Research

My primary research area is partial differential equations and dynamical systems. I am particularly interested in developing theoretical tools for understanding the long-time behavior of solutions to such systems, for example by analyzing the existence, bifurcation, and stability of nonlinear waves and coherent structures. Local seminars of interest to me include the BU Dynamics Seminar and the New England Dynamics Seminar. 

Refereed publications:

• ''Validated spectral stability via conjugate points," with J. Jaquette.  SIAM J. Appl. Dyn. Syst., 21, 366-404 (2022). 

• ''A dynamical approach to semilinear elliptic equations," with G. Cox, C. Jones, Y. Latushkin, and A. Sukhtayev. Ann. Inst. H. Poincaré Anal. Non Linéaire, 38, 421-450 (2021).  

• ''Exponential dichotomies for elliptic PDE on radial domains," with G. Cox, C. Jones, Y. Latushkin, and A. Sukhtayev. ``Mathematics of Wave Phenomena," W. Dorfler et. al. (eds.), 49-69, Springer (2020). 

• ''Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus," with E. Cooper, G. Lord, and K. Spiliopoulos. J. Nonlinear Sci., 30, 1677-1702 (2020). 

• "Spectral stability and spatial dynamics in partial differential equations." Notices of the AMS, 67, no 4, 500-507 (2020). 

• "Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity," with O. Chaudhary and C. E. Wayne. Arch. Ration. Mech. Anal. 235, 1105-1149 (2020).

• "Localized radial roll patterns in higher space dimensions," with J. J. Bramburger, D. Altschuler, C. I. Avery, T. Sangsawang, P. Carter, and B. Sandstede. SIAM J. Appl. Dyn. Syst., 18, no 3,  1420–1453 (2019).

• "Selection of quasi-stationary states in the Navier-Stokes equation on the torus," with E. Cooper and K. Spiliopoulos. Nonlinearity, 32, 209-237 (2019).

• "Grassmannian flows and applications to nonlinear partial differential equations," with A. Doikou, S.J.A. Malham, and I. Stylianidis. In "Computation and Combinatorics in Dynamics, Stochastics and Control," Proceedings of the Abel Symposium, Rosendal, Norway, August 2016, Springer (2018).

• "Partial differential systems with nonlocal nonlinearities: Generation and solutions," with A. Doikou, S.J.A. Malham, and I. Stylianidis. Phil. Trans. R. Soc. A, 376, no 2117 (2018). 

• "Instability of pulses in gradient reaction–diffusion systems: A symplectic approach," with G. Cox, C. Jones, Y. Latushkin, K. McQuighan, and A. Sukhtayev. Phil. Trans. R. Soc. A, 376, no 2117 (2018).

• "Isolas versus snaking of localized rolls," with T. Aougab, P. Carter, S. Desai, B. Sandstede, M. Stadt, and A. Wheeler. J Dyn Diff Equat, 31, 1199-1222 (2019 - appeared online 2017).

• "Stability of nonlinear waves: pointwise estimates." Discrete and Continuous Dynamical Systems Series S, 10, no 2, 191-211 (2017). Part of a special issue associated with the 2015 Bremen winter school and symposium entitled ``Diffusion on fractals and non-linear dynamics."

• "Analysis of enhanced diffusion in Taylor dispersion via a model problem," with O. Chaudhary and C. E. Wayne. ``Hamiltonian PDEs and Applications," Fields Institute Communications, 75, 31-71 (2015). 

• "Computing the Maslov Index for large systems," with S. Malham. Proceedings of the AMS,  143, no. 5, 2159-2173 (2015).

• "Superadiabaticity in Reaction Waves as a Mechanism for Energy Concentration," with S. G. Mahajan, J. T. Abrahamson, S. Birkhimer, E. Friedman, Q. H. Wang, and M. S. Strano. Energy and Environmental Science 7, 3391-3402 (2014).

• "Nonlinear stability of source defects in the complex Ginzburg-Landau equation," with T. Nguyen, B. Sandstede and K. Zumbrun. Nonlinearity 27, 739-786 (2014). 

• "Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations," with C. E. Wayne.  Proc. Roy. Society. Ed., 143, 905-927 (2013).

• "Toward nonlinear stability of sources via a modified Burgers equation,'' with T. Nguyen , B. Sandstede and K. Zumbrun. Phys. D 241, 382-392 (2012). 

• "Using global invariant manifolds to understand metastability in Burgers equation with small viscosity,'' with C. E. Wayne. SIAM Rev. 53, no. 1, 129-153 (2011).

• "Stability of traveling wave solutions for coupled surface and grain boundary motion," with Z. Pan and B. Wetton. Phys. D 239, 1730-1740 (2010). 

• "Nonlinear stability of time-periodic viscous shocks,'' with B. Sandstede and K. Zumbrun. Arch. Ration. Mech. Anal. 196, 1011-1076 (2010).

• "Nonlinear stability of semi-discrete shocks for two sided schemes," with H. J. Hupkes, B. Sandstede, and K. Zumbrun. Submitted. SIAM J. Math. Anal. 42, no. 2, 857-903 (2010).

• "Using global invariant manifolds to understand metastability in Burgers equation with small viscosity,'' with C. E. Wayne. SIAM J. Appl. Dyn. Syst. 8, no. 3, 1043-1065 (2009). 

• "Snakes, ladders, and isolas of localised patterns," with J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht. SIAM J. Math. Anal. 41, 936-972 (2009).

• "Nonlinear convective stability of traveling fronts near Turing and Hopf instabilities,'' with A. Ghazaryan and B. Sandstede. J. Differential Equations 246, 4371-4390 (2009).

• "Electrical waves in a one-dimensional model of cardiac tissue," with C.K.R.T. Jones, D. Schaeffer and M. Wechselberger. SIAM J. Appl. Dyn. Syst. 7, no. 4, 1558-1581 (2008).

• "Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws,'' with C. E. Wayne. J. Differential Equations, 244, 87-116 (2008).

• "A geometric construction of traveling waves in a bioremediation model,'' with A. Doelman and T. J. Kaper. J. Nonlinear Sci., 16, no. 4, 329-349 (2006).

• "A geometric theory of chaotic phase synchronization,'' with K. Josić. Chaos, 13(1), 247-258 (2003).

Minicourses:

• A three hour minicourse on "Stability of nonlinear waves: pointwise estimates" that I gave at the 3rd Bremen Winter School and Symposium in March 2015.

• A four lecture minicourse on "Linear stability theory" that I gave at the workshop "The stability of coherent structures and patterns" in June 2012. 

PhD Thesis:

• "Topics in stability theory for partial differential equations." Advisors: Tasso J. Kaper and C. Eugene Wayne. (2006)

Slides for talks:

• "An introduction to recent applications of the Maslov index in dynamical systems and PDEs." Presentation for an hour-long talk at the 2023 Joint Mathematics Meetings.

• "Spectral stability and the Maslov index." Presentation for a 45 minute talk at Equadiff 2019.

• "Enhanced dissipation in fluids models." Presentation for an hour-long talk, 2018.

• "Vortices and the Navier-Stokes equation: understanding solutions of equations that we can't actually solve." Hour-long public lecture, given as part of the Third Bremen Winter School and Symposium, in March 2015. Note that the embedded videos won't play in the .pdf. One of the videos can be found here. 

• "Nonlinear stability of coherent structures via pointwise estimates." Presentation for an hour-long talk, 2014.

• "Rapid convergence to quasi-stationary states in the 2D Navier-Stokes equation." Presentation for an hour-long talk, 2012.

• "Nonlinear stability of semi-discrete shocks for two sided schemes." Presentation for a half-hour long talk, 2010.

• "Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities." Presentation for an hour long talk, 2009. 

• "Using global invariant manifolds to understand metastability in Burgers equation with small viscosity." Presentation for an hour long talk, 2009.

• "Nonlinear stability of time-periodic viscous shocks." Presentation for an hour long talk, 2009. 

• "Snakes, ladders, and isolas of localised patterns." This talk was presented at the SIAM student conference at the University of Oxford, 2008.

Videos of talks:

• Watch a video of a talk I gave at MSRI in October, 2018, on "Stability, the Maslov Index, and Spatial Dynamics." The corresponding slides are here. 

• Watch a video of the talk I gave in the Topology Seminar at the University of Edinburgh on November 14, 2013, on using the Maslov Index to analyze stability in PDEs. 

• Watch a video of the talk I gave at the BIRS workshop entitled "Localized Multi-Dimensional Patterns in Dissipative Systems" in 2011. The talk title is "Rapid convergence to quasi-stationary states in the 2D Navier-Stokes equation." 

Miscellany:

• Review of the book "Nonlinear PDEs: A Dynamical Systems Approach" by Guido Schneider and Hannes Uecker for DS Web (2018). 

• Review of the book "Multiple Time Scale Dynamics" by Christian Kuehn for SIAM Review (2017). 

• Article entitled "Analyzing multiple timescales in two-dimensional fluids using dynamical systems," with C. Eugene Wayne, SIAM News (2017). 

• Entry on "Burgers equation" in the Encyclopedia of Applied and Computational Mathematics, Springer (print version 2015 - published online in 2013). 

• Oberwolfach report for the workshop I spoke at entitled "Dynamics of Patterns," organized by Wolf-Jürgen Beyn, Bernold Fiedler, and Björn Sandstede, in December 2012. Published in the European Mathematical Society, Volume 9, Issue 4 (2012), pg 3585-3588. 

• A short article I wrote for SIAM's DSWeb Magazine in 2010.