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Gideon Simpson

Publications and source records attributed to Gideon Simpson.

At least 37 records · Page 2Linked to original sources
arXiv2015

Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

math.AP↗
arXiv2015

A Theoretical Examination of Diffusive Molecular Dynamics

Diffusive molecular dynamics is a novel model for materials with atomistic resolution that can reach diffusive time scales. The main ideas of diffusive molecular dynamics are to first minimize an approximate variational Gaussian free energy of the system with respect to the mean atomic coordinates (averaging over many vibrational periods), and to then to perform a diffusive step where atoms and vacancies (or two species in a binary alloy) flow on a diffusive time scale via a master equation. We present a mathematical framework for studying this algorithm based upon relative entropy, or Kullback-Leibler divergence. This adds flexibility in how the algorithm is implemented and interpreted. We then compare our formulation, relying on relative entropy and absolute continuity of measures, to existing formulations. The main difference amongst the equations appears in a model for vacancy diffusion, where additional entropic terms appear in our development.

math.NA↗
arXiv2014

Petviashvilli's Method for the Dirichlet Problem

We examine the Petviashvilli method for solving the equation $ ϕ- Δϕ= |ϕ|^{p-1} ϕ$ on a bounded domain $Ω\subset \mathbb{R}^d$ with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on $\mathbb{R}$ by Pelinovsky & Stepanyants, 2004. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.

math.AP↗
arXiv2014

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.

math.NA↗
arXiv2014

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

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance; and through Schrödinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs, and to a conditioned diffusion process. We also show how the Gaussian approximations we obtain may be used to produce improved pCN-MCMC methods which are not only well-adapted to the high-dimensional setting, but also behave well with respect to small observational noise (resp. small temperatures) in the inverse problem (resp. conditioned diffusion).

math.NA↗
arXiv2014

The parallel replica method for simulating long trajectories of Markov chains

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 properties of quasistationary distributions, we show that our algorithm reproduces exactly (in some limiting regime) the law of the original trajectory, coarsened over the metastable states.

math.NA↗
arXiv2014

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 $μ$ 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 $ν$, from within a simple class of measures, which approximates $μ$. This problem is studied in the case where the Kullback-Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted and the particular case where $ν$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimising sequences. Furthermore, parameterisation of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularisation is explained, and a regularised minimisation is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms.

math.PR↗
arXiv2014

Local Existence Theory for Derivative Nonlinear Schrödinger Equations with Non-Integer Power Nonlinearities

We study a derivative nonlinear Schrödinger equation, allowing non-integer powers in the nonlinearity, $|u|^{2σ} u_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial data in the energy space, $H^1$. For more regular initial data, we establish not just existence of solutions, but also well-posedness of the initial value problem. These results hold for real-valued $σ\geq 1,$ while prior existence results in the literature require integer-valued $σ$ or $σ$ sufficiently large ($σ\geq 5/2$), or use higher-regularity function spaces.

math.AP↗
arXiv2013

Dynamics near a minimal-mass soliton for a Korteweg-de Vries equation

We study soliton solutions to a generalized Korteweg - de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of the authors for the nonlinear Schrödinger equation (NLS). KdV 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, which models the behavior of the perturbation for short times. This connects nicely to a work of Comech, Cuccagna & Pelinovsky (2007). These ODEs form a simple dynamical system with a single unstable hyperbolic fixed point with two possible dynamical outcomes. A particular feature of the dynamics are that they are non-oscillatory. This distinguishes the KdV problem from the analogous NLS one.

nlin.PS↗
arXiv2013

Focusing Singularity in a Derivative Nonlinear Schrödinger Equation

We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, $|ψ|^{2σ}ψ_x$. In the $L^2$-supercritical regime, $σ>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.

math.AP↗
arXiv2012

Behavior of a Model Dynamical System with Applications to Weak Turbulence

We experimentally explore solutions to a model Hamiltonian dynamical system derived in Colliander et al., 2012, 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 generic phenomenon. We also identify stationary solutions, periodic solutions in an associated problem and find experimental evidence of hyperbolic behavior. Many of our results rely upon reframing the dynamical system using a hydrodynamic formulation.

math.AP↗
arXiv2012

Numerical Analysis of Parallel Replica Dynamics

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.

math.NA↗
arXiv2011

Analytical and Numerical Results on the Positivity of Steady State Solutions of a Thin Film Equation

We consider an equation for a thin-film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh, & Taranets.

math.AP↗
arXiv2011

Broad Band Solitons in a Periodic and Nonlinear Maxwell System

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so called it extended nonlinear coupled mode equations, xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic and numerical methods, the existence and properties of spatially localized structures of the xNLCME system, which arises for a refractive index profile consisting of periodic array of Dirac delta functions. We consider the limit of small amplitude solutions with frequencies near a band-edge. In this case, stationary xNLCME is well-approximated by an infinite system of coupled, stationary, nonlinear Schrödinger equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS$^ε$, which interpolates between infinitely many decoupled NLS equations ($ε=0$) and xNLS ($ε=1$). Using bifurcation methods we show existence of solutions for a range of $ε\in(-ε_0,ε_0)$ and, by a numerical continuation method, establish the continuation of certain branches all the way to $ε=1$. Finally, we perform time-dependent simulations of truncated xNLCME and find the small-amplitude solitons to be robust to both numerical errors and the NLS approximation.

nlin.PS↗
arXiv2011

Ill-posedness of degenerate dispersive equations

In this article we provide numerical and analytical evidence that some degenerate dispersive partial differential equations are ill-posed. Specifically we study the K(2,2) equation $u_t = (u^2)_{xxx} + (u^2)_{x}$ and the "degenerate Airy" equation $u_t = 2 u u_{xxx}$. For K(2,2) our results are computational in nature: we conduct a series of numerical simulations which demonstrate that data which is very small in $H^2$ can be of unit size at a fixed time which is independent of the data's size. For the degenerate Airy equation, our results are fully rigorous: we prove the existence of a compactly supported self-similar solution which, when combined with certain scaling invariances, implies ill-posedness (also in $H^2$).

math.AP↗
arXiv2011

Embedded Eigenvalues and the Nonlinear Schrodinger Equation

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.

math.SP↗
arXiv2010

Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations

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 (time-nonlocality of the polarization) is neglected. Numerical simulations show that for initial conditions of wave-packet type (a plane wave of fixed carrier frequency multiplied by a slow varying, spatially localized function) very long-lived spatially localized coherent soliton-like structures emerge, whose character is that of a slowly varying envelope of a train of shocks. We call this structure an envelope carrier-shock train. The structure of the solution violates the oft-assumed nearly monochromatic wave packet structure, whose envelope is governed by the nonlinear coupled mode equations (NLCME). The inconsistency and inaccuracy of NLCME lies in the neglect of all (infinitely many) resonances except for the principle resonance induced by the initial carrier frequency. We derive, via a nonlinear geometrical optics expansion, a system of nonlocal integro-differential equations governing the coupled evolution of backward and forward propagating waves. These equations incorporate effects of all resonances. In a periodic medium, these equations may be expressed as a system of infinitely many coupled mode equations, which we call the extended nonlinear coupled mode system (xNLCME). Truncating xNLCME to include only the principle resonances leads to the classical NLCME. Numerical simulations of xNLCME demonstrate that it captures both large scale features, related to third harmonic generation, and fine scale carrier shocks features of the nonlinear periodic Maxwell equations.

nlin.PS↗