About

Jared Bronski is a Professor in the Department of Mathematics at the University of Illinois at Urbana-Champaign. His field is Applied Mathematics, with a particular interest in nonlinear wave propagation and the stability of waves and coherent structures. He earned his Ph.D. in Applied and Computational Mathematics from Princeton University in 1994. As a faculty member, he also serves as the Director of Graduate Studies in the Mathematics department. His research focuses on differential equations and applied mathematics, contributing to the understanding of wave phenomena and stability analysis through rigorous mathematical analysis and computation.

Research topics

• Mathematics
• Physics
• Mathematical analysis
• Mathematical physics
• Statistical physics

Selected publications

• The Evans function as a lower bound on the spectral distance function

  arXiv (Cornell University) · 2026-04-21

  preprintOpen accessSenior author

  The Evans function is an analytic function that encodes information about the intersection of certain subspaces in ODE boundary value problems. As such it is a useful tool for computing the spectrum of boundary value problems arising in the stability of coherent structures. In typical applications one is interested in the roots of the Evans function, but the overall normalization is somewhat arbitrary. We present a natural normalization of the Evans function on compact domains such that the magnitude of the Evans function provides a lower bound on the distance to the nearest point in the spectrum. In other words the magnitude of the Evans function at a point in the resolvent set implies that a ball about the point in question lies in the resolvent set. Thus, when appropriately normalized, not only does the Evans function $E(λ)$ vanish if and only if $λ$ lies in the spectrum of the operator in question, but a non-zero value for the Evans function guarantees that a disk of radius $|E(λ^*)|$ about the point $λ^*$ lies in the resolvent set. We present some calculations for some common sets of boundary conditions on a compact interval, and present some numerical experiments for 2nd and 4th order self-adjoint operators and for a linearized modified Korteweg-De Vries equation.

  PublisherDOI

• The Evans function as a lower bound on the spectral distance function

  ArXiv.org · 2026-04-21

  articleOpen accessSenior author

  The Evans function is an analytic function that encodes information about the intersection of certain subspaces in ODE boundary value problems. As such it is a useful tool for computing the spectrum of boundary value problems arising in the stability of coherent structures. In typical applications one is interested in the roots of the Evans function, but the overall normalization is somewhat arbitrary. We present a natural normalization of the Evans function on compact domains such that the magnitude of the Evans function provides a lower bound on the distance to the nearest point in the spectrum. In other words the magnitude of the Evans function at a point in the resolvent set implies that a ball about the point in question lies in the resolvent set. Thus, when appropriately normalized, not only does the Evans function $E(λ)$ vanish if and only if $λ$ lies in the spectrum of the operator in question, but a non-zero value for the Evans function guarantees that a disk of radius $|E(λ^*)|$ about the point $λ^*$ lies in the resolvent set. We present some calculations for some common sets of boundary conditions on a compact interval, and present some numerical experiments for 2nd and 4th order self-adjoint operators and for a linearized modified Korteweg-De Vries equation.

  PublisherOA PDF

• Asymptotic Stability of Sharp Fronts: Analysis and Rigorous Computations

  SSRN Electronic Journal · 2025-01-01

  preprintOpen access1st authorCorresponding
  PublisherDOI

• Asymptotic stability of sharp fronts: Analysis and rigorous computation

  Journal of Differential Equations · 2025-06-19

  articleCorresponding
  PublisherDOI

• Bounds on Unstable Spectrum for Dispersive Hamiltonian Pdes

  SSRN Electronic Journal · 2024-01-01

  preprintOpen access1st authorCorresponding
  PublisherDOI

• Floquet theory and stability analysis for Hamiltonian PDEs

  Nonlinearity · 2024-11-04 · 2 citations

  articleOpen access1st authorCorresponding

  Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

  PublisherDOI

• Bounds on unstable spectrum for dispersive Hamiltonian PDEs

  arXiv (Cornell University) · 2024-10-24

  preprintOpen access1st authorCorresponding

  We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the imaginary axis, and estimates for the number of such off-axis eigenvalues. These relations hold when the dispersion relation grows sufficiently rapidly in the wavenumber. The proofs involve a Gershgorin disk argument together with the Hamiltonian symmetry of the spectrum. The results are applicable to a broad class of nonlinear dispersive equations including the generalized Korteweg--de Vries, Benjamin--Bona--Mahoney, and Kawanhara equations.

  PublisherOA PDFDOI

• Floquet theory and stability for Hamiltonian partial differential equations

  arXiv (Cornell University) · 2023-09-07

  preprintOpen access1st authorCorresponding

  We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

  PublisherOA PDFDOI

• Superharmonic instability for regularized long-wave models*

  Nonlinearity · 2022-11-25 · 4 citations

  article1st authorCorresponding

  Abstract We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney–Luke, and Benjamin–Bona–Mahony equations. Of particular interest is a striking new instability phenomenon—spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg–de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We perform rigorous spectral asymptotics for short wavelength perturbations to establish conditions under which the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments which corroborate our analytical findings.

  PublisherDOI

• Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

  arXiv (Cornell University) · 2022-02-22

  preprintOpen access1st authorCorresponding

  We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an $n$-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the $N$-point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.

  PublisherOA PDFDOI

Recent grants

• Stability, Instability and Geometry in Applied Spectral Problems.

  NSF · $290k · 2016–2020

• Eigenvalues, geometry and instability in conservative models in applied mathematics.

  NSF · $218k · 2012–2016

• Eigenvalue and Stability Problems in Applied Mathematics

  NSF · $146k · 2008–2012

• FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory

  NSF · $216k · 2004–2008

Frequent coauthors

• J. Nathan Kutz

  16 shared

• Mathew A. Johnson

  12 shared

• Lee DeVille

  University of Illinois Urbana-Champaign

  11 shared

• Vera Mikyoung Hur

  11 shared

• Andrea K. Barreiro

  Southern Methodist University

  8 shared

• Zoi Rapti

  8 shared

• Timothy Ferguson

  Arizona State University

  7 shared

• Lincoln D. Carr

  Colorado School of Mines

  6 shared

Education

• Ph. D., Applied Mathematics

  Princeton University

  1994

• B.S. , Applied Math

  California Institute of Technology

  1989