Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons.

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multidimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows one to predict the stability and instability strength.

Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams.

We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.

Drift instability and tunneling of lattice solitons.

We derive an analytic formula for the lateral dynamics of solitons in a general inhomogeneous nonlinear media, and show that it can be valid over tens of diffraction lengths. In particular, we show that solitons centered at a lattice maximum can be "mathematically unstable" but "physically stable." We also derive an analytic upper bound for the critical velocity for tunneling, which is valid even when the standard Peierls-Nabarro potential approach fails.

Waves in nonlinear lattices: ultrashort optical pulses and Bose-Einstein condensates.

The nonlinear Schrödinger equation i (partial differential)(z)A(z,x,t)+(inverted Delta)(2)(x,t)A+[1+m(kappax)]|A|2A=0 models the propagation of ultrashort laser pulses in a planar waveguide for which the Kerr nonlinearity varies along the transverse coordinate x, and also the evolution of 2D Bose-Einstein condensates in which the scattering length varies in one dimension. Stability of bound states depends on the value of kappa=beamwidth/lattice period. Wide (kappa>>1) and kappa=O(1) bound states centered at a maximum of m(x) are unstable, as they violate the slope condition. Bound states centered at a minimum of m(x) violate the spectral condition, resulting in a drift instability. Thus, a nonlinear lattice can only stabilize narrow bound states centered at a maximum of m(x). Even in that case, the stability region is so small that these bound states are "mathematically stable" but "physically unstable."

Modeling and simulations of the pharmacokinetics of fluorophore conjugated antibodies in tumor vicinity for the optimization of fluorescence-based optical imaging.

BACKGROUND AND OBJECTIVES: One of the methods to detect and localize tumors in tissue is to use fluorophore conjugated specific antibodies as tumor surface markers. The goals of this study are to understand and quantify the pharmacokinetics of fluorophore conjugated antibodies in the vicinity of a tumor. This study concludes another stage of the development of a non-invasive fluorescenated antibody-based technique for imaging and localization of tumors in vivo. STUDY DESIGN/MATERIALS AND METHODS: A mathematical model of the pharmacokinetics of fluorophore conjugated antibodies in the vicinity of a tumor was developed based on histological staining experiments. We present the model equations of concentrations of antibodies and free binding sites. We also present a powerful simulation tool that we developed to simulate the imaging process. We analyzed the model and studied the effects of various independent parameters on the imaging result. These parameters included initial volume of markers (injected volume), total number of binding sites, tumor size, binding and dissociation rate constants, and the diffusion coefficient. We present the relations needed between these parameters in order to optimize the imaging results. RESULTS AND CONCLUSIONS: A powerful and accurate tool was developed which may assist in optimizing the imaging system results by setting the injection volume and concentration of fluorophore conjugated antibodies in tissue and approximating the time interval where maximum specific binding occurs and the tumor can be imaged.

Determination of a unique solution to parallel proton transfer reactions using the genetic algorithm.

Kinetic analysis of the dynamics as measured in multiequilibria systems is readily attained by curve-fitting methodologies, a treatment that can accurately retrace the shape of the measured signal. Still, these reconstructions are not related to the detailed mechanism of the process. In this study we subjected multiple proton transfer reactions to rigorous kinetic analysis, which consists of solving a set of coupled-nonlinear differential rate equations. The manual analysis of such systems can be biased by the operator; thus the analysis calls for impartial corroboration. What is more, there is no assurance that such a complex system has a unique solution. In this study, we used the Genetic Algorithm to investigate whether the solution of the system will converge into a single global minimum in the multidimensional parameter space. The experimental system consisted of proton transfer between four proton-binding sites with seven independent adjustable parameters. The results of the search indicate that the solution is unique and all adjustable parameters converge into a single minimum in the multidimensional parameter space, thus corroborating the accuracy of the manual analysis.

Multiple filamentation of circularly polarized beams.

We derive a new system of equations that describes the propagation of circularly polarized laser beams in a Kerr medium. Analysis and simulations of this system show that multiple filamentation is suppressed for circularly polarized beams.

Deterministic vectorial effects lead to multiple filamentation.

The standard explanation for multiple filamentation of laser pulses is that it is caused by noise in the input beam. We propose an alternative explanation that is based on deterministic vectorial (polarization) effects. We present numerical simulations in support of the vectorial-effects explanation and suggest a simple experiment for deciding whether multiple filamentation is due to vectorial effects.

Critical power for self-focusing in bulk media and in hollow waveguides.

We determine the threshold power for self-focusing collapse both in a bulk medium and in a hollow-core waveguide for various spatial profiles. We find that the threshold power for collapse in the waveguide is always equal to the lower-bound prediction for a bulk medium.

Self-focusing in the presence of small time dispersion and nonparaxiality.

We analyze the combined effect of small time dispersion and nonparaxiality on self-focusing and its ability to arrest the blowup of laser pulses by deriving reduced equations that depend on only the propagation distance and time. We calculate the pulse duration for which time dispersion dominates over nonparaxiality, or vice versa. We identify additional terms (shock term, group-velocity nonparaxiality, etc.)that should be retained when time dispersion and nonparaxiality are of comparable magnitude. These additional terms lead to temporal asymmetry, and in the visible spectrum they can dominate over both time dispersion and nonparaxiality.

Adiabatic law for self-focusing of optical beams.

An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schr odinger equation that is valid from the early stages of self-focusing until the blowup point. The adiabatic law leads to an analytical formula for the location of the blowup point and can be used to estimate the effects of various small perturbations on self-focusing. The results of the analysis are confirmed by numerical simulations.

Mathematical model of blood flow in a coronary capillary.

The coronary capillary flow is analyzed theoretically based on continuum mechanics. The capillary is a long, elastic, and permeable vessel loaded externally by tissue pressure, and it is subject to possible periodic length changes, together with adjacent myocytes. Capillary flow is driven by arteriolar-venular pressure difference. Ultrafiltration due to transmural hydrostatic and osmotic gradients is included, and consideration of mass conservation leads to a nonlinear flow equation. The results show that under physiological conditions ultrafiltration is of minor importance, and the analysis predicts regional differences in capillary flow. In regions with high tissue pressure (subendocardium), capillaries undergo significant periodic volume changes, giving rise to intramyocardial pumping. In those regions, capillary wall elasticity is of major importance. In regions with low tissue pressure (subepicardium), the possible periodic capillary length changes are predominant. The predicted flow patterns are in good qualitative agreement with measured epicardial phasic flow. In conclusion, the methodological advantage of a distributive analysis is demonstrated by its ability to elucidate and evaluate the role of flow determinants and their complex interactions.

Modeling of coronary capillary flow.

The coronary capillary flow is analyzed theoretically based on the laws of continuum mechanics. The capillary is considered as a long, elastic and permeable vessel loaded externally by tissue pressure. It is subjected to periodic length changes, together with adjacent myocytes. Capillary flow is driven by arteriolar-venular pressure differences. Ultrafiltration due to transmural hydrostatic and osmotic pressure gradients is included in the model. Consideration of mass conservation leads to a nonlinear flow equation. The results show that under stable physiological conditions ultrafiltration is of minor importance. The analysis of untethered capillaries predicts regional differences in capillary flow. In all regions, but more so in the subendocardium, capillaries undergo significant periodic volume changes, giving rise to intramyocardial pumping. In the deeper layers, capillary wall elasticity is of major importance. In the subepicardium, the possible capillary length-changes with adjacent myocytes tend to enhance systolic/diastolic volume differences. The predicted patterns of the overall capillary flow in the left ventricular (LV) wall are in good qualitative agreement with measured coronary phasic flow, showing systolic retrograde arterial inflow, accelerated venal outflow, and diastolic rapid filling accompanied by venal retrograde flow. Analysis of the flow in tethered capillary shows significant effect of the collagen attachments between the surrounding myocytes and the capillary wall. The advantage of the continuum analysis is demonstrated in the present study by its ability to elucidate and evaluate the role of flow controlling mechanisms and their complex interactions.