M. Gregory Forest

Spectral theory for the periodic sine‐Gordon equation: A concrete viewpoint

A summary of the spectral theory for quasiperiodic sine‐ and sinh‐Gordon equations is given. Analogies with whole‐line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine‐Gordon equation and physical characteristics of sine‐Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.

A Biophysical Basis for Mucus Solids Concentration as a Candidate Biomarker for Airways Disease

In human airways diseases, including cystic fibrosis (CF) and chronic obstructive pulmonary disease (COPD), host defense is compromised and airways inflammation and infection often result. Mucus clearance and trapping of inhaled pathogens constitute key elements of host defense. Clearance rates are governed by mucus viscous and elastic moduli at physiological driving frequencies, whereas transport of trapped pathogens in mucus layers is governed by diffusivity. There is a clear need for simple and effective clinical biomarkers of airways disease that correlate with these properties. We tested the hypothesis that mucus solids concentration, indexed as weight percent solids (wt%), is such a biomarker. Passive microbead rheology was employed to determine both diffusive and viscoelastic properties of mucus harvested from human bronchial epithelial (HBE) cultures. Guided by sputum from healthy (1.5–2.5 wt%) and diseased (COPD, CF; 5 wt%) subjects, mucus samples were generated in vitro to mimic in vivo physiology, including intermediate range wt% to represent disease progression. Analyses of microbead datasets showed mucus diffusive properties and viscoelastic moduli scale robustly with wt%. Importantly, prominent changes in both biophysical properties arose at ∼4 wt%, consistent with a gel transition (from a more viscous-dominated solution to a more elastic-dominated gel). These findings have significant implications for: (1) penetration of cilia into the mucus layer and effectiveness of mucus transport; and (2) diffusion vs. immobilization of micro-scale particles relevant to mucus barrier properties. These data provide compelling evidence for mucus solids concentration as a baseline clinical biomarker of mucus barrier and clearance functions.

Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation

We describe the integrable structure of solutions of the nonlinear Schrodinger (NLS) equation under periodic and quasiperiodic boundary conditions. We focus on those aspects of the exact theory which reveal the behavior of these solutions under perturbations of initial conditions (i.e. linearized instabilities), and the effects of slow modulations in space and time, perhaps in the presence of external perturbations. These results and methods continue the investigations of Ercolani, Flaschka, Forest and McLaughlin [1–7] on Korteweg-deVries (KdV), sine-Gordon (sG) and sinh-Gordon wavetrains. Our purpose here is to document the corresponding features of NLS solutions; the rigorous analysis that underlies this paper derives from [1–7] and will appear in the thesis of Lee [8].

On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system

Abstract The Backlund-gauge transformation for a system of coupled NLS (nonlinear Schrodinger) equations with a degenerate associated spectral operator is derived from an algebraic perspective, extending aspects of other results [M. Boiti, Tu. Guizhang, Il Nuovo Cimento 71B (1982) 253–264; D.H. Sattinger, V.D. Zurkowski, Physica D 26 (1–3) (1987) 225–250] that apply in the context of non-degenerate spectral operators. Moreover, we demonstrate how the Backlund-gauge transformation can be used to explicitly construct the entire unstable manifold (via superpositions of homoclinic orbits) of a plane wave solution with both self-phase instabilities and coupling instabilities. This work builds on the results of Ercolani et al. [N. Ercolani, M.G. Forest, D.W. McLaughlin, Physica D 18 (1986) 472–474; N. Ercolani, M.G. Forest, D.W. McLaughlin, Physica D 43 (2–3) (1990) 349–384] for the sine-Gordon equation, and Forest et al. [M.G. Forest, D.W. McLaughlin, D.J. Muraki, O.C. Wright, J. Nonlinear Sci., in press; M.G. Forest, S.P. Sheu, O.C. Wright, Phys. Lett. A, in press] for the integrable coupled NLS system.

Centromere tethering confines chromosome domains.

The organization of chromosomes into territories plays an important role in a wide range of cellular processes, including gene expression, transcription, and DNA repair. Current understanding has largely excluded the spatiotemporal dynamic fluctuations of the chromatin polymer. We combine in vivo chromatin motion analysis with mathematical modeling to elucidate the physical properties that underlie the formation and fluctuations of territories. Chromosome motion varies in predicted ways along the length of the chromosome, dependent on tethering at the centromere. Detachment of a tether upon inactivation of the centromere results in increased spatial mobility. A confined bead-spring chain tethered at both ends provides a mechanism to generate observed variations in local mobility as a function of distance from the tether. These predictions are realized in experimentally determined higher effective spring constants closer to the centromere. The dynamic fluctuations and territorial organization of chromosomes are, in part, dictated by tethering at the centromere.

Pericentric chromatin loops function as a nonlinear spring in mitotic force balance

During mitosis, cohesin- and condensin-based pericentric chromatin loops function as a spring network to balance spindle microtubule force.

Correlations between chaos in a perturbed sine-Gordon equation and a truncated model system

The purpose of this paper is to present a first step toward providing coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde. This is achieved by correlating: (i) Numerical results on the bifurcations to temporal chaos with spatial coherence of the damped, periodically forced sine-Gordon equation with periodic boundary conditions; (ii) An interpretation of the spatial and temporal bifurcation structures of this perturbed integrable system with regard to the exact structure of the sine-Gordon phase space; (iii) A model dynamical systems problem, which is itself a perturbed integrable Hamiltonian system, derived from the perturbed sine-Gordon equation by a finite mode Fourier truncation in the nonlinear Schrodinger limit; and (iv) The bifurcations to chaos in the truncated phase space.In particular, a potential source of chaos in both the pde and ...

The geometry of real sine-Gordon wavetrains

The characterization ofreal, N phase, quasiperiodic solutions of the sine-Gordon equation has been an open problem. In this paper we achieve this result, employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature. A significant by-product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds, and an understanding of how the real isospectral manifolds are embedded. By placing the problem in this general context, these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions.

Connections Between Stability, Convexity of Internal Energy, and the Second Law for Compressible Newtonian Fluids

In this note we provide proofs of the following statements for a compressible Newtonian fluid: (i) internal energy being a convex function of entropy and specific volume is equivalent to nonnegativity of both specific heat at constant volume and isothermal bulk modulus; (ii) convexity of internal energy together with the second law of thermodynamics imply linear stability of the rest state; and (iii) linear stability of the rest state together with the second law imply convexity of internal energy.

A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation

We consider equilibrium solutions of the Smoluchowski equation for rodlike nematic polymers with a Maier–Saupe excluded volume potential. The purpose of this paper is to present a new and simplified proof of classical well-known results: (1) all equilibria are axisymmetric and (2) modulo rotational symmetry, the number and type of axisymmetric equilibria are characterized with respect to the strength of the excluded volume potential. These results confirm the phase diagram of equilibria obtained previously by numerical simulations (Faraoni et al 1999 J. Rheol. 43 829–43, Forest et al 2004 Rheol. Acta 43 17–37, Larson and Ottinger 1991 Macromolecules 24 6270–82).