In the present work, utilizing the non-linear equations of a pre-stressed thin elastic tube filled with an incompressible inviscid fluid the propagation of weakly non-linear waves in such a medium is studied. Considering that the arteries are initially subjected to a large static transmural pressure P-0 and an axial stretch lambda(z) and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the non-linear equations governing the motion of the tube in the radial and axial directions are obtained. Utilizing the reductive perturbation technique the amplitude modulation of weakly non-linear but strongly dispersive waves is examined. The localized travelling wave solution to the evolution equation is given and the stability condition is discussed. (C) 2000 Elsevier Science Ltd. All rights reserved.