# American Institute of Mathematical Sciences

September  2008, 10(4): 903-924. doi: 10.3934/dcdsb.2008.10.903

## On a Hamiltonian PDE arising in magma dynamics

 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States 2 Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd, 500 W. 120th St., New York City, NY 10027, United States 3 School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Received  September 2007 Revised  March 2008 Published  August 2008

In this article we discuss a new Hamiltonian PDE arising from a class of equations appearing in the study of magma, partially molten rock in the Earth's interior. Under physically justifiable simplifications, a scalar, nonlinear, degenerate, dispersive wave equation may be derived to describe the evolution of $\phi$, the fraction of molten rock by volume, in the Earth. These equations have two power nonlinearities which specify the constitutive realitions for bulk viscosity and permeability in terms of $\phi$. Previously, they have been shown to admit solitary wave solutions. For a particular relation between exponents, we observe the equation to be Hamiltonian; it can be viewed as a generalization of the Benjamin-Bona-Mahoney equation. We prove that the solitary waves are nonlinearly stable, by showing that they are constrained local minimizers of an appropriate time-invariant Lyapunov functional. A consequence is an extension of the regime of global in time well-posedness for this class of equations to (large) data which includes a neighborhood of a solitary wave. Finally, we observe that these equations have compactons, solitary traveling waves with compact spatial support.
Citation: Gideon Simpson, Michael I. Weinstein, Philip Rosenau. On a Hamiltonian PDE arising in magma dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 903-924. doi: 10.3934/dcdsb.2008.10.903
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