Three-dimensional solitons in cross-combined linear and nonlinear optical lattices

The existence and stability of three-dimensional (3D) solitons, in cross-combined linear and nonlinear optical lattices, are investigated. In particular, with a starting optical lattice (OL) configuration such that it is linear in the x-direction and nonlinear in the y-direction, we consider the z-direction either unconstrained (quasi-2D OL case) or with another linear OL (full 3D case). We perform this study both analytically and numerically: analytically by a variational approach based on a Gaussian ansatz for the soliton wavefunction and numerically by relaxation methods and direct integrations of the corresponding Gross-Pitaevskii equation. We conclude that, while 3D solitons in the quasi-2D OL case are always unstable, the addition of another linear OL in the z-direction allows us to stabilize 3D solitons both for attractive and repulsive mean interactions. From our results, we suggest the possible use of spatial modulations of the nonlinearity in one of the directions as a tool for the management of stable 3D solitons.

Tricritical-like behavior of the nonlinear optical refraction at the nematic-isotropic transition in the E7 thermotropic liquid crystal

We use Z-scan technique to investigate the nonlinear optical response of the thermotropic liquid crystal E7 in the neighborhood of the nematic-isotropic phase transition. The analysis of the data for the nonlinear optical birefringence is compatible with an effective critical exponent of the order parameter, beta = 0.28 +/- 0.03, which is close to the classical value, beta = 0.25, for a tricritical point. The nonlinear optical absorption in the nematic range depends on the geometrical configuration of the nematic director with respect to the polarization beam, and vanishes in the isotropic phase.

Remarks on Parametric Surface Waves in a Nonlinear and Non-Ideally Excited Tank

We studied free surface oscillations of a fluid in a cylinder tank excited by an electric motor with limited power supply. We investigated the possibility of parametric resonance in this system, showing that the excitation mechanism can generate chaotic response. Numerical experiments are carried out to present the existence of several types of regular and chaotic attractors. For the first time powers (power of the motor, power consumed by the damping force under fluid free surface oscillations, and a total power) are calculated, investigated, and shown for different regimes, regular and chaotic ones for parametric resonance interactions. [DOI: 10.1115/1.4005844]

SUFFICIENT CONDITIONS ON DIFFUSIVITY FOR THE EXISTENCE AND NONEXISTENCE OF STABLE EQUILIBRIA WITH NONLINEAR FLUX ON THE BOUNDARY

A reaction-diffusion equation with variable diffusivity and non-linear flux boundary condition is considered. The goal is to give sufficient conditions on the diffusivity function for nonexistence and also for existence of nonconstant stable stationary solutions. Applications are given for the main result of nonexistence.

Analysis of nonlinear dynamics of vocal folds using high-speed video observation and biomechanical modeling

Primary voice production occurs in the larynx through vibrational movements carried out by vocal folds. However, many problems can affect this complex system resulting in voice disorders. In this context, time-frequency-shape analysis based on embedding phase space plots and nonlinear dynamics methods have been used to evaluate the vocal fold dynamics during phonation. For this purpose, the present work used high-speed video to record the vocal fold movements of three subjects and extract the glottal area time series using an image segmentation algorithm. This signal is used for an optimization method which combines genetic algorithms and a quasi-Newton method to optimize the parameters of a biomechanical model of vocal folds based on lumped elements (masses, springs and dampers). After optimization, this model is capable of simulating the dynamics of recorded vocal folds and their glottal pulse. Bifurcation diagrams and phase space analysis were used to evaluate the behavior of this deterministic system in different circumstances. The results showed that this methodology can be used to extract some physiological parameters of vocal folds and reproduce some complex behaviors of these structures contributing to the scientific and clinical evaluation of voice production. (C) 2010 Elsevier Inc. All rights reserved.

A transmission problem for beams on nonlinear supports

A transmission problem involving two Euler-Bernoulli equations modeling the vibrations of a composite beam is studied. Assuming that the beam is clamped at one extremity, and resting on an elastic bearing at the other extremity, the existence of a unique global solution and decay rates of the energy are obtained by adding just one damping device at the end containing the bearing mechanism.

Sensitivity, specificity and predictive values of linear and nonlinear indices of heart rate variability in stable angina patients

Abstract Background Decreased heart rate variability (HRV) is related to higher morbidity and mortality. In this study we evaluated the linear and nonlinear indices of the HRV in stable angina patients submitted to coronary angiography. Methods We studied 77 unselected patients for elective coronary angiography, which were divided into two groups: coronary artery disease (CAD) and non-CAD groups. For analysis of HRV indices, HRV was recorded beat by beat with the volunteers in the supine position for 40 minutes. We analyzed the linear indices in the time (SDNN [standard deviation of normal to normal], NN50 [total number of adjacent RR intervals with a difference of duration greater than 50ms] and RMSSD [root-mean square of differences]) and frequency domains ultra-low frequency (ULF) ≤ 0,003 Hz, very low frequency (VLF) 0,003 – 0,04 Hz, low frequency (LF) (0.04–0.15 Hz), and high frequency (HF) (0.15–0.40 Hz) as well as the ratio between LF and HF components (LF/HF). In relation to the nonlinear indices we evaluated SD1, SD2, SD1/SD2, approximate entropy (−ApEn), α1, α2, Lyapunov Exponent, Hurst Exponent, autocorrelation and dimension correlation. The definition of the cutoff point of the variables for predictive tests was obtained by the Receiver Operating Characteristic curve (ROC). The area under the ROC curve was calculated by the extended trapezoidal rule, assuming as relevant areas under the curve ≥ 0.650. Results Coronary arterial disease patients presented reduced values of SDNN, RMSSD, NN50, HF, SD1, SD2 and -ApEn. HF ≤ 66 ms2, RMSSD ≤ 23.9 ms, ApEn ≤−0.296 and NN50 ≤ 16 presented the best discriminatory power for the presence of significant coronary obstruction. Conclusion We suggest the use of Heart Rate Variability Analysis in linear and nonlinear domains, for prognostic purposes in patients with stable angina pectoris, in view of their overall impairment.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Numerical combination for nonlinear analysis of structures coupled to layered soils

This paper presents an alternative coupling strategy between the Boundary Element Method (BEM) and the Finite Element Method (FEM) in order to create a computational code for the analysis of geometrical nonlinear 2D frames coupled to layered soils. The soil is modeled via BEM, considering multiple inclusions and internal load lines, through an alternative formulation to eliminate traction variables on subregions interfaces. A total Lagrangean formulation based on positions is adopted for the consideration of the geometric nonlinear behavior of frame structures with exact kinematics. The numerical coupling is performed by an algebraic strategy that extracts and condenses the equivalent soil's stiffness matrix and contact forces to be introduced into the frame structures hessian matrix and internal force vector, respectively. The formulation covers the analysis of shallow foundation structures and piles in any direction. Furthermore, the piles can pass through different layers. Numerical examples are shown in order to illustrate and confirm the accuracy and applicability of the proposed technique.

Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of saddle-node equilibrium points

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddle-node equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.

Influence of nanoparticle size on the nonlinear optical properties of magnetite ferrofluids

The nonlinear index of refraction (n(2)) and the two-photon absorption coefficient (beta) of water-based ferrofluids made of magnetite nanocrystals of different sizes and with different coatings have been measured through the Z-scan technique, with ultrashort (femtoseconds) laser pulses. Their third-order susceptibility is calculated from the values of n(2) and beta. The influence of different particles' coatings and sizes on these nonlinear optical properties are investigated. The values of n(2) and beta depend more significantly on the nanoparticles' size than on the particular coating. We observe a decrease of beta as the nanoparticles' diameters decrease, although the optical gap is found to be the same for all samples. The results are interpreted considering modifications in the electronic orbital shape due to the particles' nanosize effect.

Solitons in nonlinear Schrödinger equations.

Using a mathematical approach accessible to graduate students of physics and engineering, we show how solitons are solutions of nonlinear Schrödinger equations. Are also given references about the history of solitons in general, their fundamental properties and how they have found applications in optics and fiber-optic communications.

Nonlinear optical characterizations of dibenzoylmethane in solution

This work reports on the two-photon absorption (2PA) cross-section and first hyperpolarizability of dibenzoylmethane solutions using femtosecond Z-scan and hyper-Rayleigh scattering techniques. The 2PA spectrum, spanning the wavelength range from 460 to 740 nm, presents a band centered at 510 nm, with a cross-section value estimated as 37 GM at this wavelength. Owing to the molecular symmetry, this band is not observed in the linear absorption spectrum. The sum-over-state approach was adopted to evaluate various spectroscopic parameters. Experimental and theoretical values of the first hyperpolarizability values were estimated in ethanol and DMSO solutions.

On some Banach space constants arising in nonlinear fixed point and eigenvalue theory

[EN] As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps.