Research Interests

My primary interest is determining the nonlinear stability and large-time behavior of solutions to dissipative PDEs, such as reaction-diffusion equations and viscous conservation laws. This includes studying nonlinear waves such as traveling waves and spatially and/or temporally periodic patterns. I typically view these PDEs as infinite-dimensional dynamical systems, and I analyze them using a variety of mathematical techniques, for example invariant manifolds, similarity variables, geometric singular perturbation theory, exponential dichotomies, and pointwise estimates.

In much of my work, the main mathematical difficulty arises from the fact that the linear operator lacks a so-called spectral gap. For example, the spectrum could be as in the above figure on the left, where a zero eigenvalue is embedded in the continuous spectrum. As a result, one cannot use standard spectral decomposition methods to separate the solutions that decay from those that simply remain bounded. Furthermore, estimates necessary for the related nonlinear analysis can become extremely delicate, as can the properties of associated bifurcations.

One particular topic I've been interested in recently is known as "metastability." Roughly speaking, this refers to long transients in the dynamics. For example, solutions could spend large periods of time near unstable states before settling down to their stable, asymptotic limit. This is interesting mathematically, because there are far fewer techniques available for analyzing transient behaviors than there are for analyzing asymptotic behaviors. Also, in real world systems, the transient time-scale may be so long that one will never actually see the limiting behavior. Thus, metastability is important in applications, as well. The type of behavior arises, for example, in the two-dimensional Navier-Stokes equation.

The above picture on the right is of me in the Catalan town of Vic. It doesn't really have anything to do with my research. However the statue, known as "Merma," is said to come alive during the festival of de Patron and chase the children while carrying a lash. Occaisionally, while working on my research, I feel as if Merma is chasing me around in circles.

Publications and Preprints (or via google scholar)
• "Nonlinear stability of source defects in oscillatory media," with T. Nguyen, B. Sandstede and K. Zumbrun. Preprint. arXiv
• "Localized radial roll patterns in higher space dimensions," with J. J. Bramburger, D. Altschuler, C. I. Avery, T. Sangsawang, P. Carter, and B. Sandstede. Submitted. [.pdf]
• "Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity," with O. Chaudhary and C. E. Wayne. Submitted. arXiv
• "Selection of quasi-stationary states in the Navier-Stokes equation on the torus," with E. Cooper and K. Spiliopoulos. To appear in Nonlinearity. arXiv
• "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 (published 2018). arXiv
• "Partial differential systems with nonlocal nonlinearities: Generation and solutions," with A. Doikou, S.J.A. Malham, and I. Stylianidis. Philosophical Transactions A, 376, no 2117 (2018). arXiv
• "Instability of pulses in gradient reaction-diffusion systems: a symplectic approach," with G. Cox, C. Jones, Y. Latushkin, K. McQuighan, and A. Sukhtayev. Philosophical Transactions A, 376, no 2117 (2018). arXiv
• "Isolas versus snaking of localized rolls," with T. Aougab, P. Carter, S. Desai, B. Sandstede, M. Stadt, and A. Wheeler. Journal of Dynamics and Differential Equations (2017), https://doi.org/10.1007/s10884-017-9624-0. [.pdf]
• "Stability of nonlinear waves: pointwise estimates." Discrete and Continuous Dynamical Systems Series S, 10, 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". [.pdf]
• "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) . arXiv
• "Computing the Maslov Index for large systems," with S.J.A. Malham. Proceedings of the AMS 143, no. 5, 2159-2173 (2015). arXiv
• "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 Environ. Sci., 7, 3391-3402 (2014) [Journal]
• "Nonlinear stability of source defects in the complex Ginzburg-Landau equation," with T. Nguyen, B. Sandstede and K. Zumbrun. Nonlinearity, 27, 739-786 (2014). [.pdf]
• "Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations," with C. E. Wayne. Proc. Roy. Soc. Edinburgh Sect. A., 143, 905-927 (2013). arXiv
• "Toward nonlinear stability of sources via a modified Burgers equation,'' with T. Nguyen, B. Sandstede and K. Zumbrun. Physica D, 241, 382-392 (2012). arXiv
• "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). [.pdf]
• "Stability of traveling wave solutions for coupled surface and grain boundary motion," with Z. Pan and B. Wetton. Phys. D 239, 1730-1740 (2010). [.pdf]
• "Nonlinear stability of time-periodic viscous shocks,'' with B. Sandstede and K. Zumbrun. Arch. Ration. Mech. Anal. 196, 1011-1076 (2010). arXiv
• "Nonlinear stability of semi-discrete shocks for two sided schemes," with H. J. Hupkes, B. Sandstede, and K. Zumbrun. SIAM J. Math. Anal. 42, no. 2, 857-903 (2010). [.pdf]
• "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). arXiv
• "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). [.pdf]
• "Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities," with A. Ghazaryan, and B. Sandstede. J. Differential Equations 246, 4371-4390 (2009). [.pdf]
• "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). [.pdf]
• "Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws," with C. E. Wayne . J. Differential Equations 244, 87-116 (2008). [.pdf]
• "A geometric construction of traveling waves in a bioremediation model," with A. Doelman and T. J. Kaper . J. Nonlinear Sci. 16, 329-349 (2006). [.pdf]
• "A geometric theory of chaotic phase synchronization," with K. Josic . Chaos 13, 247-258 (2003). [.pdf]

Lecture Notes, Thesis, Slides for Talks, Poster Presentations, and Miscellany

Lecture notes
PhD Thesis
Slides for talks - research
• "Stability, the Maslov Index, and Spatial Dynamics": Presentation for an hour-long talk. [.pdf]
• "Enhanced dissipation in fluids models": Presentation for an hour-long talk. [.pdf]
• "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. [.pdf]
There were embedded videos in this talk that do not play in the above .pdf version. Links to two of them can be found here:
• "Nonlinear stability of coherent structures via pointwise estimates": Presenatation for an hour-long talk. [.pdf]
• "Rapid convergence to quasi-stationary states in the 2D Navier-Stokes equation": Presentation for an hour-long talk. [.pdf]
• "Nonlinear stability of semi-discrete shocks for two sided schemes": Presentation for a half-hour long talk. [.pdf]
• "Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities": Presentation for an hour long talk. [.pdf]
• "Using global invariant manifolds to understand metastability in Burgers equation with small viscosity": Persentation for an hour long talk. [.pdf]
• "Nonlinear stability of time-periodic viscous shocks": Presentation for an hour long talk. [.pdf]
• "Snakes, ladders, and isolas of localised patterns": This talk was presented at the SIAM student conference at the University of Oxford, April 25, 2008. [.pdf]
Slides for talks - other
• Slides from a presentation on "Social factors and (under)representation in mathematics", given by myself and Moon Duchin during the 2016 Summer@ICERM REU Program at ICERM at Brown University. [.pdf]
• Slides from a job search seminar at the IMA in November, 2012. [.pdf]
Poster presentations
• "Nonlinear stability of time-periodic viscous shocks": This poster was presented at the conference "Geometric Analysis, Elasticity and PDE" at Heriot-Watt University, June 23-27, 2008. [.pdf]
• "A geometric analysis of traveling waves in a bioremediation model": This poster was presented at the conferences "SIAM conference on Application of Dynamical Systems," in Snowbird, UT, May 22-26, 2005, and "Connections for Women: Dynamical Systems" at MSRI, January 18-19, 2007. [.pdf]
Miscellany
• 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.
• Wrote review of the book "Nonlinear PDEs: A Dynamical Systems Approach" by Guido Schneider and Hannes Uecker for DS Web (2018).
• Wrote 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).
• Watch a video of the talk I gave in the Topology Seminar at the University of Edinburgh on November 14, 2013, on the potential use of the Maslov Index when studying stability in PDEs.
• 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.
• 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."
• Watch a video of the talk I gave at the 17th German-American Frontiers of Science Symposium at the US National Academy of Sciences in Irvine, CA, in 2011. The talk title is "Stability of Patterns."
• A short article I wrote for SIAM's DSWeb Magazine in 2010.