Approximate solutions for magmon propagation from a reservoir

A 1D partial differential equation (pde) describing the flow of magma in the Earth's mantle is considered, this equation allowing for compaction and distension of the surrounding matrix due to the magma. The equation has periodic travelling wave solutions, one limit of which is a solitary wave, called a magmon. Modulation equations for the magma equation are derived and are found to be either hyperbolic or of mixed hyperbolic/elliptic type, depending on the specific values of the wave number, mean height and amplitude of the underlying modulated wave. The periodic wave train is stable in the hyperbolic case and unstable in the mixed case. Solutions of the modulation equations are found for an initial-boundary value problem on the semi-infinite line, these solutions representing the influx of magma from a large reservoir. The modulation solutions are found to consist of a full or partial undular bore. Excellent agreement with numerical solutions of the governing pde is obtained, except in the limit where the wave train becomes a train of magmons. An alternative approximation based on the assumption that the wave train is a series of uniform magmons is also derived and is found to be superior to modulation theory in this limit.