On a Hamiltonian PDE arising in Magma Dynamics
| dc.creator | Simpson, Gideon | |
| dc.creator | Weinstein, Michael I. | |
| dc.creator | Rosenau, Philip | |
| dc.date | 2008-01-03 | |
| dc.date | 2008-01-17 | |
| dc.date.accessioned | 2026-07-07T08:54:46Z | |
| dc.date.available | 2026-07-07T08:54:46Z | |
| dc.description | 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 $ϕ$, 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 $ϕ$. 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 include a neighborhood of a solitary wave. Finally, we observe that these equations have {\it compactons}, solitary traveling waves with compact spatial support at each time. | |
| dc.description | 27 pages, submitted to DCDS-B | |
| dc.identifier | https://arxiv.org/abs/0801.0555 | |
| dc.identifier | http://arxiv.org/abs/0801.0555 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146055 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.subject | Analysis of PDEs | |
| dc.title | On a Hamiltonian PDE arising in Magma Dynamics | |
| dc.type | text |