Research has been supported by the Simons Foundation Collaboration Grant
number 714021,
the National Science Foundation by NSF Grant
DMS-1614785, NSF Grant DMS-1211183,
the University of Kansas General Research Fund
allocation 2302278, and by an NSF Postdoctoral Fellowship under DMS-0902192.
Preprints:
- Diffusive Stability & Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
With Mariana Haragus, Wesley R. Perkins and Bjorn de Rijk.
- Subharmonic Dynamics of Wave Trains in Systems of Viscous Conservation Laws
With Wesley R. Perkins
- Nonlinear Stability of Periodic Waves in the Hamiltonian
Magma Equations
With Wesley R. Perkins
- A Center Manifold Approach to the Existence of Solutions in the Capillary-Gravity
Whitham Equation
With Tien Troung and Miles H. Wheeler.
- Instabilities in KP and Rotation Modified KP Type Equations
With Vera Mikyoung Hur and Ashish K. Pandey.
- Stability of Roll-Waves in the Vanishing Viscosity
Limit
With Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun.
- Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems
With Wesley R. Perkins
Submitted. (arXiv)
- Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
With Mariana Haragus and Wesley R. Perkins.
Submitted. (arXiv)
- Stability of Traveling Wave Solutions of Nonlinear Dispersive
Equations of NLS Type
With Katelyn P. Leisman, Jared C. Bronski, and Robert Marangell.
Submitted. (arXiv)
In Press:
- Generalized Solitary Waves in a
Capillary-Gravity Whitham Equation
With J. Douglas Wright.
Studies in Applied Mathematics, to appear. (arXiv)
- On the Bifurcation Diagram of the Capillary-Gravity
Whitham Equation
With Mats Ehrnstrom, Ola Isaac Hogasen Mahlen, and Filippo Remonato.
Water Waves,
to appear. (arXiv)
Refereed Publications:
(In reverse order of date of publication)
- Modulational Instability of Viscous Fluid Conduit Periodic Waves
With Wesley R. Perkins
SIAM Journal on Mathematical Analysis, 52: 277-305, 2020. (arXiv)
- On the Dynamics of Traveling Fronts Arising in Nanoscale Pattern Formation
With Gregory Lyng and Connor Smith.
Physica D, 401: 16pp, 2020. (arXiv)
- Nondegeneracy and Stability of Antiperiodic Bound States
for Fractional Nonlinear Schrodinger Equations
With Kyle M. Claassen.
Journal of Differential Equations, 266: 5664-5712, 2019.
(arXiv)
- Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow
Water Model
With Mats Ehrnstrom and Kyle M. Claassen.
Archive for Rational Mechanics and Analysis, 231: 1635-1673, 2019.
(doi, arXiv)
- Spetral Stability of Inviscid Roll Waves
With Pascal Noble, L. Miguel Rodrigues, Zhao Yang, and Kevin Zumbrun.
Communications in
Mathematical Physics, 367:265-316, 2019.
(doi, arXiv)
- Numerical Bifurcation and Spectral Stability of Wavetrains
in Bidirectional Whitham Models
With Kyle M. Claassen.
Studies in
Applied Mathematics, 141: 205-246, 2018.
(arXiv)
- Oscilltion Estimates of Eigenfunctions Via the
Combinatorics of Noncrossing Partitions
With Vera Mikyoung Hur and Jeremy L. Martin.
Discrete Analysis,
13: 20pp, 2017 (Journal Posting)
- Note on the Stability of Viscous Roll-Waves
With Blake Barker, Pascal Noble, L. Miguel Rodrigues, and Kevin
Zumbrun.
Comptes Rendus Mecanique, 345:125-129, 2017. (arXiv)
- Stability of Viscous St. Venant Roll-Waves: From Onset
to the Infinite-Froude Number Limit
With Blake Barker, Pascal Noble, L. Miguel Rodrigues, and Kevin
Zumbrun.
Journal of Nonlinear Science, 27: 285-342, 2017. (arXiv)
- Modulational Instability in Equations of KdV Type
With Jared C. Bronski and Vera Mikyoung Hur
Book Chapter in New Approaches in Nonlinear Waves, Lecture
Notes in Physics, 908, Springer: 83-133, 2016.
(arXiv)
- Modulational Instability in the Whitham Equation with
Surface Tension and Vorticity
With Vera Mikyoung Hur
Nonlinear Analysis: Theory, Methods & Applications, 129: 104-118, 2015. (arXiv)
- Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations
With Vera Mikyoung Hur.
SIAM Journal on Mathematical Analsis, 47 no. 5: 3528-3554,
2015. (reprint)
- Spectral Stability of Periodic Wave Trains of the Korteweg-de Vries/Kuramoto-Sivashinsky Equation in the Korteweg-de Vries Limit
With Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun.
Transactions
of the AMS, 367 no. 3: 2159-2212, 2015. (preprint)
- Modulational Instability in the Whitham Equation for
Water Waves
With Vera Mikyoung Hur.
Studies in Applied Mathematics, 134 no. 3: 120-143, 2015. (arXiv)
- Behavior of Periodic Solutions of Viscous Conservation
Laws Under Localized and Nonlocalized Perturbations
With Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun.
Inventiones
mathematicae, 197 no. 1: 115-213, 2014.
(arXiv)
See here for an
announcement of these results from CNRS (in French).
- An Instability Index Theory for Quadratic Pencils and Applications
With Jared C. Bronski and Todd Kapitula.
Communications
in Mathematical Physics, 327 no. 2: 521-550, 2014. (arXiv)
- Stability of Small Periodic Waves in Fractional KdV Type Equations
SIAM Journal on
Mathematical Analysis, 45 no. 5: 2597-3228, 2013. (reprint)
- Nonlinear Modulational Stabilty of Periodic Traveling Wave Solutions of the
Generalized Kuramoto-Sivashinsky Equation
With Blake Barker, Pascal Noble, L.Miguel Rodrigues, and Kevin Zumbrun.
Physica D, 258 no. 1: 11-46, 2013.
(arXiv)
- Non-Localized Modulation of Periodic Reaction Diffusion Waves: The Whitham Equation
With Pascal Noble, L.Miguel Rodrigues, and Kevin Zumbrun.
Archive for Rational Mechanics and Analysis, 207 no. 2: 669-692, 2013.
(arXiv)
- Non-Localized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability
With Pascal Noble, L.Miguel Rodrigues, and Kevin Zumbrun.
Archive for Rational Mechanics and Analysis, 207 no. 2: 693-715, 2013.
(arXiv)
- Stability of Periodic Kuramoto-Sivashinsky Waves
With Blake Barker, Pascal Noble, L.Miguel Rodrigues, and Kevin Zumbrun.
Applied Mathematics Letters, 25 no. 5: 824-829, 2012.
(arXiv)
- Convergence of Hill's Method for Nonselfadjoint Operators
With Kevin Zumbrun.
SIAM Journal on Numerical Anaylsis, 50 no. 1: 64-78, 2012.
(reprint)
- Metastability of Solitary Roll Wave Solutions of the St. Venant Equations
with Viscosity
With Blake Barker, L. Miguel Rodrigues, and Kevin Zumbrun.
Physica D, 240 no. 16: 1289-1310, 2011.
(arXiv)
- An Index Theorem for the Stability of Periodic Traveling Waves of KdV Type
With Jared C. Bronski and Todd Kapitula.
Proceedings of the Royal Society of Edinburgh: Section A, 141 Issue 6: 1141-1173, 2011.
(preprint)
- Nonlinear Stability of Periodic Traveling Wave Solutions of
Viscous Conservation Laws in Dimensions One and Two
With Kevin Zumbrun.
SIAM Journal on Applied Dynamical Systems, 10 no. 1: 189-211, 2011.
(reprint)
- Nonlinear Stability of Viscous Roll Waves
With Kevin Zumbrun and Pascal Noble.
SIAM Journal on Mathematical Analysis, 43 no. 2: 557-611, 2011.
(reprint)
- Nonlinear Stability of Spatially-Periodic Traveling-Wave Solutions of
Systems of Reaction Diffusion Equations
With Kevin Zumbrun.
Annales de l'Institut Henri Poincare - Analyse non lineaire, 28 Issue 4: 471-483, 2011.
(arXiv)
- Whitham Averaged Equations and Modulational Stability of Periodic
Traveling Waves of a Hyperbolic-Parabolic Balance Law
With Blake Barker, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun.
Journées équations aux dérivées partielles (2010), Exp. No. 3, 24 p.
(arXiv).
- Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
With Kevin Zumbrun.
SIAM Journal on Mathematical Analysis, 42 no. 6: 2681-2702, 2010.
(reprint)
- On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
With Kevin Zumbrun and Jared C. Bronski.
Physica D, 239: 2037-2065, 2010.
(preprint)
- On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
Physica D, 239 no. 19: 1892-1908, 2010.
(preprint)
- Nonlinear Stability of Periodic Traveling Wave Solutions
of Systems of Viscous Conservation Laws in the Generic Case
With Kevin Zumbrun.
Journal of Differential Equations, 249 no. 5: 1213-1240, 2010.
(arXiv)
- Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation
With Kevin Zumbrun.
Studies in Applied Mathematics, 125 no. 1: 69-89, 2010.
(preprint)
- The Transverse Instability of Periodic Waves in Zakharov-Kuznetsov Type Equations
Studies in Applied Mathematics, 124 no. 4: 323-345, 2010.
(preprint)
- The Modulational Instability for a Generalized KdV Equation
With Jared C. Bronski.
Archive for Rational Mechanics and Analysis, 197 no. 2: 357-400, 2010.
(arXiv)
- Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation
SIAM Journal on Mathematical Analysis, 41 no. 5: 1921-1947, 2009.
(reprint)
- Krein Signatures for the Faddeev-Tahktajan Eigenvalue Problem
With Jared C. Bronski.
Communications in Mathematical Physics, 288 no. 3: 821-846, 2009.
(arXiv)
Ph. D. Thesis
- On the Stability of Periodic Solutions of Nonlinear Dispersive Equations (.pdf).
Selected Talks and Presentations
"On the Instability of Periodic Wave Trains in the Whitham Equation"
(.pdf)
The Eighth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory, March 24-28, 2013, Athens, GA.
"Stability of Periodic Traveling Waves in a KdV/Kuramoto-Sivashinsky Equation"
(.pdf)
2012 SIAM Conference on Nonlinear Waves and Coherent Structures, June 13-16, 2012, Seattle, WA.
"Index Theorems for the Stability of Periodic Traveling Waves of KdV Type"
(.pdf)
University of Kansas Computational and Applied Mathematics Seminar, September 21, 2011.
"The Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili (KP) Equation"
(.pdf).
2010 SIAM Conference on Nonlinear Waves and Coherent Structures, August 16-19, 2010, Philadelphia, PA.
"Rigorous Stability Theory for Nonlinear Modulated Waves (Justification of the Physicists Intuition)"
(.pdf).
Ball State University Faculty Colloquium, November 19, 2009.
"Krein Signatures for Eigenvalue Problems Associated to Integrable Systems" (.pdf).
The XIXth International Workshop on Operator Theory and its Applications, July 22-26 2008, College of William and Mary.
Undergraduate Publications:
- Non-Destructive Testing of Thermal Resistances for a Single Inclusion in a
2-D Domain
With N. Christian
31 pages. Rose-Hulman Institute of Technology Undergraduate Math Journal, Vol. 6, Issue 1, 2005.
Download .pdf
- Non-Destructive Testing of Thermal Resistances for a Single Inclusion in a 2-D Domain
With N. Christian
9 pages. Ball State Undergraduate Mathematics Exchange, Fall 2004, Vol. 2, No. 2.
Note: This is a shortened version
of the full technical report written
in partial fulfillment of the requirements of the 2004 REU at Rose Hulman
Institute of Technology.
- Quantum Mechanics in Quantum Computing
8 pages. Ball State Undergraduate Mathematics Exchange, Fall 2003, Vol. 1, No. 1.
Undergraduate Research Experiences
REU at Rose Hulman Institute of Technology, Summer 2004
Advisor: Kurt Bryan
ERULF with DOE at Argonne National Laboratory, Summer 2003
Advisor: Dmitry Karpeev