简介:InthispaperweconsiderpolynomialsplinesS(x)withequidistantnodeswhichmaygrowa5O(|x|~5).Wepresentanintegralrepresentationofsuchsplineswithadistributionkernel.Thisrepre-sentationisrelatedtotheFourierintegralofslowlygrowingfunctions.ThepartoftheFourierex-ponentialsherewithplaythesocalledexponentialsplinesbySchoenberg.Theintegralrepresentationprovidesaflexibletoolfordealingwiththegrowingequidistantsplines.First.itallowsustocon-structarichlibraryofsplinespossessingthepropertythattranslationsofanysuchsplineformaba-sisofcorrespondingsplinespace.Itisshownthatanysuchsplineisassociatedwithadualsplinewhosetranslationsformabiorthogonalbasis.Asexampleswepresentsolutionsoftheproblemsofprojectionofagrowingfunctionontosplinespacesandofsplineinterpolationofgrowingfunc-tion.Wederiveformulasforapproximateevaluationofsplinesprojectingafunctionontothesplinespaceandestablishtherewithexactestimationsoftheapproximationerrors.
简介:Existingfarfieldexpressionsofsecondorderpotentialsarebynomeanscomplete.Hencetherehasbeennoexactfarfieldexpressionofsecondorderpotentials.InthispaperthefarfieldexpressionforΦd(2)ispurposelyavoidedindeducingtheformulaeofsecondorderforcesandaseriesoffunctionsΦdRnareused.Thefarfieldexpressionofisgiven,whichfor(x,U,z)∈Σ,φdRn(2)φd(2).Usingthesepropertiesformulaeforcalculatingsecondorderdiffractionforcesareobtained.Tocalculatetheintegral∫∫1/gfΨdsitisdividedintotwoparts.Oneistheintegraloverafinitedomainandthefunctionundertheintegraliscontinuous,sotheusualapproximateintegrationformulaemaybeused.Theotheristheintegraloveraninfinitedomain.Usingthefarfieldexpressionoffirstorderpotentials,formulaeforcalculatingtheintegraltomeetgivenaccuraciesaregiven.Themooringforceinsurgedirectionisusedforcomparisonbetweennumericalpredictionsandexperimentalmeasurements.Thepredictedresultsarecheckedagainstthemeasuredvalueinaspeciallydesignedtest.Inthelowfrequencydomainofinterest,themooringforcesinsurge,forcalculatedandexperimentalspectraareingoodconsistencysolongasthedampingcoefficientsischoosenappropriately.
简介:Wesystematicallyinvestigatethemotionofslowlymovingmatter-wavegapsolitonsinanonlinearpotential,producedbytheweakrandomspatialvariationoftheatomicscatteringlength.Withtheweakrandomness,weconstructaneffective-particletheorytostudythemotionofgapsolitons.Basedontheeffective-particletheory,theeffectoftherandomnessongapsolitonsisobtained,andthemotionofgapsolitonsisfinallysolved.Moreover,theanalyticresultsforthegeneralbehavioursofgapsolitonmotion,suchastheensemble-averagespeedandthereflectionprobabilitydependingontheweakrandomnessareobtained.Wefindthatwiththeincreaseoftherandomstrengththeensemble-averagespeedofgapsolitonsdecreasesslowlywherethereductionisproportionaltothevarianceoftheweakrandomness,andthereflectionprobabilitybecomeslarger.ThetheoreticalresultsareingoodagreementwiththenumericalsimulationsbasedontheGross-Pitaevskiiequation.
简介:TheMelnikovmethodwasextendedtoperturbedplanarnon-Hamiltonianintegrablesystemswithslowly-varyingangleparameters.Basedontheanalysisofthegeometricstructureofunperturbedsystems,theconditionoftransverselyhomoclinicintersectionwasestablished.ThegeneralizedMelnikovfunctionoftheperturbedsystemwaspresentedbyapplyingthetheoremonthedifferentiabilityofordinarydifferentialequationsolutionswithrespecttoparameters.ChaosmayoccurinthesystemifthegeneralizedMelnikovfunctionhassimplezeros.