Computational and Applied Mathematics (CAM) Seminar

Spring, 2007

CAM seminar usually runs from 2:00-3:00PM, Wednesday.

(Link to CAM seminar talks in Fall, 2006)

(Link to CAM seminar talks in other years.)

(Link to Numerical Analysis Group Webpage)

Date	Speaker/Institution	Title

Jan 31	Organizing meeting
Feb 14	John Albert, University of Oklahoma	Stability of multi-soliton solutions of the KdV equation (abstract)
Mar 14	Erik Van Vleck	Exponential Dichotomy for a Class of Schrodinger Operators
Mar 28	Weishi Liu	A general framework for steady-state Poisson-Nernst-Planck systems (I)
Apr 4	Weishi Liu	A general framework for steady-state Poisson-Nernst-Planck systems (II)
Apr 11	Anna Ghazaryan, University of North Carolina at Chapel Hill	Nonlinear convective instability of Turing-unstable fronts near onset: a case study (abstract)
Apr 18	Myunghyun Oh	Stability of cnoidal waves for the focusing nonlinear Schrodinger equation with a potential (abstract)
April 20	Bob Eisenberg, Rush Medical School at Chicago	Ion Channels (abstract)
Apr 25	Bob Russell, Simon Fraser University	An Elementary View of Adaptive Methods for the Solution of Time-Dependent PDEs (abstract)
May 2	Nicole Abaid	Asymptotic expansions of the PNP system for the ion channel problem (abstract)
May 9	Charles Lamb	A Gap Junction and Ephaptic Coupling Model for Smooth Muscle Tissues

1. John Albert, University of Oklahoma.

Explicit multi-soliton solutions of the Korteweg-de Vries (KdV) equation have been known since the 1960's, but there remain some open questions about how they fit into the general landscape of solutions of KdV in energy space. In this talk we consider the question of whether a small initial perturbation of a multi-soliton solution in H^1 gives rise to a solution which stays close to a multi-soliton for all time. We review existing results, and describe a variational approach to the problem based on the method of concentration compactness.

2. Anna Ghazaryan, University of North Carolina at Chapel Hill.

We consider a model system, consisting of two nonlinearly coupled partial differential equations, which exhibits a family of front solutions. In the coordinate frame that moves with a front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: the essential spectrum of the linearization at the front crosses the imaginary axis and, thus, spatially periodic patterns arise. Using exponentially weighted norms, we see that, in the co-moving frame and on the linear level the perturbations are pushed by the front away from the interface of the front. The problem with capturing this phenomenon on a nonlinear level is that the nonlinearity is not defined on exponentially weighted spaces. For our model equation we prove nonlinear stability of the front in exponentially weighted spaces, thereby establishing that the instability of the front is of a convective nature. The situation considered and the methods used in the proof are specific for non-self-adjoint problems. This is a joint work with B. Sandstede.

3. Myunghyun Oh, University of Kansas.

We study the stability of the pulse solutions and the periodic solutions with large spatial periods of the focusing nonlinear Schrodinger equation with a potential. We play with symmetries of the equation and consider edge bifurcation of the pulse solutions.

4. Bob Eisenberg, Rush Medical School at Chicago.

Protein channels conduct ions through a narrow tunnel of fixed charge and act as gatekeepers for cells and cell compartments. Hundreds of types of channels are studied every day in thousands of laboratories because of their biological and medical importance: a substantial fraction of all drugs used by physicians act directly or indirectly on channels.

The function of open channels can be described if the protein is described as an arrangement of charges—not as an invariant potential of mean force or set of rate constants—and the electric field and current flow are computed by the Poisson-Nernst-Planck (PNP) equations. The PNP equations can be derived from a nonequilibrium analysis of Langevin trajectories of charged particles moving in a an electric and concentration field created by the particles themselves (and boundary conditions). PNP describes the flux of ions in a selfconsistent mean electric field specified in traditional (nonlinear) Gouy-Chapman/Debye-Hückel/Poisson-Boltzmann equilibrium theories of electrolyte solutions and proteins. PNP is nearly the the Vlasov equations of plasma physics and the drift diffusion equation of semiconductor physics used there to describe the diffusion and migration of quasi-particles, holes and electrons.

The dramatic selectivity of the cardiac Ca channel of clinical fame arises naturally if correlations are introduced into PNP in the chemical tradition used to describe concentrated salt solutions. The fixed charge of the selectivity channel forces the channel to hold four positive mobile charges, making a concentration of some 17 molar univalent charge! Four (monovalent) Na+ ions occupy twice the volume of two (divalent) Ca++; the resulting difference in excluded volume produces calcium selectivity. This crowded charge model of selectivity predicts many selectivity properties in a wide range of ions and conditions (e.g., concentrations ranging over 5 orders of magnitude) after two adjustable parameters are set to optimal (unchanging) values.

Taken together, these results suggest that open ionic channels are natural nanotubes with properties dominated by the enormous fixed charge lining their walls and the consequent crowding of ions in their tiny volume. Other atomic detail is unexpectedly unimportant.

Highly charged nonequilibrium systems of this sort are hard to describe by direct simulations of molecular dynamics because those simulations are usually too brief to compute flux or current. Traditional simulations also have difficulty with the electric field since they use periodic boundary conditions and equilibrium boundary conditions, at best. Biomolecules are usually controlled by the number density of modulators present in trace amounts, often 10-6 ´ number density of water. Simulations have inherent difficulty in estimating such densities.

An opportunity exists to apply the well established methods of physics to the central problems of biology. In my opinion, the plasmas of biology need to be analyzed like the plasmas of physics. The mathematics of semiconductors and ionized gases should be the starting point for the mathematics of ions and proteins as well. It seems likely that the energetics of the compressible plasma of ions near active sites of proteins is an important determinant of their function. Of course, the plasma of ions and proteins differs significantly from those of physics. The definite structure of proteins, the small size of protein active sites, the spherical shape of metallic ions, and the pervasive presence of water are novel features of biological plasmas. So challenges abound, but I believe they can be addressed by existing tools and traditions of applied mathematics and physics.

5. Robert D. Russell, Simon Fraser university.

We shall give a discussion of the basic principles which motivate the use of adaptive methods for solving time-dependent PDEs --- including MMPDE (moving mesh PDE) methods. A key feature of MMPDEs is that they can be interpreted and analysed a continuous changes of variables from physical to computational space. Reaction diffusion (RD) equations have proven to be particularly amenable to solution in this way since one can often choose the MMPDE such that the underlying qualitative solution structure is automatically preserved by the numerical solution, even as singular solution behaviour is approached.

In this context we shall consider several fourth order RD problems which have been of interest in a variety of fields of late. We show how the interplay of analysis and numerics has led to a number of interesting discoveries. As well, we shall discuss some of the computational challenges, including some fascinating ones relating to the numerical solution of ODEs and of interest to researchers in dynamical systems.

6. Nicole Abaid, Asymptotic expansions of the PNP system for the ion channel problem.

Consider the equations for the electric field from continuity found by Nernst and Planck, and from concentration of ions found by Poisson. This system, used to model flow of ions through channels in a cell's membrane, can be viewed as a singularly perturbed system. We will discuss a geometric approach to solving the system in the zeroth order, and using asymptotic expansion, solve the first order system.

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