SolutionstoNonlinearSystems BolingGuo,DongfenBian,FangfangLi,XiaoyuXi,2017 ISBN978-3-11-049528-7,e-ISBN978-3-11-049427-3 StochasticPDEsandDynamics BolingGuo,HongjunGao,XuekePu,2016 ISBN978-3-11-049510-2,e-ISBN978-3-11-049388-7 NonlinearEquationswithSmallParameter NikolaiTarkhanov,SergeiGlebov,OlegM.Kiselev,2017 ISBN978-3-11-033554-5,e-ISBN978-3-11-033568-2 NonlinearDynamics.Non-IntegrableSystemsandChaoticDynamics AlexanderBorisov,2017 ISBN978-3-11-043938-0,e-ISBN978-3-11-043058-5 DeGruyterSeriesinNonlinearAnalysisandApplications JürgenAppelletal.(Eds.) ISSN0941-813X Boling Guo, Lixin Tian, Zhenya Yan, Liming Ling, Yu-Feng Wang Rogue Waves Mathematical Theory and Applications in Physics MathematicsSubjectClassification2010 Primary:37K10,37J35,37K15;Secondary:70H06,81R12 Authors Prof.BolingGuo Prof.ZhenyaYan LaboratoryofComputationalPhysics ChineseAcademyofSciences InstituteofAppliedPhysicsand InstituteofSystemsScience ComputationalMathematics KeyLabMathematicsMechanization 6HuayuanRoad 55ZhongguancunEastRoad HaidianDistrict HaidianDistrict 100088Beijing 100190Beijing People’sRepublicofChina People’sRepublicofChina [email protected] [email protected] Prof.LixinTian AssociateProf.LimingLing NanjingNormalUniversity SouthChinaUniversityofTechnology SchoolofMathematicalScience WushanRD.,TianheDistrict CenterforEnergyDevelopment 510641Guangzhou 210023Nanjing People’sRepublicofChina People’sRepublicofChina [email protected] [email protected] Dr.Yu-FengWang InstituteofAppliedPhysicsand ComputationalMathematics 6HuayuanRoad HaidianDistrict 100088Beijing People’sRepublicofChina [email protected] ISBN978-3-11-046942-4 e-ISBN(PDF)978-3-11-047057-4 e-ISBN(EPUB)978-3-11-046969-1 Set-ISBN978-3-11-047058-1 LibraryofCongressCataloging-in-PublicationData ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress. BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;detailed bibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2017WalterdeGruyterGmbH,Berlin/Boston Typesetting:IntegraSoftwareServicesPvt.Ltd. Printingandbinding:CPIbooksGmbH,Leck Coverimage:ChongGuo @Printedonacid-freepaper PrintedinGermany www.degruyter.com Contents 1 TheResearchProcessforRogueWaves 1 1.1 TheResearchProcessforRogueWavePhenomenon 1 1.2 SomeFamousExperimentsofRogueWaves 5 1.3 ResearchMethodandPhysicalMechanismofRogueWaves 9 1.3.1 MethodologyofRogueWaves 9 1.3.2 PhysicalMechanismofRogueWaves 10 1.4 MechanismsofRogueWaves 12 1.4.1 LinearMechanismsofRogueWaves 12 1.4.2 NonlinearMechanismsofRogueWaves 17 1.5 RogueWaveSolutionsforNonlinearPartialDifferentialEquations 21 1.6 OpticalRogueWaves 26 1.7 FinancialRogueWaves 28 1.8 NonautonomousRogueWaveSolutions 28 2 ConstructionofRogueWaveSolutionbytheGeneralizedDarboux Transformation 30 2.1 TheClassicalDarbouxTransformation 30 2.2 GeneralizedDarbouxTransformationfortheClassicalKdVEquation 32 2.3 DarbouxTransformationforN-CoupledFocusingNLSEquation 35 2.4 RogueWaveSolutionsfortheTwo-ComponentNLSEquation 37 2.4.1 RogueWaveSolutionsfortheTwo-ComponentNLS Equation 37 2.4.2 Bright-DarkBreatherandRogueWaveSolutions 41 2.5 GeneralizedDarbouxTransformationforNLSEquation 45 2.5.1 GeneralizedDarbouxTransformation 46 2.5.2 Higher-OrderRogueWavesinDeterminantForms 50 2.5.3 MathematicalCharactersoftheRogueWaveSolutionsfor StandardNLSEquations 55 2.6 GeneralizedDarbouxTransformationforDNLSEquation 59 2.6.1 DarbouxTransformation-I 59 2.6.2 DarbouxTransformation-II 63 2.6.3 Reductions 66 2.6.4 GeneralizedDarbouxTransformations 67 2.6.5 GeneralizedDarbouxTransformation-II 69 2.6.6 High-OrderSolutionsforDNLSEquation 72 3 ConstructionofRogueWaveSolutionbyHirotaBilinearMethod, Algebro-geometricApproachandInverseScatteringMethod 81 3.1 HirotaBilinearMethod 81 3.1.1 RogueWaveSolutionfortheNLSEquation 81 3.1.2 RogueWaveSolutionfortheDS-IEquation 98 VI Contents 3.2 ReductionfromtheKPEquation 105 3.3 Algebro-geometricReductionApproach 109 3.3.1 Relationship Between Fredholm Determinant and (-Function 110 3.3.2 WronskianSolutions 112 3.3.3 ConstructionofRogueWaveSolution 116 3.4 InverseScatteringMethodandRogueWaveSolution 118 3.4.1 DirectProblem 119 3.4.2 ScatteringMatrix 120 3.4.3 InvolutionRelation 120 3.4.4 JumpsoftheEigenfunctionsandScatteringDataAcrossthe BranchCut 122 3.4.5 TimeEvolution 123 3.4.6 InverseProblem 124 3.4.7 DarbouxTransformationandRogueWaveSolutions 127 4 TheRogueWaveSolutionandParameters ManaginginNonautonomousPhysicalModel 135 4.1 IntroductiontotheRogueWaveSolution 135 4.2 Space-TimeModulationNonlinearSchrödingerEquation 137 4.2.1 One-DimensionalNonlinearPhysicalModel 137 4.2.2 SymmetryAnalysis-SimilarityTransformationandSimilarity Solution 137 4.2.3 OneDimensionalSelf-SimilarityOpticalRogueWave SolutionandItsParameterAnalysis 139 4.3 (3+1)-DimensionalSpace-TimeModulationGross–Pitaevskii/NLS Equation 144 4.3.1 Three-DimensionalNonlinearPhysicalModel 144 4.3.2 SymmetryAnalysis-SimilarityTransformationandReduction System 146 4.3.3 SimilarityVariable,ConstraintConditionandVelocity Field 147 4.3.4 Three-DimensionalSelf-SimilarRogueWaveSolutionsand ItsParametersRegulation 148 4.4 GeneralizedInhomogeneousHigher-OrderNonlinearSchrödinger EquationwithModulatingCoefficients 152 4.4.1 SymmetryReductions–TransformationandHirota Equation 154 4.4.2 Determining Similarity Variables and Controlled Coefficients 156 4.4.3 DarbouxTransformationfortheHirotaEquation 159 4.4.4 OpticalRogueWaveSolutions 160 Contents VII 4.5 Two-DimensionalBinaryMixturesofBose–Einstein—Condensates 165 4.5.1 Two-ComponentGross–PitaevskiiEquations 165 4.5.2 SymmetryReductionAnalysis 166 4.5.3 Determining Similarity Variables and Controlled Coefficients 168 4.5.4 TypesofNonlinearInteraction 170 4.5.5 Self-SimilarVectorRogueWaveSolution 171 4.6 Two-DimensionalNonlocalNonlinearSchrödingerEquation 176 4.6.1 Two-DimensionalNonlocalNonlinearModel 176 4.6.2 Two-DimensionalVariableSeparationReduction 177 4.6.3 Two-DimensionalRogueWave-LikeSolution 178 4.7 TheGeneralizedAblowitz–Ladik–HirotaLatticewithVariable Coefficients 184 4.7.1 DiscreteNonlinearPhysicalModel 184 4.7.2 Differential-Difference Similarity Reductions and Constraints 186 4.7.3 Determining the Similarity Transformation and Coefficients 187 4.7.4 NonautonomousDiscreteRogueWaveSolutionsand Interaction 188 Bibliography 193 1 The Research Process for Rogue Waves 1.1 TheResearchProcessforRogueWavePhenomenon Roguewaveisalsonamedasfreakwave,monsterwave,extremewave,killerwave, giantwave,etc.Itisdifficulttogiveafullexplanationofroguewave,duetoitscom- plex phenomenon. Figure 1.1 shows us the dramatic appearance of the rogue wave whichwasdownloadedfromtheInternet.Theauthoritystatementinoceanography isthatitdevelopsfromtheoceansuddenlywithhighamplitude,whichappearsfrom nowhereanddisappearswithoutatrace.Nowadays,suchnonlinearphenomenonhas beenobservedinnonlinearoptics,Bose–Einsteincondensates(BEC),atmospherics, superfluidandevenfinance. Ithasarousedtheattentionofscientistsfrom oceano- graphy, physics and other nonlinear fields, since the term “freak rogue waves” was firstcoinedbyDraper[1]in1965. Forcenturies,roguewavehasbeenseenasthesealegendandapartofthemarine folkculture. Intheocean,rogue waves,like adeep-seamonster,engulfsailorsand shipswithoutanytrace,whichcausedalotofmaritimedisasters. In 1933, in the Pacific Northwest, USA warship Ramapo encountered the tallest roguewaveeverrecorded,with34m(112ft)height,whichwasvisuallymeasuredby thecrewsonthedeck[2]. Since 1952, in the Indian Ocean, at least 12 cases of ships encountering rogue wave have been recorded, near the Aga DeGeneres stream and along the coast of South Africa. One miserable case is that, on June 13, 1968, the tanker World Glory encountered a rogue wave, which broke the tanker into two parts (as seen in Fig- ure1.2)andcausedthedeathof22crews,whenthetankerwasalongtheSouthAfrican coast[3]. In1966,theMichelangelocruiseship,duringthevoyagefromItalytotheUnited States, suddenly encountered a huge wave with 24 m height. The wave tore a hole inthesuperstructure,smashedtheheavyglass, andmadeacrewmemberandtwo passengersdie. In1978,Munich,aGermanbarge,sankintheAtlanticOcean.Thetwistedwreck- ageindicatesthatitwasdestroyedbyahugewave. In1980,aroguewavewith25mheightwasphotographedbyPhilippeLijour,the firstmateoftheFrenchtankerEssoLanguedoc,neartheseaoftheeasternportcity DurbanofSouthAfrica(Azania).AsseeninFigure1.3,ahugewaterholewasobserved clearly. In 1984, a rogue wave attacked the 2/4-A oil platform of Norwegian Ekofisk oil field,whichwaslocatedaboveaveragesealevelof20m.Inthisaccident,thewallof theplatform’scontrolroomwasdestroyed,producingastandstillfor24hours[4]. In 1986, aUS warship SS Spray suffered three consecutive burstsof waves with approximately25mheightintheseaofCharleston[5].Ahighlyasymmetricalrogue wavewasalsoobservedasshowninFigure1.4,whichwasshotfromthewarship.

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