Perturbation solution for small amplitude solitary waves in two-phase fluid flow of compacting media

Abstract

A perturbation solution for small-amplitude solitary waves is derived for the third-order nonlinear partial differential equation due to Scott and Stevenson which describes the one-dimensional migration of melt through the Earth's mantle. The straightforward perturbation expansion breaks down and a coordinate stretching transformation is performed to render the perturbation expansion uniformly valid. The lowest-order perturbation solution has the same form as the single-soliton solution of the Korteweg-de Vries equation. The perturbation solution is derived to second order in implicit form. It is found to be extremely accurate when compared with known exact solutions for specific values of the exponents n and m. The zero- and first-order perturbation solutions are found to be accurate when n = m. The properties of small-amplitude solitary waves are investigated using the perturbation solution.