Research Interests for David G. Schaeffer
Research Interests: Applied Mathematics, especially Partial Differential Equations
Granular flow
Although I worked in granular flow for 15 years, I largely stopped working in this area around 5 years ago. Part of my fascination with this field derived from the fact that typically constitutive equations derived from engineering approximations lead to illposed PDE. However, I came to believe that the lack of wellposed governing equations was the major obstacle to progress in the field, and I believe that finding appropriate constitutive relations is a task better suited for physicsts than mathematicians, so I reluctantly moved on.
One exception: a project analyzing periodic motion in a model for landslides as a Hopf bifurcation. This work is joint with Dick Iverson of the Cascades Volcanic Laboratory in Vancouver Washington. This
paper
[1] was a fun paper for an old guy because we were able to solve the problem with techniques I learned early in my careerseparation of variables and one complex variable.
Fluid mechanics
In my distant bifurcationtheory past I studied finitelength effects in Taylor vortices. Questions of this sort were first raised by Brooke Benjamin. My
paper [2] shed some light on these issues, but some puzzles remained. Over the past few years I have conducted a leisurely collaboration with Tom Mullin trying to tie up the loose ends of this problem. With the recent addition of Tom Witelski to the project, it seems likely that we will soon complete it.
Mathematical problems in electrocardiology
About 10 years ago I began to study models for generation of cardiac rhythms. (Below I describe how I got interested in this area.) This work has been in collaboration with Wanda Krassowska (BME), Dan Gauthier (Physics) and Salim Idress (Med School). Postdocs Lena Tolkacheva and Xiaopeng Zhou contributed greatly to the projects, as well as grad students John Cain and Shu Dai. The first
paper [3], with Colleen Mitchell was a simple cardiac model, similar in spirit and complexity to the FitzHughNagumo model, but based on the heart rather than nerve fibers. Other
references
[49] are given below.
A general theme of our group's work has been trying to understand the origin of alternans. This term refers to a response of the heart at rapid periodic pacing in which action potentials alternate between short and long durations. This bifurcation is especially interesting in extended tissue because during propagation the shortlong alternation can suffer phase reversals at different locations, which is called discordant alternans.
Alternans is considered a precursor to more serious arrythmias.
Let me describe one current
project
[9]. My student, Shu Dai, is analyzing a weakly nonlinear modulation equation modeling discordant alternans that was proposed by Echebarria and Karma. First we show that, for certain parameter values, the system exhibits a degenerate (codimension 2) bifurcation in which Hopf and steadystate bifurcations occur simultaneously. Then we show, as expected on grounds of genericity (see Guckenheimer and Holmes, Ch. 7) that chaotic solutions can appear. The appearance of chaos in this model is noteworthy because it contains only one space dimension; by contrast the usual route to chaos in cardiac systems is believed to be through the breakup of spiral or scroll waves, which of course requires two or more dimensions.
Other biologogical problems
Showing less caution than appropriate for a person my age, I have recently begun to supervise a student, Kevin Gonzales, on a project modeling gene networks. Working with Paul Magwene (Biology), we seek to understand the network through which yeast cells, if starved for nitrogen, choose between sporulation and pseudohyphal growth. (Whew!) This work is an outgrowth of my participation in the recently funded Center for Systems Biology at Duke.
I have gotten addicted to applying bifurcation theory to differential equations describing biological systems. For example, my colleagues Harold and Anita Layton are tempting my with some fascinating bifurcations exhibited by the kidney. Here is a whimsical catch phrase that describes my addiction: "Have bifurcation theory but won't travel". (Are you old enoughand sufficiently tuned in to American popular cultureto understand the reference?)
Research growing out of teaching
Starting in 1996 I have sometimes taught a course
that led to an expansion of my research.
The process starts by my sending a memo to the science
and engineering faculty at Duke, asking if they would
like the assistance of a group of math graduate
students working on mathematical problems arising in
their (the faculty member's) research.
I choose one area from the responses, and I teach a
casestudy course for math grad students focused on
problems in that area.
In broad terms, during the first half of the course I
lecture on scientific and mathematical background for
the area; and during the second half student teams do
independent research, with my collaboration, on the
problems isolated earlier in the semester.
I also give supplementary lectures during the second
half, and at the end of the semester each team lectures
to the rest of the class on what it has discovered.
This course was written up in the
SIAM Review [11].
Topics and their proposers have been:
Lithotripsy 
L. Howle, P. Zhong (ME) 
Population models in ecology 
W. Wilson (Zoology) 
Electrophysiology of the heart I 
C. Henriquez (BME) 
Electrophysiology of the heart II 
D. Gauthier (Physics). 
Lithotripsy is an alternative to surgery for treating
kidney stonesfocused ultrasound pulses are used to
break the stones into smaller pieces that can be passed
naturally.
Multiple research publications, including a PhD. thesis, have come out of these courses, especially my work in electrophysiology.
I hope to offer this course in the future. Duke faculty: Do you have a problem area to propose?
References
 [1] D.G. Schaeffer and R. Iverson, Steady and intermittent slipping in a model of landslide motion regulated by porepressure feedback, SIAM Applied Math 2008 (to appear)
 [2] Schaeffer, David G., Qualitative analysis of a model for boundary effects in the Taylor problem, Math. Proc. Cambridge Philos. Soc., vol. 87, no. 2, pp. 307337, 1980 [MR81c:35007]
 [3] Colleen C. Mitchell, David G. Schaeffer, A twocurrent model for the dynamics of cardiac membrane, Bulletin Math Bio, vol. 65 (2003), pp. 767793
 [4] D.G. Schaeffer, J. Cain, E. Tolkacheva, D. Gauthier, Ratedependent waveback velocity of cardiac action potentials in a donedimensional cable, Phys Rev E, vol. 70 (2004), 061906
 [5] D.G. Schaeffer, J. Cain, D. Gauthier,S. Kalb, W. Krassowska, R. Oliver, E. Tolkacheva, W. Ying, An ionically based mapping model with memory for cardiac restitution, Bull Math Bio, vol. 69 (2007), pp. 459482
 [6] D.G. Schaeffer, C. Berger, D. Gauthier, X. Zhao, Smallsignal amplification of perioddoubling bifurcations in smooth iterated mappings, Nonlinear Dynamics, vol. 48 (2007), pp. 381389
 [7] D.G. Schaeffer, X. Zhao, Alternate pacing of bordercollision perioddoubling bifurcations, Nonlinear Dynamics, vol. 50 (2007), pp. 733742
 [8] D.G. Schaeffer, M. Beck, C. Jones, and M. Wechselberger, Electrical waves in a onedimensional model of cardiac tissue, SIAM Applied Dynamical Systems (Submitted, 2007)
 [9] D.G. Schaeffer and Shu Dai, Spectrum of a linearized amplitude equation for alternans in a cardiac fiber, SIAM Analysis 2008 (to appear)
 [10] D.G. Schaeffer, A. Catlla, T. Witelski, E. Monson, A. Lin, Annular patterns in reactiondiffusion systems and their implications for neuralglial interactions (Preprint, 2008)
 [11] L. Howle, D. Schaeffer, M. Shearer, and
P. Zhong, Lithotripsy, The treatment of kidney stones with shock waves, SIAM Review vol. 40 (1998), pp356371
 Keywords:
 Action Potentials, Animals, Cell Membrane, Computer Simulation, Dogs, Electrophysiology, Heart, Heart Conduction System, Humans, Ions, Models, Cardiovascular, Models, Theoretical, Muscle Cells, Myocytes, Cardiac, Numerical Analysis, ComputerAssisted, Rana catesbeiana
 Recent Publications
 S. Payne, B. Li, H. Song, D.G. Schaeffer, and L. You, Selforganized pattern formation by a pseudoTuring mechanism
(Submitted, Winter, 2010)
 Gonzales, K; Kayikci, O; Schaeffer, DG; Magwene, P, Modeling mutant phenotypes and oscillatory dynamics in the \emph{Saccharomyces cerevisiae} cAMPPKA pathway,
Plos Computational Biology, vol. 7
(Winter, 2010),
pp. 40 [doi] [abs]
 Dai, S; Schaeffer, DG, Bifurcations in a modulation equation for alternans in a cardiac fiber,
Esaim: Mathematical Modelling and Numerical Analysis, vol. 44 no. 6
(Winter, 2010),
pp. 12251238, E D P SCIENCES, ISSN 0764583X [Gateway.cgi], [doi] [abs]
 Farjoun, Y; Schaeffer, DG, The hanging thin rod: a singularly perturbed eigenvalue problem,
Siam Sppl. Math.
(July, 2010)
 Dai, S; Schaeffer, DG, Chaos in a onedimensional model for cardiac dynamics,
Chaos, vol. 20 no. 2
(June, 2010)
