
David G. Schaeffer, James B. Duke Professor Emeritus of Mathematics and CNCS: Center for nonlinear and complex systems
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
 Contact Info:
 Education:
Ph.D.  Massachusetts Institute of Technology  1968 
B.S.  University of Illinois  1963 
 Specialties:

Applied Math
 Research Interests: Applied Mathematics, especially Partial Differential Equations
Granular flow is the major focus of my
research, and this is what I describe in the first part
of this page.
In the second part, I discuss a course I began teaching
in 1996 that has expanded my research horizons.
Part I: Granular flow
Besides finding granularflow problems fascinating, I
am interested in them because
of their practical importance. Manufacturing
industries must handle
heaps (literally) of raw materials in granular form,
and difficulties in
handling them are very expensive.
There are also important geophysical applications
involving granular
flow: e.g., avalanches, earthquakes, beach erosion, etc.
(Incidentally, [B97] and [JNB96] in the
references
contain surveys of experiments in granular flow.)
Illposed partial differential equations are one of the
mathematical challenges which arise in granular flow.
(Although it is more than a decade old, I still have
occasion to refer to my original demonstration of this
behavior, item [S87] in the
references.)
Physically, illposedness appears at the point when
sheared material cracks and then breaks into separate
pieces which move independently: in technical terms,
when a shear band forms.
A model problem exhibiting this behavior was developed
in [S92] in the
references.
Currently I am struggling to understand what
mathematical sense can be made of these illposed problems.
The equations are illposed in a manner analogous to
u_{tt} = u_{xx}  u_{yy} ,
the wave equation with a spacelike direction appearing
as the time axis.
Specifically, there is a wedge of directions with the
property that any plane wave
exp{i(k_{1}x+k_{2}y)}
whose wave vector k belongs to this wedge
suffers catastrophic amplification.
(As with this example, the steadystate
equations of granular flow are hyperbolic.)
With a linear illposed problem such as above, no
solution is possible unless the initial data are
unrealistically smooth (analytic, to be precise).
However, in the nonlinear equations describing granular
flow, the wedge of illposed directions rotates as the
solution varies, and this may control growth.
Working with the model problem of [S92] in the
references,
we have found two tentative conclusions:
(i) Although transients are unpredictable,
nevertheless at large times the solution tends to a
well defined steadystate. (This is an appealing
result for industry since they study only steadystate
equations.)
(ii) Under some circumstances the longterm,
steadystate solution appears to "ignore" some of the
boundary conditions. So far we have been unable to
predict this behavior reliably. In applications, it is
vital to understand such issues in order to know what
boundary conditions should be imposed for the
(hyperbolic) steadystate equations of granular flow.
I hope that the model problem will shed light on this
question.
Below are annotated references to work in various other
directions in granular flow. The first five items,
like the work described above, relate to a continuum
description of granular material; the latter two relate
to discrete models for granular materials. Industry
uses continuum models because they are simpler.
However, a granular medium consists of many individual
particles, and the separation between microscales and
macroscales is incomparably smaller than in a fluid.
Thus, a fundamental investigation of granular flow must
include its discrete nature.
(a) In [GM99a,b] of the
references,
an initialvalue problem is solved for the (hyperbolic)
steadystate equations for granular flow in a conical
hopper, with Cauchy data specified at the top of the
hopper. (As hinted above, it is not known whether this
is the most physical boundary problem.)
(b) In [GSS99] of the
references
we developed an approximation for the analogue of
Jenike's radial solution in a conical hopper with an
inverted conical insert. In the near future we plan to
study numerically solutions of the steadystate
equations in this geometry.
(c) Describing the unloading that occurs near a shear
band leads to an interesting freeboundary problem for
the wave equation. This problem is analyzed in
[SS93,94] in the
references.
(d) The dramatic role of imperfections near the onset
of shear banding is demostrated in [SS97] in the
references.
(e) Models for liquefication of soils are analyzed in
[HS98,99] in the
references.
In one of these problems the solution of an
initial/boundary value problem depends on the boundary
data in a discontinuous, fractal, manner.
(f) Although continuum models for granular materials
deal exclusively with
the average behavior of materials, recent
experiments by Behringer and collaborators have
emphasized that fluctuations from the average may be
substantial.
Two simple probabilistic models are studied in
[SS98,SSS99] in the
references;
these are preliminary attempts to understand
fluctuations theoretically.
(g) Several people (e.g., [A99] in the
references)
in the group at Duke are working on moleculardynamics
(MD) simulations of granular flow.
Although MD calculations provide complete information
about the flow,
the calculations are too lengthy for practical
situations.
Ultimately we hope to merge continuum and MD methods,
using MD at the finest level of an adaptive mesh
refinement code and
continuum equations at all higher levels; but this
remains an elusive goal.
My collaborators in studying granular flow include Tom
Witelski,
Bill
Allard, and Bob
Behringer at Duke; Michael
Shearer,
and
Pierre
Gremaud
at NC State; Brian Hayes at Stevens; and Tony Royal at
Jenike & Johanson, Inc
Part II: Research growing out of teaching
Every spring semester since 1996 I have taught a course
that has 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.
The topics and their proposers for each year have been:
1996  Lithotripsy 
L. Howle, P. Zhong (ME) 
1997  Population models in ecology 
W. Wilson (Zoology) 
1998  Electrophysiology of the heart I 
C. Henriquez (BME) 
1999  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.
This course was written up in [HSSZ98] in the
references;
the research of one team was published in [MTHZ97] in the
references.
Population models in ecology became the PhDthesis
topic for A. Ashih, working jointly with myself and
Will Wilson.
Some of his results will appear in [AW99] in the
references,
and two more publications are in preparation.
The 1999 heart course focused on the experiments of
[HBG99] in the
references,
in which a small piece of cardiac tissue was subjected
to periodic stimuli at various frequencies much above
the normal pacing rate.
Martin Hall included some of the ideas developed in the
course in his Physics PhD thesis, [H99] in the
references.
Based on what I learned teaching this course, I have
undertaken a joint research project with Dan Gauthier
(Physics) and Wanda Krassowska (BME) studying
mathematical models for the electrical response of the
heart, in particular comparing their predictions with
experiment.
Although there are many models for the electrical
response of the heart, the simplest model involves a
system of only two ODE, analogous to the ODE resulting
from suppression of spatial extent in the
FitzHughNagumo equations.
Such a model is well within the reach of
undergraduates, and this fall I taught an undergraduate
seminar on it.
The course is culminating in research projects by three
separate student teams.
The problems are elementary by usual research
standards, but their successful completion will further
the research begun in [HBG99].
 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
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 John W. Cain
 Aaron Ashish
 Shu Dai
 Kevin E. Gonzales
 Matthew M Bowen
 Michael Gordon
 Lianjun An
 Feng Wang
 Risto Lehtinen
 Maija Kuusela
 Joseph Fehribach
 E. Bruce Pitman
 John Goodrich
 Postdocs Mentored
 Anne Catlla (2006  2008)
 Xiaopeng Zhao (2005  2007)
 Wenjun Ying (2005  2008)
 Elena Tolkacheva (2004  2006)
 J. Matthews (2000/092003/06)
 Recent Publications
(More 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
(Submitted, Winter, 2010)
 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]
 Farjoun, Y; Schaeffer, DG, The hanging thin rod: a singularly perturbed eigenvalue problem,
SIAM Sppl. Math.
(Submitted, July, 2010)
 Dai, S; Schaeffer, DG, Chaos in a onedimensional model for cardiac dynamics,
Chaos, vol. 20 no. 2
(June, 2010)
