Solitary waves in an inhomogeneous rod composed of a general hyperelastic material

In this paper, we study analytically the development of a soliton propagating along a circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material density. The purpose is to provide analytical descriptions for the following two phenomena found, respectively, in numerical and perturbation studies: (1) Fission of a soliton. When a soliton moves from a part of the rod with thick cross-sections to a part with thin cross-sections, it will split into two or more solitons; (2) When a soliton propagates along a rod with slowly decreasing radius, it will develop into a solitary wave with a shelf behind. By using a nondimensionalization process and the reductive perturbation technique, we derive a variable-coefficient Korteweg-de Vries (vcKdV) equation as the model equation. The inverse scattering transforms are used to study the vcKdV equation. By considering the associated isospectral problem, the phenomenon of soliton fission is successfully explained. We are able to provide a condition that exactly how many solitons will emerge when a single soliton moves from a thick section to a thin section. Then, by introducing suitable variable transformations, we successfully manage to transform the vcKdV equation into a cylindrical KdV equation. As a result, several exact bounded solutions in terms of Airy function Ai and Bi are obtained. One of the solutions has the shape of a solitary wave with a shelf behind. Thus, it provides an analytical description for the perturbation and experimental results in literature. Comparisons are also made between the analytical solutions and numerical results, and good agreement is found. © 2002 Elsevier Science B.V. All rights reserved.