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:

  1. Diffusive Stability & Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
    With Mariana Haragus, Wesley R. Perkins and Bjorn de Rijk.
  2. Subharmonic Dynamics of Wave Trains in Systems of Viscous Conservation Laws
    With Wesley R. Perkins
  3. Nonlinear Stability of Periodic Waves in the Hamiltonian Magma Equations
    With Wesley R. Perkins
  4. A Center Manifold Approach to the Existence of Solutions in the Capillary-Gravity Whitham Equation
    With Tien Troung and Miles H. Wheeler.
  5. Instabilities in KP and Rotation Modified KP Type Equations
    With Vera Mikyoung Hur and Ashish K. Pandey.
  6. Stability of Roll-Waves in the Vanishing Viscosity Limit
    With Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun.
  7. Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems
    With Wesley R. Perkins
    Submitted. (arXiv)
  8. Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
    With Mariana Haragus and Wesley R. Perkins.
    Submitted. (arXiv)
  9. 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:

  1. Generalized Solitary Waves in a Capillary-Gravity Whitham Equation
    With J. Douglas Wright.
    Studies in Applied Mathematics, to appear. (arXiv)
  2. 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)
  1. Modulational Instability of Viscous Fluid Conduit Periodic Waves
    With Wesley R. Perkins
    SIAM Journal on Mathematical Analysis, 52: 277-305, 2020. (arXiv)
  2. On the Dynamics of Traveling Fronts Arising in Nanoscale Pattern Formation
    With Gregory Lyng and Connor Smith.
    Physica D, 401: 16pp, 2020. (arXiv)
  3. 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)
  4. 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)
  5. 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)
  6. Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models
    With Kyle M. Claassen.
    Studies in Applied Mathematics, 141: 205-246, 2018. (arXiv)
  7. 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)
  8. 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)
  9. 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)
  10. 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)
  11. 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)
  12. 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)
  13. 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)
  14. Modulational Instability in the Whitham Equation for Water Waves
    With Vera Mikyoung Hur.
    Studies in Applied Mathematics, 134 no. 3: 120-143, 2015. (arXiv)
  15. 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).
  16. 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)
  17. Stability of Small Periodic Waves in Fractional KdV Type Equations
    SIAM Journal on Mathematical Analysis, 45 no. 5: 2597-3228, 2013. (reprint)
  18. 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)
  19. 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)
  20. 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)
  21. 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)
  22. Convergence of Hill's Method for Nonselfadjoint Operators
    With Kevin Zumbrun.
    SIAM Journal on Numerical Anaylsis, 50 no. 1: 64-78, 2012. (reprint)
  23. 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)
  24. 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)
  25. 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)
  26. Nonlinear Stability of Viscous Roll Waves
    With Kevin Zumbrun and Pascal Noble.
    SIAM Journal on Mathematical Analysis, 43 no. 2: 557-611, 2011. (reprint)
  27. 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)
  28. 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).
  29. 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)
  30. 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)
  31. On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
    Physica D, 239 no. 19: 1892-1908, 2010. (preprint)
  32. 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)
  33. 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)
  34. The Transverse Instability of Periodic Waves in Zakharov-Kuznetsov Type Equations
    Studies in Applied Mathematics, 124 no. 4: 323-345, 2010. (preprint)
  35. 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)
  36. 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)
  37. 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


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:

    1. 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
    2. 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.
    3. 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