Gideon Simpson

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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)
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)
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)
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)