An investigation of the mathematical model of a compacting medium proposed by McKenzie (1984) for the purpose of understanding the migration and segregation of melts in the Earth is presented. The numerical observation that the governing equations admit solutions in the form of nonlinear one-dimensional waves of permanent shape is confirmed analytically. The properties of these solitary waves are presented, namely phase speed as a function of melt content, nonlinear interaction and conservation quantities. The information at hand suggests that these waves are not solitons.
We investigate the stability of the one-dimensional solitary waves solutions of the equations proposed by McKenzie to model the ascent of melts in the Earth interior. We show that for small porosity and two-dimensional horizontal disturbances with long wavelength, these solitary waves are unstable. We also exhibit two-and threedimensional solitary-wave solutions of the McKenzie equations.
An iterative construction of the potential function entering in a regular Sturm-Liouville problem is discussed. The given data is in the form of two spectra associated with distinct boundary conditions at one end point.
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