Exp-function method for nonlinear wave equations

Exp-functionmethodfornonlinearwaveequationsJi-HuanHe*,Xu-HongWuCollegeofScience,DonghuaUniversity,1882Yan-anXiluRoad,Shanghai20051,PRChinaAccepted7March2006CommunicatedbyProf.G.IovaneAbstractInthispaper,anewmethod,calledExp-functionmethod,isproposedtoseeksolitarysolutions,periodicsolutionsandcompacton-likesolutionsofnonlineardifferentialequations.ThemodifiedKdVequationandDodd–Bullough–Mikhailovequationarechosentoillustratetheeffectivenessandconvenienceofthesuggestedmethod.2006ElsevierLtd.Allrightsreserved.1.IntroductionRecentlymanynewapproachestononlinearwaveequationshavebeenproposed,forexample,tanh-functionmethod[1–6],F-expansionmethod[7–9],Jacobianellipticfunctionmethod[10–12],variationaliterationmethod[13,14],Adomianmethod[15–18],variationalapproach[19–21],andhomotopyperturbationmethod[22–24].Allmeth-odsmentionedabovehavelimitationintheirapplications.InthispaperwesuggestanovelmethodcalledExp-functionmethod(orExp-methodforshort)tosearchforsolitarysolutions,compact-likesolutionsandperiodicsolutionsofvar-iousnonlinearwaveequations.2.BasicideaofExp-functionmethodInordertoillustratethebasicideaofthesuggestedmethod,weconsiderfirstthefollowingnonlineardispersiveequationoftheform[13,25–27]:utþu2uxþuxxx¼0.ð1ÞThisequationiscalledmodifiedKdVequation,whicharisesintheprocessofunderstandingtheroleofnonlineardis-persionandintheformationofstructureslikeliquiddrops,anditexhibitscompactons:solitonswithcompactsupport.Introducingacomplexvariationgdefinedasg¼kxþxt.ð2Þ0960-0779/$-seefrontmatter2006ElsevierLtd.Allrightsreserved.doi:10.1016/j.chaos.2006.03.020*Correspondingauthor.E-mailaddresses:[e-mail](J.-H.He),[e-mail](X.-H.Wu).Chaos,SolitonsandFractals30(2006)700–708þku2u0þk3u000¼0;ð3Þwhereprimedenotesthedifferentialwithrespecttog.TheExp-functionmethodisverysimpleandstraightforward,itisbasedontheassumptionthattravelingwavesolu-tionscanbeexpressedinthefollowingform:uðgÞ¼Pdn¼canexpðngÞPqm¼pbmexpðmgÞ;ð4Þwherec,d,p,andqarepositiveintegerswhichareunknowntobefurtherdetermined,anandbmareunknownconstants.WesupposethatthesolutionofEq.(3)canbeexpressedasuðgÞ¼acexpðcgÞþþadexpðdgÞapexpðpgÞþþaqexpðqgÞ.ð5ÞTodeterminevaluesofcandp,webalancethelineartermofhighestorderinEq.(3)withthehighestordernonlinearterm.Bysimplecalculation,wehaveu000¼c1exp½ð7pþcÞgþc2exp½8pgþð6Þandu2u0¼c3exp½ðpþ3cÞgþc4exp½4pgþ¼c3exp½ð5pþ3cÞgþc4exp½8pgþ;ð7Þwhereciaredeterminedcoefficientsonlyforsimplicity.BalancinghighestorderofExp-functioninEqs.(6)and(7),wehave7pþc¼5pþ3c;ð8Þwhichleadstotheresultp¼c.ð9ÞSimilarlytodeterminevaluesofdandq,webalancethelineartermoflowestorderinEq.(3)u000¼þd1exp½ð7qþdÞgþd2exp½8qgð10Þandu2u0¼þd3exp½ðqþ3dÞgþd4exp½4qg¼þd3exp½ð5qþ3dÞgþd4exp½8qg;ð11Þwherediaredeterminedcoefficientsonlyforsimplicity.BalancinglowestorderofExp-functioninEqs.(10)and(11),wehaveð7qþdÞ¼ð5qþ3dÞ;ð12Þwhichleadstotheresultq¼d.ð13ÞForsimplicity,wesetp=c=1andq=d=1,soEq.(5)reducestouðgÞ¼a1expðgÞþa0þa1expðgÞexpðgÞþb0þa1expðgÞ.ð14ÞSubstitutingEq.(14)intoEq.(3),andbythehelpofMatlab,wehave1A½C3expð3gÞþC2expð2gÞþC1expðgÞþC0þC1expðgÞþC2expð2gÞþC3expð4gÞ¼0;ð15ÞJ.-H.He,X.-H.Wu/Chaos,SolitonsandFractals30(2006)700–708701whereA¼ðexpðgÞþb1expðgÞþb0Þ4;C3¼xa1b0þka31b0k3a0wa0ka21a0þk3a1b0;C2¼8k3a1b1þ2ka31b14k3a1b202wa12ka1a20þ2wa1b1þ4k3a0b02ka21a1þ2ka21a0b0þ2xa1b202xa0b08k3a1;C1¼xa1b30þ6xa1b0b1xa0b20k3a0b2018k3a1b0b16ka1a0a1þka1a20b0ka30þ23k3a0b1xa0b15xa1b0þk3a1b305k3a1b0þka21a1b0þ5ka21a0b1;C0¼4xa1b214ka1a21þ32k3a1b1þ4ka1a20b132k3a1b21þ4k3a1b20b14xa1b14k3a1b204ka20a14xa1b20þ4ka21a1b1þ4xa1b20b1;C1¼18k3a1b0b16xa1b0b1k3a1b30þk3a0b1b20þxa0b215ka0a21þ5xa1b0b21þxa0b1b20xa1b30ka1a21b023k3a0b21ka20a1b0þ5k3a1b0b21þka30b1þ6ka1a0a1b1;C2¼2xa0b21b02xa1b212ka31þ2ka1a21b1þ2xa1b314k3a0b21b02xa1b20b1þ4k3a1b20b18k3a1b21þ2ka20a1b12ka0a21b0þ8k3a1b31;C3¼ka0a21b1þxa0b31ka31b0þk3a0b31xa1b0b21k3a1b0b21.Equatingthecoefficientsofexp(ng)tobezero,wehaveC3¼0;C2¼0;C1¼0;C0¼0;C2¼0;C3¼0;C4¼0:8:ð16ÞSolvingthesystem,Eq.(16),simultaneously,weobtaina0¼a1b0þ3k2b0a1;a1¼b20ð3k2þ2a21Þ8a1;b1¼b20ð3k2þ2a21Þ8a21;x¼ka21k3;8:ð17Þwherea1andb0arefreeparameters.We,therefore,obtainthefollowingsolution:uðx;tÞ¼a1exp½kxðka21þk3Þtþa1b0þ3k2b0a1þb20ð3k2þ2a21Þ8a1expðkxþðka21þk3ÞtÞexp½kxðka21þk3Þtþb0þb20ð3k2þ2a21Þ8a21expðkxþðka21þk3ÞtÞ¼a1þ3k2b0a1exp½kxðka21þk3Þtþb0þb20ð3k2þ2a21Þ8a21expðkxþðka21þk3ÞtÞ.ð18ÞGenerallya1,b0,andkarerealnumbers,andtheobtainedsolution,Eq.(18),isageneralizedsolitonarysolution.Incasekisanimaginarynumber,theobtainedsolitonarysolutioncanbeconvertedintoperiodicsolutionorcom-pact-likesolution.Wewritek¼iK.ð19ÞUsethetransformationexp½kxðka21þk3Þt¼exp½iKxiðKa21K3Þt¼cos½KxðKa21K3Þtþisin½KxðKa21K3Þtandexp½kxþðka21þk3Þt¼exp½iKxþiðKa21K3Þt¼cos½KxðKa21K3Þtisin½KxðKa21K3Þt.Eq.(18)becomesuðx;tÞ¼a1þ3K2b0a1ð1þpÞcos½KxðKa21K3Þtþb0þið1pÞsin½KxðKa21K3Þt;ð20Þ