On Study of Nonlinear Partial Differential Equations in Mathematical Physics

Abstract

Theobjectiveofthisthesisistostudythenonlinearpartialdi¤erentialequa- tions(PDEs)inmathematicalphysicsinorderto ndnewexactsolutionsincluding Jacobiellipticfunctionsolutions,hyperbolicfunctionsolutions,trigonometricfunc- tionsolutions,rationalfunctionsolutions,solitonsandothersolutionsforthemby applyingvariousmethods.Thesemethodsare:thenewextendedauxiliaryequation method,thenewmappingmethod,theJacobiellipticequationmethod,thegener- alizedRiccatiequationmappingmethod,thegeneralizedKudryashovmethod,the 6-modelexpansionmethod,the rstintegralmethod,therational (G0=G)-expansion method,thegeneralizedsub-ODEmethod,thesolitonansatzmethod,thesine-cosine method,theRiccatiequationexpansionmethod,the (G0=G)-expansionmethod,the Bernoulliequationmethod,themodi edsimpleequationmethod,theexp-function method,thegeneralExpa-functionmethod,themultipleexp-functionmethodand thelinearsuperpositionprinciple. Thisthesisconsistsofsevenchapters,togetherwithanArabicandEnglishsum- maries.Itisorganizedasfollows: Inchapterone, wecompriseanintroductioncontainedinabriefsurvey.De- velopmentandsurveyoftheavailablemathematicaltoolsusedtogenerateexact solutionsfornonlinearPDEsgiveninchaptersII-VII. Inchaptertwo, weapplyfourdi¤erentmethodsto ndmanynewexactsolutions ofeightnonlinearPDEs.Firstlyinsections2.1-2.3,weapplyanewextendedauxil- iaryequationmethodto ndmanynewJacobiellipticfunctionsolutions,hyperbolic functionsolutionsandtrigonometricfunctionsolutionsofthehigher-ordernonlin- earSchrödingerequationwithderivativenon-Kerrnonlinearterms,thehigher-order