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

In this paper, we study analytically the development of a soliton propagati ng along a circular rod composed of a general compressible hyperelastic mat erial with variable cross-sections and variable material density. The purpo se is to provide analytical descriptions for the following two phenomena fo und, respectively, in numerical and perturbation studies: (1) Fission of a soliton. When a soliton moves from a part of the rod with thick cross-secti ons to a part with thin cross-sections, it will split into two or more soli tons; (2) When a soliton propagates along a rod with slowly decreasing radi us, it will develop into a solitary wave with a shelf behind. By using a no ndimensionalization process and the reductive perturbation technique, we de rive a variable-coefficient Korteweg-de Vries (vcKdV) equation as the model equation. The inverse scattering transforms are used to study the vcKdV eq uation. By considering the associated isospectral problem, the phenomenon o f soliton fission is successfully explained. We are able to provide a condi tion that exactly how many solitons will emerge when a single soliton moves from a thick section to a thin section. Then, by introducing suitable vari able transformations, we successfully manage to transform the vcKdV equatio n into a cylindrical KdV equation. As a result, several exact bounded solut ions 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 li terature. Comparisons are also made between the analytical solutions and nu merical results, and good agreement is found. (C) 2002 Elsevier Science B.V . All rights reserved.