In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is di¯cult to directly tackle the fully nonlinear rod equations. Here, by using a non-dimensionalization process and the reductive perturbation technique, we derive a new type of nonlinear dispersive equation as the model equation. We then focus on the travelling-wave solutions of this new equation. As a result, we obtain a system of ordinary di¬erential equations. An important feature of this system is that there is a vertical singular line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the singular line, we are able to determine all qualitatively di¬erent phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with nite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors' knowledge, such a bifurcation has not been found in any other dynamical systems.

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In this paper, we study nonlinear dispersive waves in a slender tube composed of an incompressible elastic material. One of the purposes is to show that solitary waves can propagate in such a structure. A major difficulty associated with the geometry of a tube is that logarithm terms can arise. By using a novel approach involving splitting the unknowns into two parts and series expansions, we manage to overcome this difficulty. A dimension reduction is successfully carried out, and as a result a set of one-dimensional model equations are established. It is also shown that the dispersion relation of these model equations matches with the exact dispersion relation of the three-dimensional field equations up to the right order. Then, the reductive perturbation method is used to deduce the far-field equation, which turns out to be the KdV equation. Since this equation admits a solitary-wave solution, this shows that solitary waves can propagate in an elastic tube. The influence of the inner radius on the solitary wave is then discussed.