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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