22 citations · 134 across the 18 of their papers we have counts for
20 papers
Exponential Dichotomies for Elliptic Equations on Multidimensional Domains
Margaret Beck, Ryan Goh, Alanna Haslam-Hyde
The existence of exponential dichotomies has been well-established as a powerful tool to study existence, stability, and bifurcations of coherent structures. Currently, the applica…
Localized radial roll patterns in higher space dimensions
Jason J. Bramburger, Dylan Altschuler, Chloe I. Avery +4
Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the len…
Validated spectral stability via conjugate points
Margaret Beck, Jonathan Jaquette
Classical results from Sturm-Liouville theory state that the number of unstable eigenvalues of a scalar, second-order linear operator is equal to the number of associated conjugate…
Spectral stability and spatial dynamics in partial differential equations
Margaret Beck
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analy…
Exponential dichotomies for elliptic PDE on radial domains
Margaret Beck, Graham Cox, Christopher Jones +2
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one…
A dynamical approach to semilinear elliptic equations
Margaret Beck, Graham Cox, Christopher Jones +2
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in is given in terms of an infinite-dimensional dynamical system.…
Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus
Margaret Beck, Eric Cooper, Gabriel Lord +1
The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on is considered for …
Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity
Margaret Beck, Osman Chaudhary, C. Eugene Wayne
Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences…
Nonlinear stability of source defects in oscillatory media
Margaret Beck, Toan T. Nguyen, Björn Sandstede +1
In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core…
Partial differential systems with nonlocal nonlinearities: Generation and solutions
Margaret Beck, Anastasia Doikou, Simon J. A. Malham +1
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions a…
Instability of pulses in gradient reaction-diffusion systems: A symplectic approach
Margaret Beck, Graham Cox, Christopher Jones +3
In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's c…
Grassmannian flows and applications to nonlinear partial differential equations
Margaret Beck, Anastasia Doikou, Simon J. A. Malham +1
We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, fro…
Selection of quasi-stationary states in the Navier-Stokes equation on the torus
Margaret Beck, Eric Cooper, Konstantinos Spiliopoulos
The two dimensional incompressible Navier-Stokes equation on with , periodic boundary conditions, and viscosity is consider…
Analysis of enhanced diffusion in Taylor dispersion via a model problem
Margaret Beck, Osman Chaudhary, C. Eugene Wayne
We consider a simple model of the evolution of the concentration of a tracer, subject to a background shear flow by a fluid with viscosity in an infinite channel. Taylor o…
Computing the Maslov index for large systems
Margaret Beck, Simon J. A. Malham
We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given referenc…
Nonlinear stability of source defects in the complex Ginzburg-Landau equation
Margaret Beck, Toan T. Nguyen, Bjorn Sandstede +1
In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave train…
Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
Margaret Beck, C. Eugene Wayne
Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time sc…
Toward nonlinear stability of sources via a modified Burgers equation
Margaret Beck, Toan Nguyen, Bjorn Sandstede +1
Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at to spatially periodi…
Nonlinear stability of time-periodic viscous shocks
Margaret Beck, Bjorn Sandstede, Kevin Zumbrun
In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedd…
Using global invariant manifolds to understand metastability in Burgers equation with small viscosity
Margaret Beck, C. Eugene Wayne
The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a pheno…