**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 ill-posed PDE. However, I came to believe that the lack of well-posed 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 career--separation of variables and one complex variable.

**Fluid mechanics**

In my distant bifurcation-theory past I studied finite-length 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 FitzHugh-Nagumo model, but based on the heart rather than nerve fibers. Other references [4--9] 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 short-long 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 steady-state 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 enough--and sufficiently tuned in to American popular culture--to 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 case-study 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). |

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?

- [1] D.G. Schaeffer and R. Iverson, Steady and intermittent slipping in a model of landslide motion regulated by pore-pressure 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. 307--337, 1980 [MR81c:35007]
- [3] Colleen C. Mitchell, David G. Schaeffer, A two-current model for the dynamics of cardiac membrane, Bulletin Math Bio, vol. 65 (2003), pp. 767--793
- [4] D.G. Schaeffer, J. Cain, E. Tolkacheva, D. Gauthier, Rate-dependent waveback velocity of cardiac action potentials in a done-dimensional 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. 459--482
- [6] D.G. Schaeffer, C. Berger, D. Gauthier, X. Zhao, Small-signal amplification of period-doubling bifurcations in smooth iterated mappings, Nonlinear Dynamics, vol. 48 (2007), pp. 381--389
- [7] D.G. Schaeffer, X. Zhao, Alternate pacing of border-collision period-doubling bifurcations, Nonlinear Dynamics, vol. 50 (2007), pp. 733--742
- [8] D.G. Schaeffer, M. Beck, C. Jones, and M. Wechselberger, Electrical waves in a one-dimensional 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 reaction-diffusion systems and their implications for neural-glial 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), pp356--371

Office Location: | 231 Physics Bldg, Durham, NC 27708 |

Office Phone: | (919) 660-2814 |

Email Address: | |

Web Page: | http://www.math.duke.edu/~dgs |

**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 granular-flow 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.)

Ill-posed 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, ill-posedness 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 ill-posed problems. The equations are ill-posed in a manner analogous to

the wave equation with a space-like direction appearing as the time axis. Specifically, there is a wedge of directions with the property that any plane wave*u*_{tt}= u_{xx}- u_{yy},*exp{i(k*whose wave vector_{1}x+k_{2}y)}*k*belongs to this wedge suffers catastrophic amplification. (As with this example, the*steady-state*equations of granular flow are hyperbolic.) With a linear ill-posed 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 ill-posed 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 steady-state. (This is an appealing result for industry since they study only steady-state equations.)

(ii) Under some circumstances the long-term, steady-state 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) steady-state 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 initial-value problem is solved for the (hyperbolic) steady-state 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 steady-state equations in this geometry.

(c) Describing the unloading that occurs near a shear band leads to an interesting free-boundary 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 molecular-dynamics (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 case-study 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 stones--focused 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 PhD-thesis 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 FitzHugh-Nagumo 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, Computer-Assisted • Rana catesbeiana

**Current Ph.D. 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 (2000/09-2003/06)

**Recent Publications**- Gonzales, K; Kayıkçı, O; Schaeffer, DG; Magwene, PM,
*Modeling mutant phenotypes and oscillatory dynamics in the Saccharomyces cerevisiae cAMP-PKA pathway.*, Bmc Systems Biology, vol. 7 (Winter, 2013), pp. 40 [doi] [abs] - S. Payne, B. Li, H. Song, D.G. Schaeffer, and L. You,
*Self-organized pattern formation by a pseudo-Turing mechanism*(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. 1225-1238, E D P SCIENCES, ISSN 0764-583X [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 one-dimensional model for cardiac dynamics*, Chaos, vol. 20 no. 2 (June, 2010)

- Gonzales, K; Kayıkçı, O; Schaeffer, DG; Magwene, PM,