Gideon Simpson

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.

Spectral analysis for matrix Hamiltonian operators

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrodinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Although we focus on a proof of the 3D cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability.Source code for verification of our computations, and for further experimentation, is available at http://hdl.handle.net/1807/25174.

Degenerate dispersive equations arising in the study of magma dynamics

An outstanding problem in Earth science is understanding the method of transport of magma in the Earths 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 partial differential equations of the form: where (z, 0) > 0 and (z, t) → 1 as z → ±∞.Although we treat arbitrary n and m, the exponents are physically expected to be between 2 and 5 and 0 and 1, respectively.In the case of percolation, the magma moves via the buoyant ascent of a less dense phase, treated as a fluid, through a denser, porous phase, treated as a matrix. In contrast to classical porous media problems where the matrix is fixed and the fluid is compressible, here the matrix is deformable, with a viscous constitutive relation, and the fluid is incompressible. Moreover, the matrix is modelled as a second, immiscible, compressible fluid to mimic the process of dilation of the pores. Flow via a conduit is modelled as a viscously deformable pipe of magma, fed from below.Analogue and numerical experiments suggest that these equations behave akin to KdV and BBM; initial conditions evolve into a collection of solitary waves and dispersive radiation. As → 0, the equations become degenerate. A general local well-posedness existence theory is given for a physical class of data (roughly H1) via fixed point methods. The strategy requires positive lower bounds on (z, t). The key to global existence is the persistence of these bounds for all time. Furthermore, we construct a Lyapunov energy functional, which is locally convex about the uniform porosity state, ≡ 1, and prove (global in time) nonlinear dynamic stability of the uniform state for any m and n. For data which are large perturbations of the uniform state, we prove global in time well-posedness for restricted ranges of m and n. This includes, for example, the case n = 4,m = 0, where an appropriate uniform in time lower global on can be proved using the conservation laws. We compare the dynamics with that of other problems and discuss open questions concerning a larger range of exponents, for which we conjecture global existence.

A generalized parallel replica dynamics

Metastability is a common obstacle to performing long molecular dynamics simulations. Many numerical methods have been proposed to overcome it. One method is parallel replica dynamics, which relies on the rapid convergence of the underlying stochastic process to a quasi-stationary distribution. Two requirements for applying parallel replica dynamics are knowledge of the time scale on which the process converges to the quasi-stationary distribution and a mechanism for generating samples from this distribution. By combining a Fleming-Viot particle system with convergence diagnostics to simultaneously identify when the process converges while also generating samples, we can address both points. This variation on the algorithm is illustrated with various numerical examples, including those with entropic barriers and the 2D Lennard-Jones cluster of seven atoms.

Kullback-Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure

Asymptotic Stability of Ascending Solitary Magma Waves

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Solitary Wave Benchmarks in Magma Dynamics

on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure

Arrest of Langmuir wave collapse by quantum effects

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Asymptotic Stability of High-dimensional Zakharov-Kuznetsov Solitons

, from within a simple class of measures, which approximates

Algorithms for Kullback--Leibler Approximation of Probability Measures in Infinite Dimensions

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