Solitary waves in elastic rods have been generated in experiments (see Samsonov [1]).There are some similarities between wave propagation in a slender tube and that in an elastic rod.Since the 1970s, nonlinear dispersive waves in elastic rods have been the subject of many studies.For a review, we refer the readers to Dai and Fan [2].Here we shall adopt a method of coupled series-asymptotic expansions to study nonlinear dispersive waves in a slender elastic tube.Compared with a rod.a tube has an extra inner surface, which implies that two extra boundary conditions have to be satisfied.Also.a major difficulty is that as the domain for the radial variable does not contain R =0 for a tube, the terms such as log(R) can arise in the expansions.The key is to split the unknowns into two parts with one involving a power series and another involving the logarithm terms.