Numerical studies of fluid flow in the mantle suggest that magma migration is an inherently time-dependent process that produces magmatic solitary waves from obstructions in melt flux. Previous work has considered one and two dimensional problems. Here we present the results of three dimensional calculations that utilize a new, efficient multigrid scheme. We demonstrate that one and two dimensional solitary waves are unstable and break up into sets of 3-D solitary waves which are perfectly spherical when propagating through a uniform porosity medium. While these waves are not solitons, their non-linear interactions are qualitatively similar. The solitary waves are highly opportunistic and establish efficient pathways for migration by linking up with nearby waves. When the initial condition is a random distribution of porosity, the porosity structure can organize into elongate, time-dependent channels formed from chains of solitary waves. These results are natural consequences of the assumptions that the matrix is permeable and viscously deformable. We suggest that solitary waves are likely to exist in the mantle and may contribute to the episodicity of mantle magmatism. ¿ American Geophysical Union 1995