International Journal of Non-Linear Mechanics 40 (2005) 1041–1048 www.elsevier.com/locate/nlm Longitudinalstrainsolitarywavesinpresenceof cubicnon-linearity A.V.Porubov a , ∗ ,G.A.Maugin b a Ioffe Physico-Technical Institute of the Russian Academy of Sciences, Polytekhnicheskaya st., 26, St. Petersburg 194021, Russia b Laboratoire de Modélisation en Mécanique associé au CNRS, Université Pierre et Marie Curie, Paris, France Received 29 January 2005; accepted 2 March 2005 Abstract A new governing equation with combined quadratic and cubic non-linearities is obtained to account for longitudinal strain solitary waves in an elastic rod. It is shown that a strain solitary wave solution of this equation arises as a result of balance betweenquadraticnon-linearityanddispersionandexistsevenintheabsenceofcubicnon-linearity.However,theamplitude, thewidthandthevelocityofthewaveareaffectedbythecubicnon-linearitycausing,inparticular,anarrowingofthewave. This allows to agree better with experiments on strain solitary wave generation.  2005 Elsevier Ltd. All rights reserved. PACS: 62.30.+d; 03.40.kf; 43.25.+y Keywords: Non-linear waves; Elastic rod; Murnaghan’s materials 1. Introduction The study of non-linear strain waves in an elastic rodhasattractedconsiderableinterestsincetherodis a natural one-dimensional waveguide which may be a part of any construction. Certainly simplified one- dimensional governing equations may be obtained in thiscase,andevolutionofnon-linearstrainwavesmay bedescribedanalytically.Ofspecialinterestarelocal- izedlongitudinalwavesthatmaypropagatealongthe ∗ Corresponding author. Tel.: +78122479352; fax: +78122471017. E-mail address: porubov@math.ioffe.ru (A.V. Porubov). 0020-7462/$-see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2005.03.001 rod keeping their shape. Recent findings in the area maybefoundin [1–6]. Despite numerous theoretical works, only a few experimental data are available on longitudinalstrainsolitarywavesdetection [7].More- over, the theory developed by now predicts higher width of the wave than the observed one. It points to a possible lack of correct theory. Existence of localized or solitary waves in a rod is provided by a balance between non-linearity and dis- persion.Non-linearityinapureelasticrodiscausedby thefinitestressvaluesandtheelasticmaterialproper- tieswhiledispersionresultsfromthefinitetransverse size of the rod. Certainly the leading non-linearity for longitudinal waves is quadratic, and usually