On a Hamiltonian PDE arising in magma dynamics

Gideon Simpson*, Michael I. Weinstein, Philip Rosenau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


USA. 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 Benjainin-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.

Original languageEnglish
Pages (from-to)903-924
Number of pages22
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number4
StatePublished - Nov 2008


  • Compactons
  • Magma
  • Solitary waves
  • Stability
  • Variational methods
  • Viscously deformable porous media
  • Well-posedness


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