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Abstract. An outstanding problem in Earth science is understanding the method of transport of magma in the Earth’s mantle. Two proposed methods for this transport are percolation through porous rock and flow up conduits. Under reasonable assumptions and simplifications, both means of transport can be described by a class of degenerate nonlinear dispersive… (More)

- Andrew Binder, Tony Lelièvre, Gideon Simpson
- J. Comput. Physics
- 2015

Article history: Received 24 April 2014 Received in revised form 9 October 2014 Accepted 3 January 2015 Available online 9 January 2015

- Jeremy L. Marzuola, Sarah Raynor, Gideon Simpson
- J. Nonlinear Science
- 2010

We study soliton solutions to the nonlinear Schrödinger equation (NLS) with a saturated nonlinearity. NLS with such a nonlinearity is known to possess a minimal mass soliton. We consider a small perturbation of a minimal mass soliton and identify a system of ODEs extending the work of Comech and Pelinovsky (Commun. Pure Appl. Math. 56:1565–1607, 2003),… (More)

- James E. Colliander, Jeremy L. Marzuola, Tadahiro Oh, Gideon Simpson
- Experimental Mathematics
- 2013

We experimentally explore solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a… (More)

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrödinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3D cubic problem, this work presents a new… (More)

- Gideon Simpson, Michael I. Weinstein
- Multiscale Modeling & Simulation
- 2011

We consider the one-dimensional propagation of electromagnetic waves in a weakly nonlinear and low-contrast spatially inhomogeneous medium with no energy dissipation. We focus on the case of a periodic medium, in which dispersion enters only through the (Floquet-Bloch) spectral band dispersion associated with the periodic structure; chromatic dispersion (… (More)

The parallel replica dynamics, originally developed by A. F. Voter, efficiently simulates very long trajectories of metastable Langevin dynamics. We present an analogous algorithm for discrete time Markov processes. Such Markov processes naturally arise, for example, from the time discretization of a continuous time stochastic dynamics. Appealing to… (More)

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… (More)

- David M. Ambrose, Gideon Simpson
- SIAM J. Math. Analysis
- 2015

- Xiao Liu, Gideon Simpson, Catherine Sulem
- J. Nonlinear Science
- 2013