Solitary waves in a slender tube composed of an incompressible nonlinear elastic material

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.