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

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

We experimentally explore solutions to a model Hamil-tonian 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… (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)

- Gideon Simpson, Michael I. Weinstein
- SIAM J. Math. Analysis
- 2008

Coherent structures, such as solitary waves, appear in many physical problems, including fluid mechanics , optics, quantum physics, and plasma physics. A less studied setting is found in geophysics, where highly viscous fluids couple to evolving material parameters to model partially molten rock, magma, in the Earth's interior. Solitary waves are also found… (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)

- 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 (Com-mun. which models the behavior of the… (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)

The summer school will introduce graduate students to the theoretical and computational tools necessary to study problems at the intersection of stochastic processes, probability and nonlinear waves. The three main areas to be covered are: randomly forced differential and partial differential equations (a.k.a. stochastic differential equations and… (More)

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

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