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2005

Volume 65, Issue 6, pp. 1839-2198


Channel Formation in Gels

N. G. Cogan and James P. Keener

SIAM J. Appl. Math. 65, pp. 1839-1854 (16 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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We derive and give an analysis of a model of gel dynamics based on a two-phase description of the gel, where one phase consists of networked polymer and the second phase is the fluid solvent. It is found that for the gel to maintain an edge in a poor solvent, the function describing the osmotic pressure must be of a particular form. The model is used to study the behavior of a gel forced by a pressure gradient to move between two flat plates. The distribution of the network phase under these conditions is found to be nonuniform and dependent on the pressure gradient. There is a range of pressure gradients for which the network has regions of high and low volume fraction separated by a sharp boundary, indicative of a channel. We provide the bifurcation analysis of how these novel, singularly perturbed, channeled solutions occur.

Force Density Function Relationships in 2-D Granular Media

Robert C. Youngquist, Philip T. Metzger, and Kelly N. Kilts

SIAM J. Appl. Math. 65, pp. 1855-1869 (15 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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An integral transform relationship is developed to convert between two important probability density functions (distributions) used in the study of contact forces in granular physics. Developing this transform has now made it possible to compare and relate various theoretical approaches with one another and with the experimental data, despite the fact that one may predict the Cartesian probability density and another the force magnitude probability density. Also, the transforms identify which functional forms are relevant to describing the probability density observed in nature, and so the modified Bessel function of the second kind has been identified as the relevant form for the Cartesian probability density corresponding to exponential forms in the force magnitude distribution. Furthermore, it is shown that this transform pair supplies a mathematical framework sufficient for describing the evolution of the force magnitude distribution under shearing. Apart from the choice of several coefficients, whose evolution of values must be explained in the physics, this framework successfully reproduces the features of the distribution that are taken to be an indicator of jamming and unjamming in a granular packing.

A School-Oriented, Age-Structured Epidemic Model

Viggo Andreasen and Thomas Frommelt

SIAM J. Appl. Math. 65, pp. 1870-1887 (18 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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A model of childhood epidemics focusing on the impact of the school-year is presented. At the onset of the epidemic season, a new cohort of susceptible students enter the school, all other age-classes advance one year, while the oldest age-group leaves the mixing pool. If the susceptible pool is sufficiently large at the onset of the season, an epidemic will arise and run to its conclusion prior to the end of the school-year. The system is expressed in terms of a discrete dynamical system giving the changes in the age-dependent immunity structure on a year-to-year basis. If disease transmission is independent of age, the system settles at epidemics of constant size in each season. If disease transmission is age-dependent, more complicated dynamics may occur, including multiple stable states and chaos.

Wavelet Mie Representations for Solenoidal Vector Fields with Applications to Ionospheric Geomagnetic Data

Thorsten Maier

SIAM J. Appl. Math. 65, pp. 1888-1912 (25 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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A wavelet technique, the wavelet Mie representation, is introduced for the analysis and modeling of the earth's magnetic field and corresponding electric current distributions from geomagnetic data obtained within the ionosphere. The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz decomposition of spherical vector fields and (ii) the Mie representation of solenoidal vector fields in terms of poloidal and toroidal parts. The wavelet Mie representation is shown to provide an adequate tool for geomagnetic modeling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features. An important example is ionospheric currents flowing radially onto or away from the earth. To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.

Retrieving Topological Information for Phase Field Models

Qiang Du, Chun Liu, and Xiaoqiang Wang

SIAM J. Appl. Math. 65, pp. 1913-1932 (20 pages) | Cited 11 times

Online Publication Date: July 31, 2006

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The phase field approach has become a popular tool in modeling interface motion, microstructure evolution, and more recently the shape transformation of vesicle membranes under elastic bending energy. While it is advantageous to employ phase field models in numerical simulations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes, detecting topological events may also become important for many applications such as those in the simulation of blood cells. Motivated by such considerations, a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events. It can also be used as a control method to avoid unphysical changes of topology due to the numerical methods, should it become necessary for particular practical applications. Through numerical experiments, we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes.

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

Emilio Freire

SIAM J. Appl. Math. 65, pp. 1933-1951 (19 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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The birth of limit cycles in 3D (three-dimensional) piecewise linear systems for the relevant case of symmetrical oscillators is considered. A technique already used by the authors in planar systems is extended to cope with 3D systems, where a greater complexity is involved.
Under some given nondegeneracy conditions, the corresponding theorem characterizing the bifurcation is stated. In terms of the deviation from the critical value of the bifurcation parameter, expressions in the form of power series for the period, amplitude, and the characteristic multipliers of the bifurcating limit cycle are also obtained.
The results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3D electronic circuit genealogically related to the classical Van der Pol oscillator.

On a Mathematical Model of the Productivity Index of a Well from Reservoir Engineering

Akif Ibragimov, Dinara Khalmanova, Peter P. Valko, and Jay R. Walton

SIAM J. Appl. Math. 65, pp. 1952-1980 (29 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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Motivated by the reservoir engineering concept of the productivity index of a producing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann, or third boundary condition) is investigated using a known variational principle and confirmed numerically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and uniqueness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.

Webster's Horn Equation Revisited

Sjoerd W. Rienstra

SIAM J. Appl. Math. 65, pp. 1981-2004 (24 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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The problem of low-frequency sound propagation in slowly varying ducts is systematically analyzed as a perturbation problem of slow variation. Webster's horn equation and variants in bent ducts, in ducts with nonuniform soundspeed, and in ducts with irrotational mean flow, with and without lining, are derived, and the entrance/exit plane boundary layer is given. It is shown why a varying lined duct in general does not have an (acoustic) solution.

Revealing Pairwise Coupling in Linear-Nonlinear Networks

Duane Q. Nykamp

SIAM J. Appl. Math. 65, pp. 2005-2032 (28 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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Through an asymptotic analysis of a simple network, we derive an estimate of the coupling between a pair of units when all other units are unobservable. The analysis is based on a model where the response of each unit is a linear-nonlinear function of a white noise stimulus. The results accurately determine the coupling when all unmeasured units respond to the stimulus differently than the measured pair. To account for the possibility of unmeasured units similar to the measured pair, we cast our results in the framework of ``subpopulations,' which are defined as a group of units who respond to the stimulus similarly. We demonstrate that we can determine when correlations between two units are caused by a connection between their subpopulations, although the precise identity of the units involved in the connection may remain ambiguous. The result is rigorously valid only when the coupling is sufficiently weak to justify a second-order approximation in the coupling strength. We demonstrate through simulations that the results are still valid even with stronger coupling and in the presence of some deviations from the linear-nonlinear model. The analysis is presented in terms of neuronal networks, although the general framework is more widely applicable.

Mathematical Analysis of the Generalized Natural Modes of an Inhomogeneous Optical Fiber

E. M. Kartchevski, A. I. Nosich, and G. W. Hanson

SIAM J. Appl. Math. 65, pp. 2033-2048 (16 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber without a sharp boundary is formulated as a problem for the set of time-harmonic Maxwell equations with the Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues (including,as subsets, the well-known guided and leaky modes) of this problem are the complex propagation constants on a logarithmic Riemann surface. A theorem on spectrum localization is proved, and then the original problem is reduced to a nonlinear spectral problem with a compact integral operator. It is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Riemann surface, and it is also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Riemann surface.

Inverse Medium Scattering Problems for Electromagnetic Waves

Gang Bao and Peijun Li

SIAM J. Appl. Math. 65, pp. 2049-2066 (18 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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Consider a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in $\mathbb{R}^3$. In this paper, existence and uniqueness of the variational problem for forward scattering are established. An energy estimate for the scattered field with a uniform bound with respect to the wavenumber is obtained in the case of low frequency on which the Born approximation is based. A continuation method for the inverse medium scattering problem, which reconstructs the scatterer of an inhomogeneous medium from boundary measurements of the scattered wave, is developed. The algorithm requires multifrequency scattering data. Using an initial guess from the Born approximation, each update is obtained via recursive linearization on the wavenumber k by solving one forward problem and one adjoint problem of Maxwell's equations.

Stimulus-Locked Traveling Waves and Breathers in an Excitatory Neural Network

Stefanos E. Folias and Paul C. Bressloff

SIAM J. Appl. Math. 65, pp. 2067-2092 (26 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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We analyze the existence and stability of stimulus-locked traveling waves in a one-dimensional synaptically coupled excitatory neural network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights, and the output firing rate of a neuron is taken to be a Heaviside function of activity. Given an inhomogeneous moving input of amplitude I0 and velocity v, we derive conditions for the existence of stimulus-locked waves by working in the moving frame of the input. We use this to construct existence tongues in (v,I0)-parameter space whose tips at I0 = 0 correspond to the intrinsic waves of the homogeneous network. We then determine the linear stability of stimulus-locked waves within the tongues by constructing the associated Evans function and numerically calculating its zeros as a function of network parameters. We show that, as the input amplitude is reduced, a stimulus-locked wave within the tongue of an unstable intrinsic wave can undergo a Hopf bifurcation, leading to the emergence of either a traveling breather or a traveling pulse emitter.

Energy Maximizers, Negative Temperatures, and Robust Symmetry Breaking in Vortex Dynamics on a Nonrotating Sphere

Chjan C. Lim

SIAM J. Appl. Math. 65, pp. 2093-2106 (14 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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This paper relates the existence and uniqueness of constrained energy maximizers to the occurrence of negative temperatures in a recent statistical mechanics model of the energy-enstrophy theory. We construct examples of steady state solutions of the vorticity equation which break SO(3) symmetry from the negative temperature vorticity distributions in the spherical model. These vortex states correspond to solid-body rotation flows at rotation rates $\Theta $, which depend only on the fixed value of enstrophy $\Gamma$, that is, $\Theta =\sqrt{\Gamma /(4\int_{S^2}\cos^2\theta\, dx)}$. They are robust in the sense that they constitute most probable states in a spherical model of the statistical energy-enstrophy theory at negative temperatures, and have exponentially large Gibbs probability relative to any other macrostates. The existence and uniqueness of energy maximizers in a variational formulation of the new energy-enstrophy theory also give a necessary condition for the spherical model energy-enstrophy theory to be well defined at all temperatures.

Reconstruction of a Small Inclusion in a Two-Dimensional Open Waveguide

Habib Ammari, Ekaterina Iakovleva, and Hyeonbae Kang

SIAM J. Appl. Math. 65, pp. 2107-2127 (21 pages) | Cited 8 times

Online Publication Date: July 31, 2006

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We consider wave propagation in a perturbed open waveguide. We provide a new asymptotic expansion for the scattered wave when the inclusion is of small diameter. We design a MUSIC (multiple signal classification) type of algorithm for locating the inclusion and illustrate its viability in numerical examples.

Transport of Nutrients in Bones

Guillermo H. Goldsztein

SIAM J. Appl. Math. 65, pp. 2128-2140 (13 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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Lacunar-canalicular systems are networks of pores (lacunae) interconnected by thin channels (canaliculi) that are embedded in bones. The efficient transport of nutrients within lacunar-canalicular systems is necessary to keep bones healthy. Several theories have been proposed to identify the physical phenomena responsible for this efficient transport. In this paper, we develop and study a mathematical model motivated by one of those theories.

Field-Induced Motion of Nematic Disclinations

Paolo Biscari and Timothy J. Sluckin

SIAM J. Appl. Math. 65, pp. 2141-2157 (17 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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An individual defect in a nematic liquid crystal moves not only in response to its interaction with other defects but also in response to external fields. We analyze the motion of a wedge disclination in the presence of an applied field of strength H. We neglect backflow and seek steadily traveling patterns. The stationary picture yields a semi-infinite wall of strength $\pi$, bounded by the defect line. We find that the disclination advances into the region containing the wall at velocity v(H), where v scales as H/|log H| as long as the magnetic coherence length is greater than the core radius. When the external field is applied in the presence of a pair of disclinations, their dynamics is strongly influenced. We compute the expected relative velocity of the disclinations as a function of distance and field. The natural tendency for the disclinations to annihilate each other can be overcome by a sufficiently strong field suitably directed.

Slow Passage through Resonance for a Weakly Nonlinear Dispersive Wave

Sergei Glebov, Oleg Kiselev, and Vladimir Lazarev

SIAM J. Appl. Math. 65, pp. 2158-2177 (20 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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A solution of the nonlinear Klein--Gordon equation perturbed by a small external force is investigated. The frequency of the perturbation varies slowly and passes through a resonance. The resonance generates solitary packets of waves. The full asymptotic description of this process is presented.

Instability of the Ionospheric Plasma: Modeling and Analysis

Christophe Besse, Jean Claudel, Pierre Degond, Fabrice Deluzet, Gérard Gallice, and Christian Tessieras

SIAM J. Appl. Math. 65, pp. 2178-2198 (21 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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This paper is concerned with the theory and modeling of plasma instabilities in the ionosphere. We first consider the so-called striation model, which consists of balance equations for the density and momenta of the plasma species, coupled with an elliptic equation for the potential. The linearized instability of this model is analyzed in the framework of Fourier theory, both for smooth and discontinuous steady states. Then, we show that the dissipation mechanisms at work in the more refined "dynamo model" allow us to stabilize high wave-number perturbations. We also analyze turbulence as a possible source of additional dissipation (in a similar way as in fluid mechanics). To this aim, we use the statistical approach to turbulence and derive a so-called turbulent striation model, of which we analyze the stability properties. Numerical experiments are used to support our investigations.
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