PDE arising in Magma Dynamics
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<span style="font-variant:small-caps;">Gideon Simpson</span>

<span style="font-variant:small-caps;">Michael I. Weinstein</span>

<span style="font-variant:small-caps;">Philip Rosenau</span>

Address for correspondence: grs2103@columbia.edu

Introduction {#sec:intro}
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Consistent, macroscopic models of magma, partially molten rock, in the Earth’s interior can were developed in [@McKenzie84; @Scott84; @Scott86], coupling the flow of the solid rock with that of the liquid via conservation of mass and momentum. Under appropriate assumptions (small fluid fraction, no large scale shear motions, etc.) the system may be reduced to a single scalar equation for the evolution of the fluid fraction, the porosity $\phi$, [@Barcilon86; @Barcilon89; @Wiggins95], that admit solitary waves. In one spatial dimension, the equation is $$\label{eq:magma}
\dt\f + \dx\paren{\f^n} - \dx\paren{\f^n\dx\paren{\f^{-m}\dt\f}}=0$$ with the boundary conditions that $\f(x,t) \to 1$ as $x \to \pm \infty$. The nonlinearity parameter $n$ is specified by a Darcy’s Law relationship between the permeability, $K$, of the rock and its porosity of the form $K(\f) \propto \f^n$. The other nonlinearity parameter, $m$, is related to the bulk viscosity, $\zeta$, of the porous rock, with $\zeta(\f) \propto \f^{-m}$. It is expected that $ 2\leq n\leq 5$ and $0 \leq m \leq 1$. A well-posedness theory for the initial value problem of (\[eq:magma\]) is developed in [@Simpson07]; 