Anoteonsolitarywavesinviscoelasticporousmedia Xin-SheYang Department of Fuel and Energy and Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK Accepted19July2000 Abstract Solitary waves occur in the system of wave propagation in a dispersive energy conserving medium. We show that soliton-like travellingwavesarisenaturallyinthe¯owofviscoelasticmedia. Ó 2001ElsevierScienceLtd.Allrightsreserved. 1. Introduction Density-driven¯owinviscoelasticmediaisaveryimportantandinterestingphenomenon.Itsmodelling israthercomplicatedduetothehighlynonlinearfeatureofthegoverningequations[1±4].Viscoelastic¯ow oftengenerateswavepatternsintheporousmedia[3±5].Numericalsimulationsandexperiment[5]indi- cated that the purely viscous ¯ow such as magma transport can support solitary wave under proper conditions,whileotherstudies[4]suggeststhattheporositywavestravelinasoliton-likefashion.However, wehavenotseenaclearanalysistoshowhowthesefeaturesoccur.Thus,inthisshortnote,weuseapairof modi®ednonlinearadvection±diusionequationtoshowhowthesolitarywavesarisenaturallyinvisco- elastic¯ow.Wealsoshowtheimportanceoftheviscosityinsupportingsoliton-likewaves. 2. Nonlinear advection±diusion equations Ina1-Dviscoelasticporousmediumwiththeaxis z pointingupward,theporosity / andexcesspressure w inthemediaevolvewithtime t.Byproperscalingsimilartotheformulationsin[2±4],wecanwritedown thegoverningequationsas o/ ot  o oz / m 1     ow oz   0; m P 1; 1 k o ot / ow oz   o/ oz   o 2 / oz 2  / ow oz  / 2 ; 2 where k  0:001±100 is the Doborah number [3].  is a parameter characterising the eect of eective pressure. These are highly nonlinear advection±diusion equations. The nonlinearity of porosity-depen- denceistospreadouttheenergyandthedispersivefeatureoftheexcesspressureistodrawtheenergy www.elsevier.nl/locate/chaos Chaos,SolitonsandFractals122001)2353±2356 E-mail address: xinshe@amsta.leeds.ac.ukX.-S.Yang). 0960-0779/01/$-seefrontmatter Ó 2001ElsevierScienceLtd.Allrightsreserved. PII:S0960-077900)00179-X