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1531-3492

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2007 , Volume 7 , Issue 3

A special volume in honor of Fred Wan

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2007, 7(3): i-ii
doi: 10.3934/dcdsb.2007.7.3i

*+*[Abstract](509)*+*[PDF](1623.2KB)**Abstract:**

This special issue of the Discrete and Continuous Dynamical Systems is dedicated to Professor Frederic Yui-Ming Wan on the occasion of his seventieth birthday. For this special occasion, a conference on Biology and Mechanics: Applications of Mathematics and Computations was held on May 25 and 26, 2006 at the Beckman Center for National Academies at Irvine. This volume is a compilation of contributions from friends, colleagues, and collaborators of Fred Wan. All submissions were peer-reviewed under the normal procedure of DCDS-B.

Fred Wan has made influential and pioneering contributions in many areas of applied mathematics in his long career, such as shell and elastic theory, asymptotic analysis, and modeling and analysis of biological systems. The topics covered in this volume are mostly related to Fred's research on applications of mathematics and computations in mechanics and biology.

For the full preface, please click the Full Text "PDF" button above.

2007, 7(3): 465-495
doi: 10.3934/dcdsb.2007.7.465

*+*[Abstract](456)*+*[PDF](344.5KB)**Abstract:**

This paper is concerned with the Korteweg-de Vries equation which models unidirectional propagation of small amplitude long waves in dispersive media. The two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of length $L$ of the media of propagation is considered. It is shown that the solution of the two-point boundary value problem converges as $L\rightarrow +\infty$ to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. In addition to its intrinsic interest, our result provides justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem for the KdV equation.

2007, 7(3): 497-514
doi: 10.3934/dcdsb.2007.7.497

*+*[Abstract](385)*+*[PDF](498.9KB)**Abstract:**

In this paper, we formulate the problem of phase transitions in a slender elastic cylinder induced by tension/extension as a boundary-value problem of a first-order dynamical system. One aim is to give analytical descriptions for some geometrical size effects observed in experiments. Three types of end boundary conditions corresponding to real physical situations are proposed. With the help of a phase-plane analysis analytical solutions for both a force-controlled problem and a displacement-controlled problem are obtained. It turns out that the value of the radius-length ratio has a great influence on the solutions. For a displacement-controlled problem it influences the number of all possible solutions. The engineering stress-strain curves plotted from the analytical solutions seem to capture the key features (e.g., stress peak, stress drop and stress plateau) of the curves measured in a few experiments in literature. Also, the analytical results reveal that smaller the radius is sharper the stress drop is and the width of the transformation front is of the order of the radius, which are in agreement with the experimental observations. We also compare the analytic solutions for the three types of boundary conditions, and a very interesting finding is that the engineering stress-strain curves are almost identical under these different boundary conditions.

2007, 7(3): 515-525
doi: 10.3934/dcdsb.2007.7.515

*+*[Abstract](611)*+*[PDF](164.8KB)**Abstract:**

In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their accuracy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time steps. IIF2 is found to be more accurate for systems with large diffusion while CN-MG is more efficient when the number of spatial grid points is large.

2007, 7(3): 527-541
doi: 10.3934/dcdsb.2007.7.527

*+*[Abstract](379)*+*[PDF](412.7KB)**Abstract:**

Capacitance/Voltage characteristics of the MOSFET gate region at small oxide thickness show substantial variation from the oxide capacitance. We provide a numerical and analytic approach to the standard drift-diffusion equations and show how this can be used to determine analytic formulae that can be used in simulation programs such as SPICE.

2007, 7(3): 543-551
doi: 10.3934/dcdsb.2007.7.543

*+*[Abstract](613)*+*[PDF](370.8KB)**Abstract:**

Climate models solve the equations for the conservation of momentum, mass and energy in the atmosphere and oceans, the equations of state for air and for sea water, as well as equations for energy and water exchange with the land and cryosphere. This paper reviews the basis of climate models, and illustrates the application of a new generation of climate models to predict the co-evolution of atmospheric $CO_2$ and climate.

2007, 7(3): 553-572
doi: 10.3934/dcdsb.2007.7.553

*+*[Abstract](447)*+*[PDF](365.5KB)**Abstract:**

We consider the stretching of a thin cylindrical thread with viscosity that depends on temperature. The thread is pulled with a prescribed force while receiving continuous heating from an external axially nonuniform heater. We use the canonical equations derived by Huang

*et al.*(2007) and consider the limit of large dimensionless heating rate. We show that the asymptotic solution depends only on the local properties of the heating near its maximal heating value. We derive a uniformly valid asymptotic solution for the shape and the temperature profiles during the stretching process. We use a criterion to determine when breaking will occur and derive simple analytical expressions for the shape at breaking that clearly show the influence of heating strength and the degree of localization of the heating. The asymptotic shape profiles give good agreement with numerical simulations. These results are applied to the formation of glass microelectrodes.

2007, 7(3): 573-580
doi: 10.3934/dcdsb.2007.7.573

*+*[Abstract](500)*+*[PDF](137.8KB)**Abstract:**

This paper describes some recent theoretical results pertaining to the experimentally-observed relation between the speed of a shock wave in a solid and the particle velocity immediately behind the shock. The new feature in the present analysis is the assumption that compressive strains are limited by a materially-determined critical value, and that the internal energy density characterizing the material is unbounded as this critical strain is approached. It is shown that, with this assumption in force, the theoretical relation between shock speed and particle velocity is consistent with many experimental observations in the sense that it is asymptotically linear for strong shocks of the kind often arising in the laboratory.

2007, 7(3): 581-604
doi: 10.3934/dcdsb.2007.7.581

*+*[Abstract](590)*+*[PDF](1559.3KB)**Abstract:**

We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. The 3-d method relies on accurate discretizations of singular surface integrals, a spatial rescaling and the use of an adaptive surface mesh. The discretized boundary integral equations are solved iteratively using GMRES and a discretized version of the Dirichlet-Neumann map, formulated in terms of a vector potential, is used to determine the normal velocity of the tumor surface. Explicit time stepping is used to update the tumor surface. We present simulations of the nonlinear evolution of growing tumors. At early times, good agreement is obtained between the results of a linear stability analysis and nonlinear simulations. At later times, linear theory is found to overpredict the growth of perturbations. Nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards. The morphologic instability allows the tumor to increase its surface area, relative to its volume, thereby allowing the cells in the tumor bulk greater access to nutrient. This in turn allows the tumor to overcome the diffusional limitations on growth and to grow to larger sizes than would be possible if the tumor were spherical. Consequently, instability provides a means for avascular tumor invasion.

2007, 7(3): 605-618
doi: 10.3934/dcdsb.2007.7.605

*+*[Abstract](476)*+*[PDF](473.2KB)**Abstract:**

One important problem in human brain mapping research is to locate the important anatomical features. Anatomical features on the cortical surface are usually represented by landmark curves, called sulci/gyri curves. These landmark curves are important information for neuroscientists to study brain disease and to match different cortical surfaces. Manual labelling of these landmark curves is time-consuming, especially when large sets of data have to be analyzed. In this paper, we present algorithms to automatically detect and match landmark curves on cortical surfaces to get an optimized brain conformal parametrization. First, we propose an algorithm to obtain a hypothesized landmark region/curves using the Chan-Vese segmentation method, which solves a Partial Differential Equation (PDE) on a manifold with global conformal parameterization. This is done by segmentating the high mean curvature region. Second, we propose an automatic landmark curve tracing method based on the principal directions of the local Weingarten matrix. Based on the global conformal parametrization of a cortical surface, our method adjusts the landmark curves iteratively on the spherical or rectangular parameter domain of the cortical surface along its principal direction field, using umbilic points of the surface as anchors. The landmark curves can then be mapped back onto the cortical surface. Experimental results show that the landmark curves detected by our algorithm closely resemble these manually labeled curves. Next, we applied these automatically labeled landmark curves to generate an optimized conformal parametrization of the cortical surface, in the sense that homologous features across subjects are caused to lie at the same parameter locations in a conformal grid. Experimental results show that our method can effectively help in automatically matching cortical surfaces across subjects.

2007, 7(3): 619-628
doi: 10.3934/dcdsb.2007.7.619

*+*[Abstract](499)*+*[PDF](327.6KB)**Abstract:**

Experimental work on the basic problem of resonant acoustic oscillations in a closed straight cylindrical tube goes back at least to Lettau [14]. He showed that, even for "small" piston velocities, shock waves traverse the tube. Shocks are a nonlinear phenomenon and a means of converting mechanical energy to heat. Betchov [1], followed by Chu and Ying [4]}, Gorkov [7] and Chester [2], gave the first satisfactory theoretical explanation of the phenomena. The interest at this time was in an understanding of noise excitation in jets and reciprocating engines. A completely new phenomenon emerged with the experiments of Lawrenson et al [13]. They showed that very high shockless pressures can be generated by resonant acoustic oscillations in specially shaped containers. They called this Resonant Macrosonic Synthesis (RMS) and indicated important technological applications. The first analytical results explaining RMS were given by Mortell & Seymour [18], showing good qualitative agreement with both experimental and numerical results. The challenge was to understand the interaction of the geometry with the nonlinearity. It was shown that when the geometry yields incommensurate eigenvalues, i.e. the higher modes are not integer multiples of the fundamental, the resulting motion is shockless. With no shocks, higher pressures resulted for the same energy input. Here we review the 'classical' resonance in a straight tube, and then show that shockless motions can be produced even in a straight tube by introducing a variable ambient density distribution.

2007, 7(3): 629-641
doi: 10.3934/dcdsb.2007.7.629

*+*[Abstract](463)*+*[PDF](700.4KB)**Abstract:**

The Korteweg-de Vries (KdV) equation has long been known to describe shallow water waves in an appropriate asymptotic limit. The Camassa-Holm (CH) equation, shown in 1993 [1] also to be completely integrable, allows solitons including

*peakons*with discontinuous derivative. Consequently, its relevance to shallow water theory has been much doubted until the link was established [4] in 2004. Here, the perturbation procedure in terms of the standard amplitude ($\varepsilon$) and shallow depth ($\delta$) parameters shows that, while the KdV equation applies for $\delta^4 $«$ \varepsilon $«$ \delta$ with error term the larger of $\varepsilon^3$ and $\delta^6$,. However, the formal link to the CH equation imposes additional constraints on the coordinate ranges which are applicable. The derivation procedure is also extended to account for depth variations with bed slope $\mathcal{O}(\gamma\delta)$, provided that $\gamma $«$ \varepsilon$ and $\gamma $«$ \delta^2$ for $\delta^3 $«$ \varepsilon $«$ \delta$ and, within these parameter ranges, a theory for modulated waveforms is outlined. This utilizes expressions (in terms of elliptic functions) for the general travelling wave solutions to a (non-integrable) generalization of the CH equation. These include, as limiting cases, solitary waves with adjustable, but constrained, peak curvature (of $\mathcal{O}(\varepsilon \delta^2)$). The nonlinear dispersion relation for the periodic waveforms is illustrated and sample periodic waveforms are illustrated and compared to equivalent KdV approximations.

2007, 7(3): 643-650
doi: 10.3934/dcdsb.2007.7.643

*+*[Abstract](345)*+*[PDF](152.6KB)**Abstract:**

The analog of the well-known Prager-Synge hypercircle theorem for three-dimensional, linearly elastic bodies is derived for the linear Sanders-Koiter theory of elastically anisotropic shells under surface and edge loads and then used to compute the accuracy of Novozhilov's simplified equations for an elastically isotropic general cylindrical shell. The key idea is to note that the Novozhilov equations are equivalent to the Sanders-Koiter equations with certain surface loads and distributed dislocations.

2007, 7(3): 651-660
doi: 10.3934/dcdsb.2007.7.651

*+*[Abstract](468)*+*[PDF](134.3KB)**Abstract:**

We consider a simple climate model of global warming to help understand and constrain predictions from the more comprehensive General Circulation Models (GCMs). By using observations to constrain the climate gain factor, which presents the greatest uncertainty in GCMs, we discuss the atmosphere's response to a doubling of carbon dioxide concentration in the atmosphere in both equilibrium and time-dependent states.

2007, 7(3): 661-682
doi: 10.3934/dcdsb.2007.7.661

*+*[Abstract](440)*+*[PDF](286.5KB)**Abstract:**

In this paper, we study the asymptotic behavior of the Hermite polynomials $H_{n}((2n+1)^{1/2}z)$ as $n\rightarrow \infty$. A globally uniform asymptotic expansion is obtained for $z$ in an unbounded region containing the right half-plane Re $z \geq 0$. A corresponding expansion can also be given for $z$ in the left half-plane by using the symmetry property of the Hermite polynomials. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou.

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