H. Demiray Istanbul Technical University, Faculty of Sciences and Letters, Department of Engineering Sciences, Maslak, Istanbul, Turkey e-mail: demiray@ito.edu.tr Tel:#90-212-2853283; Fax:#90-212-2856386 ARI (1999) 51 : 228 } 234  Springer-Verlag 1999 ORIGINAL ARTICLE H. Demiray Dressed solitary waves in thick-walled elastic tubes Received: 1 February 1999/Accepted: 8 February 1999 Abstract In the present work, by using the approximate nonlinear equations of an incompressible inviscid #uid contained in a prestressed thick elastic tube, the propaga- tion of a localized travelling wave solution in such a me- dium is investigated. Employing the hyperbolic tangent method and considering the long-wave limit, we showed that the lowest-order term in the perturbation expansion gives a solitary wave equivalent to the localized travelling wave solution of the Korteweg-de Vries equation. The solitary wave type of solution is also given for the second- order terms in the perturbation expansion. The correction terms in the speed of propagation are also obtained as a part of the solution of perturbation equations. The possible application of the present solution to blood #ow problems in arteries is also discussed. Key words Elastic tubes ) Dressed solitary waves ) Arterial wall 1 Introduction As pointed out by Su and Gardner (1977) and later by Ichikawa and Watanabe (1977), when some perturbation methods are applied to a class of physical systems, under the assumption of weak nonlinearities and long-wave limits, the resulting evolution equation will be the well- known Korteweg-de Vries equation. The physical inten- tion here is to balance the weak nonlinearity of the gov- erning equations with the dispersion of the system that is also supposed to be weak. In the mathematical analysis of such problems one normally uses the reductive perturba- tion method, or the multiple-scale expansion (Je!rey and Kawahara 1981), which is based on the change of indepen- dent variables, called the stretched variables. The lowest- order "eld variable in the expansion is governed by such an evolution equation. The solitary waves are the local- ized travelling wave solution of this type of evolution equation. When one wishes to consider the contributions of the higher-order terms in the expansion, it is very hard to analyse the problem, except for the problem of nonlinear ion-acoustic waves in plasma physics (see, e.g., Ichikawa and Watanabe 1977). The di$culty is due to the fact that the expansion breaks down and singular terms appear. For this reason one is forced to introduce multiple time and/or space scales or to use a renormalization procedure for the propagation velocities (Kodama and Taniuti 1978). Recently, Mal#iet and Wieers (1996) proposed an alterna- tive approach to investigate these kinds of problems and applied it to the above-mentioned problems in plasma physics. In the present work, employing the approximate non- linear equations of an inviscid #uid contained in a pre- stressed thick elastic tube, which is considered as a model for arteries, the propagation of localized travelling wave solution for these equations is investigated in the long- wave limit. Expanding the "eld quantities into an asymp- totic series of the wave number, which is supposed to be small, a set of di!erential equations governing the "eld variables of various order are obtained. Employing the hyperbolic tangent method (Mal#iet 1996), a localized travelling wave solution is obtained for the "rst- and second-order terms in the perturbation expansion. It is shown that the solution for the "rst-order term in the expansion is just sech type of solitary wave which is the same as that obtained from the solution of the corres- ponding Korteweg-de Vries equation. The solitary wave type of solution is also given for the second-order terms in the expansion. In addition, the wave speed and its correc- tion terms are also obtained as a part of the solution. Finally, the applicability of the present model to #ow problems in arteries is discussed.