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1、TrialEquationMethodtoNonlinearEvolutionEquationswithRankInhomogeneous：MathematicalDiscussionsandItsApplicationsCommun.Theor.Phys.(Beijing,China)45(2006)PP.219-223@InternationalAcademicPublishersVo1.45,No.2,February15,2006TrialEquationMethodtoNonlinearEvolutionEquationswithRankInhomogeneous:Mat
2、hematicalDiscussionsandItsApplicationsLIUCheng-ShiDepartmentofMathematics,DaqingPetroleumInstitute,Daqing163318,China(ReceivedMay17,2005)AbstractAtrialequationmethodtononlinearevolutionequationwithrankinhomogeneousisgiven.As郇catj0ns,theexacttravelingwaIresolutionstosomehigher-ordernonlinearequ
3、ationssuchasgeneralizedBoussinesqequation.generalizedPochhammer-Chreeequation,KdV-Burgersequation,andKSequationandso0n,areobtained.Amongthese,someresultsarenew.Theproposedmethodisbasedontheideaofreductionoftheordef0fODE.S0皿emathematicaldermisoftheproposedmethodarediscussed.PACSnumbers:02.30.Jr
4、,05.45.YvKeywords:trialequationmethod,solvableequation,nonlinearevolutionequation,exactsolution1IntroductionBasedontheideaofreductionoforderoftheordinarydifferentialequation(ODE),weintroducedanewmethod,thatis.trialequationmethodtoseekfortheexacttravel—ingwavesolutionstononlinearevolutionequati
5、onswithrankhomogeneous.Ix,2]Thekeyideaofthetrialequationmethodistoreducetheequationconsideredtothesolv-ableequationasfollows:t上=F()(1)Inmanycases,F(u)isapolynomia1.Inthepresentpaper,weproposeatrialequationmethodtothenonlinearevolutionequationswithrankinhomogeneous.Infact,inmanycasesweonlyneedt
6、otakeanewtrialequation.whichisdifferenttotheequa-tion(1).InSec.2,wegiveasimpletrialequationmethod.Butsometimesweneedchangeourmethodsothatitisapplicabletomoregeneralequations.InSec.3,wedis—CUSSthemathematicalaspectsofthetrialequation,andproposeamorepowerfulbutmorecomplexmethod.InSec.4,asapplica
7、tions'weobtainsomenewsolutionstosomesoliton—producingequationswithhighernonlineartermssuchascompoundKdV-Burgers—typeequationt上t+duPu￡+bu2nuz+rt上z+t上z∞=0,(2)wherer≠u,d≠0,P>0,d,b=const.,generalizedBoussinesqequationu托一(u+dupu+bu2pu+ru+
