Hyperbolic Secant Function Research Articles

Elasto-plastic constitutive model of soil-structure interface in consideration of strain softening and dilation

The behavior of soil-structure interface plays a major role in the definition of soil-structure interaction. In this paper a bi-potential surface elasto-plastic model for soil-structure interface is proposed in order to describe the interface deformation behavior, including strain softening and normal dilatancy. The model is formulated in the framework of generalized potential theory, in which the soil-structure interface problem is regard as a two-dimensional mathematical problem in stress field, and plastic state equations are used to replace the traditional field surface. The relation curves of shear stress and tangential strain are fitted by a piecewise function composed by hyperbolic functions and hyperbolic secant functions, while the relation curves of normal strain and tangential strain are fitted by another piecewise function composed by quadratic functions and hyperbolic secant functions. The approach proposed has the advantage of deriving an elastoplastic constitutive matrix without postulating the plastic potential functions and yield surface. Moreover, the mathematical principle is clear, and the entire model parameters can be identified by experimental tests. Finally, the predictions of the model have been compared with experimental results obtained from simple shear tests under normal stresses, and results show the model is reasonable and practical.

Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L 2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in ℝ2 which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ℝ2n with the property that the short-time Fourier transform defines an isometric embedding from L 2(ℝ n ) to L μ 2 (ℝ2n ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.

Diode-pumped passively Q-switched and modelocked Nd:GdVO_4 laser at 134 μm with V:YAG saturable absorber

Using V:YAG as the saturable absorber, a diode-pumped passively Q-switched and mode-locked Nd:GdVO(4) laser at 1.34 microm is realized. Nearly 100% modulation depth of mode-locking has been achieved. The width of the mode-locked pulse is estimated to be less than 460 ps with 125 MHz repetition rate within an about 1 micros-long Q-switched pulse envelope. A maximum output power of 220 mW and Q-switched pulse energy of 10.5 microJ is obtained. Using the hyperbolic secant function methods, a fluctuation rate equation model considering the Gaussian distribution of the intracavity photon density and the population inversion in the gain medium as well as the ground-state population intensity of the saturable absorber has been proposed to describe the mode-locking process of diode-pumped Nd:GdVO(4)/V(3+):YAG laser. With the space-dependent rate equations solved numerically, the theoretical calculations reproduce the laser characteristics well.

Exact solitary-wave solutions and periodic wave solutions for generalized modified Boussinesq equation and the effect of wave velocity on wave shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity’s variation.

Pulse dynamics and solitons in mode-locked lasers

A model system is presented that describes the physical properties of mode-locked lasers. This distributed system incorporates gain and filtering saturated with energy while loss is saturated with power. It is found that general initial pulses evolve to stable localized solutions which exist for wide choices of the parameters, the only requirement being sufficient gain. Moreover, these pulses are essentially solitons of the classical nonlinear Schr\"odinger (NLS) equation. In the anomalous regime, the additional terms present in the system serve to provide the mode locking mechanism. Consequently, these pulses are approximated by the classical NLS soliton, given by hyperbolic secant functions, in agreement with recent experiments.

The fundamental plane of dwarf irregular galaxies

Aims. Near-IR photometry of dwarf irregular galaxies (dIs) has been acquired to expand the sample of objects suitable for investigating the fundamental plane (FP) discovered previously by us. Data for blue compact dwarfs (BCDs), dwarf ellipticals (dEs), and dwarf spheroidal galaxies (dSphs) are amalgamated to evaluate how closely these classes of galaxies are related to dIs. Methods. Surface brightness profiles for dIs are modeled using a hyperbolic secant function (sech), which has been shown to sample the old galaxy component observed in the NIR (Ks). Also, profiles for BCDs are modeled using a sech function, but to this is added a Gaussian burst. Kinematics for dIs and BCDs are constrained by the width W20 of integrated HI line profiles, and motions within dEs and dSphs are judged from stellar velocity dispersions. Results. The existence of a FP for dIs is confirmed. Omitting extreme deviants, the scatter of dIs about the plane is 0.43 mag, somewhat higher than expected from observational errors alone. Corrections of line widths for tilt do not reduce the residuals, so motions must be predominantly random. Corrections of surface brightnesses for tilt have no effect either. BCDs and dEs lie precisely on the FP defined by dIs, suggesting strong underlying connections among dIs, BCDs, and dEs.

Modified Tartagli Method for Calculation of Jominy Hardenability Curve

Basing on the experimental results of the hardenability investigations, which employed Jominy method, the model of the neural networks was developed and fully verified experimentally. The model makes it possible to obtain Jominy hardenability curves basing on the steel chemical composition. The modified hardenability curves calculation method is presented in the paper, initially developed by Tartaglia, Eldis, and Geissler, later extended by T. Inoue. The method makes use of the similarity of the Jominy curve to the hyperbolic secant function. The empirical formulae proposed by the authors make calculation of the hardenability curve possible basing on the chemical composition of the steel. However, regression coefficients characteristic for the particular steel grade must be known. Replacing some formulae by the neural network models is proposed in the paper.

New measures of sharpness for symmetric powder diffraction peak profiles

New measures of sharpness for symmetric powder diffraction peak profiles are proposed. The sharpness parameter is defined through the \nuth-order moment of the Fourier transform of the profile function. Analytical expressions for the sharpness parameter for empirical model profile functions, namely the Gaussian, logistic distribution, hyperbolic secant, Lorentzian, Voigt, Pearson VII and pseudo-Voigt functions, and theoretical size-broadening profiles with statistical size distribution are presented. Theoretical diffraction profiles with complicated formulae can be approximated by empirical model functions assuming equivalent values of the sharpness parameter. The concept of the sharpness parameter provides a simple way to define an approximation for a theoretical diffraction peak profile with empirical model functions.

The propagation properties of symmetrical hyperbolic secant beams in weakly nonlocal nonlinear bulk media

The propagation of (2+1)D paraxial symmetrical hyperbolic secant beams in weakly nonlocal nonlinear media is studied by variational approach. A series of differential equations which describe the evolutions of the width, the phase, the phase front curvature, and the amplitude of the beam are obtained. The critical power for a beam propagating as a spatial optical soliton in weakly nonlocal nonlinear media is also obtained. A quantitative depiction of the steadying effect of nonlocality on the propagation of a beam is obtained through the steadiness analyse, which provides a self-consistent explanation of the transition from unsteady local kerr solitons to steady weakly nonlocal solitons. The results of numerical method agree with that of variational calculations, implying that hyperbolic secant function is a good approximation to a (2+1)D weakly nonlocal spatial optical soliton.

Actively $Q$-switched and Mode-Locked Diode-Pumped ${\hbox {Nd:GdVO}}_{4}{\hbox {--KTP}}$ Laser

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> By using a comparative simple configuration and a short cavity length, a diode-pumped actively <formula formulatype="inline"><tex>$Q$</tex></formula>-switched and mode-locked intracavity frequency doubled <formula formulatype="inline"> <tex>${\hbox {Nd:GdVO}}_{4}{\hbox {--KTP}}$</tex></formula> green laser with the modulation depth 100% was realized, from which the great average output power, the high efficiency were obtained and the mode-locked pulse inside the <formula formulatype="inline"><tex>$Q$</tex></formula>-switched pulse has a repetition rate of 476 MHz. Using the hyperbolic secant function methods and by considering the Gaussian distribution of the intracavity photon density, the influences of continuous pump rate, the upper state lifetime of the active medium and the turnoff time of the acousto-optic <formula formulatype="inline"> <tex>$Q$</tex></formula>-switch, we proposed a developed rate equation model for actively <formula formulatype="inline"><tex>$Q$</tex></formula>-switched and mode-locked green lasers. With this developed model, the theoretical calculations are in good agreement with the experimental results and the width of the mode-locked green pulse is estimated to be about 185 ps. </para>

Modeling a nanosecond quasi-Fourier-transform limited Ti:Sa laser source

We recently setup a high energy, tunable, injection seeded Ti:Sapphire laser source whose implementation was reported in [Rev. Sci. Instrum. 78, 033102 (2007)]. The analysis of the spectral linewidth has demonstrated that the pulses were time Fourier-transform limited. The present paper reports on the modeling of this source with the aim of understanding the energy conversion in the injected as well as in the free run regimes. The experimental lasing threshold, output energy, slope efficiencies, buildup time, and pulse duration are compared to the analytical solution obtained by solving the laser rate equations. The developed model deals with the spatial dependence of the photon and population inversion densities, the physical properties of the Ti:Sa medium, and the unstable resonator (based on a super-Gaussian output coupler). The energy-related as well as the temporal features are discussed. The use of a squared asymmetrical hyperbolic secant function to describe the temporal pulse shapes is justified. The pulse durations exhibit a simple analytical shape when they are plotted versus the output pulse energy. The model is applied to experimental data obtained at different wavelengths by tuning the laser amplifier previously described. It can readily be applied to other pulsed Ti:Sapphire sources.

Propagation properties of spatial optical solitons in different nonlocal nonlinear media with arbitrary degrees of nonlocality

We address the physical features exhibited by spatial optical solitons propagating in nonlocal Kerr-type media with Gaussian-shaped response and exponential-decay response, respectively. An iteration algorithm based on the split-step Fourier method is developed to obtain the numerical solutions of the solitons for the nonlocal nonlinear Schrödinger equation with arbitrary degrees of nonlocality. Our numerical results show that the soliton properties in the normalized system are different with the change of the degree of nonlocality and with the different responses. The profiles undergo a gradual and continuous transition from a Gaussian-shaped function in the strongly nonlocal case to a hyperbolic secant function in the local case for the Gaussian-shaped response, but for the exponential-decay response, the soliton profile is not Gaussian-shaped even in the strongly nonlocal cases. For the same response function, the stronger the nonlocality is, the higher the critical powers for solitons are and the larger of the phase shifts of the solitons. For the same degrees of nonlocality, when the degrees of nonlocality is larger enough, both the critical power and the phase shift for the Gaussian-shaped response are larger than that for the exponential-decay response.

A Generalized Method and Exact Solutions in Bose–EinsteinCondensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method ispresented to construct some exact analytical solutions of nonlinearpartial differential equations. Making use of the method, we presentrich exact analytical solutions of the one-dimensional nonlinearSchrödinger equation which describes the dynamics of solitons inBose–Einstein condensates with time-dependent atomic scatteringlength in an expulsive parabolic potential. The solutions obtainedinclude not only non-traveling wave and coefficient function'ssoliton solutions, but also Jacobi elliptic function solutions andWeierstrass elliptic function solutions. Some plots are given todemonstrate the properties of some exact solutions under theFeshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.

Analysis of a self [formula omitted]-switched mode-locking [formula omitted]-cut Nd : GdV O 4 /KTP green laser

A diode-pumped self Q -switched mode-locking intracavity frequency doubled c -cut Nd:GdV O 4/KTP green laser is presented. The self-mode-locking is self-starting and stable. About 100% modulation depth for the self-mode-locked green pulses has been achieved. Using the hyperbolic secant function methods and the theory of Kerr-lens mode-locking, considering the Kerr-lens effect of gain medium, the cascading of second-order nonlinearities of KTP crystal, and also the influences of continuous pump rate, the stimulated-radiation lifetime of the active medium, we give a developed rate equation model for self Q -switched and mode-locked laser. For this developed model, the theoretical evaluations are in good agreement with the experimental results and the width of the mode-locked green pulse was estimated to be about 210 ps.

Theoretical and experimental studies on the self-Q-switched and mode-locked Nd:GdVO4/KTP green laser

A diode-pumped self-Q-switched and mode-locked intracavity frequency doubled Nd:GdVO4/KTP green laser is presented. The self mode-locking is self-starting and stable. About 100% modulation depth for the self-mode-locked green pulses has been achieved. We propose a developed rate equation model for self-Q-switched and mode-locked lasers, using the hyperbolic secant function methods and the theory of Kerr-lens mode-locking. This will consider the cascading of second-order nonlinearities of KTP crystal, the third-order nonlinearity of the laser medium, influences of the continuous pump rate, and the stimulated radiation lifetime of the active medium. With the model developed, theoretical calculations are shown to be in good agreement with experimental results. The width of the mode-locked green pulse is estimated to be about 120 ps.

New solitary wave solutions and periodic wave solutions for the compound KdV equation

In this paper, for compound KdV equation, four new solitary wave solutions in the form of hyperbolic secant function and six periodic wave solutions in the form of cosine function are obtained by using undetermined coefficient method. On three different layers, the velocity interval which ensures that bell-shaped solitary wave solutions and periodic wave solutions exist synchronously is obtained, respectively. The length of the interval is related to coefficients of the two nonlinear terms.

Analytical and experimental studies on the diode-pumped, simultaneously Q-switched and mode-locked c-cut Nd:GdVO 4/KTP green laser with LT-GaAs absorber

A diode-pumped passively Q-switched mode-locked (QML) intracavity frequency doubled c-cut Nd:GdVO4/KTP green laser with a LT-GaAs saturable absorber is presented. More than 90% modulation depth for the mode-locked green pulses has been achieved. Using the hyperbolic secant function methods, a developed rate equation model for Q-switched and mode-locked lasers considering the Gaussian spatial distribution of the intracavity photon intensity, the influences of continuous pump rate, the upper state lifetime of the active medium, and the excited-state lifetime of the saturable absorber, was proposed. With this developed model, the theoretical results are in good agreement with the experimental results and the width of the mode-locked green pulse was estimated to be about 380 ps.

Self-mode locking in a diode-pumped self-Q-switched green laser

Simultaneous self-mode locking and self-Q switching in a diode-pumped intracavity frequency doubled Cr4+:Nd3+:YAG∕KTP green laser operating at 532nm is presented. Nearly 100% modulation depth for self-mode-locked green pulses has been achieved and a maximum average output power of 132mW was obtained at an incident pump power of 5.72W. By using a hyperbolic secant function method, considering the Gaussian spatial distribution of the intracavity photon density and the influences of continuous pump rate, as well as the stimulated radiation lifetime of the active medium and the excited-state lifetime of the saturable absorber, a rate equation model was introduced to reconstruct the Q-switched and mode-locked lasers. The theoretical results are in good agreement with the experimental results, and the width of the self-mode-locked green pulse was estimated to be about 350ps.

Analysis of a laser-diode end-pumped intracavity frequency-doubled passivelyQ-switched and mode-locked Nd:GdVO4/KTP laser

A diode-pumped intracavity frequency-doubled passively Q-switched and mode-locked c-cut Nd3+:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber is experimentally demonstrated for the first time. Over 95% modulation depth for the mode-locked green pulses has been achieved. By using hyperbolic secant function methods, considering the influences of a continuous pump rate, the upper state lifetime of the active medium, and the excited-state lifetime of the saturable absorber, a rate equation model developed for Q-switched and mode-locked lasers has been given. With this model, the theoretical evaluations are in good agreement with the experimental results, and the width of the mode-locked green pulse was estimated to be about 280 ps.

Solitary waves in finite deformation elastic circular rod

A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations were obtained. The necessary condition of these solutions existence was given also.