On a Hamiltonian PDE arising in magma dynamics

Abstract

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 <em>compactons</em>, solitary traveling waves with compact spatial support.