Hopf Bifurcation Research of Hydraulic Turbine Governing System with a Saturation Nonlinearity

A nonlinear mathematical model for hydro turbine governing system with saturation nonlinearity in small perturbation has been proposed with all the essential components, i.e. turbine, PID t^^pe governor with saturation part and generator included in the model. Existence, stability and direction of Hopf bifurcation of an example HTGS are investigated in detail and presented in forms of bifurcation diagrams and time ＂wavefomis. The analysis show,that a supercritical Hopf bifurcation may exist in hydraulic turbine systems in some certain conditions. Moreover, the dynaiidc behavior of system with different parameters such as Tw, Tab, Ty and 尺 are studied extensively. An example with numerical simulations is presented to illustrate the theoretical results. The researches provide a reasonable explanation for the Hopf phenomenon happened in operation of hydroelectric generating unit.

Topological laser on square lattice with gain–loss-induced higher-order corner modes

We investigate the higher-order topological laser in the two-dimensional(2D) coupled-cavity array. By adding staggered on-site gain and loss to the 2D Hermitian array with a trivial phase, the system will emerge degenerate topological corner modes, which are protected by bulk band gap. For such a non-Hermitian model, by adjusting the parameters of the system and introducing the pumping into the cavity at the corner, a single-mode lasing with topological protection emerges.Furthermore, single-mode lasing exists over a wide range of pumping strengths. No matter where the cavity is initially stimulated, after enough time evolution, all the cavities belonging to the topological corner mode can emit a stable laser.

Nonlinear Saturation of Baroclinic Instability in the Phillips Model: The Case of Energy

A conservation law for the Phillips model is derived. Using this law, the nonlinear saturation of purely baroclinic instability caused by the vertical velocity shear of the basic flow in the Phillips model—the case of energy—is studied within the context of Arnold’s second stability theorem. Analytic upper bounds on the energy of wavy disturbances are obtained. For one unstable region in the parameter plane, the result here is a second-order correction in ε to Shepherd’s; For another unstable region, the analytic upper bound on the energy of wavy disturbances offers an effective constraint on wavy (nonzonal) disturbances Φ′ i at any time.

Cross-phase modulation instability in optical fibres with exponential saturable nonlinearity and high-order dispersion

Utilizing the linear-stability analysis, this paper analytically investigates and calculates the condition and gain spectra of cross-phase modulation instability in optical fibres in the ease of exponential saturable nonlinearity and high-order dispersion. The results show that, the modulation instability characteristics here are similar to those of conventional saturable nonlinearity and Kerr nonlinearity. That is to say, when the fourth-order dispersion has the same sign as that of the second-order one, a new gain spectral region called the second one which is far away from the zero point may appear. The existence of the exponential saturable nonlinearity will make the spectral width as well as the peak gain of every spectral region increase with the input powers before decrease. Namely, for every spectral regime, this may lead to a unique value of peak gain and spectral width for two different input powers. In comparison with the case of conventional saturable nonlinearity, however, when the other parameters are the same, the variations of the spectral width and the peak gain with the input powers will be faster in case of exponential saturable nonlinearity.

MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRODINGER EQUATIONS WITH SATURABLE NONLINEARITY

In this paper,we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics.More precisely,for any given nonnegative integer k,by using a minimization argument,we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem,and show,by a deformation lemma and Miranda's theorem,that the minimizer is the desired solution.

NONLINEAR SATURATION OF BAROCLINIC INSTABILITY IN THE GENERALIZED PHILLIPS MODEL(Ⅱ)- THE LOWER BOUND ON THE DISTURBANCE ENERGY AND POTENTIAL ENSTROPHY TO THE NONLINEARLY UNSTABLE BASIC FLOW

On the basis of the nonlinear stability theorem in the context of Arnol'd's second theorem for the generalized Phillips model,nonlinear saturation of baroclinic instability in the generalized Phillips model is investigatedThe lower bound on the disturbance energy and potential enstrophy to the nonlinearly unstable basic flow in the generalized Phillips model is presented,which indicates that there may exist an allocation between a nonlinearly unstable basic flow and a growing disturbance

Nonlinear saturation amplitude of cylindrical Rayleigh Taylor instability

The nonlinear saturation amplitude （NSA） of the fundamental mode in the classical Rayleigh-Taylor instability with a cylindrical geometry for an arbitrary Atwood number is analytically investigated by considering the nonlinear corrections up to the third order. The analytic results indicate that the effects of the initial radius of the interface （r0） and the Atwood number （A） play an important role in the NSA of the fundamental mode. The NSA of the fundamental mode first increases gently and then decreases quickly with increasing A. For a given A, the smaller the ro/λ（λ is the perturbation wavelength）, the larger the NSA of the fundamental mode. When ro/λ is large enough （r0 〉〉 λ）, the NSA of the fundamental mode is reduced to the prediction in the previous literatures within the framework of the third-order perturbation theory.

NONLINEAR SATURATION OF BAROCLINIC INSTABILITY IN THE GENERALIZED PHILLIPS MODEL (Ⅰ) -THE UPPER BOUND ON THE EVOLUTION OF DISTURBANCE TO THE NONLINEARLY UNSTABLE BASIC FLOW

On the basis of the nonlinear stability theorem in the context of Arnol's second theorem for the generalized Phillips model, nonlinear saturation of baroclinic instability in the generalized Phillips model is investigated. By choosing appropriate artificial stable basic flows, the upper bounds on the disturbance energy and potential enstrophy to the nonlinearly unstable basic flow in the generalized Phillips model are obtained, which are analytic completely and without the limitation of infinitesimal initial disturbance.

Law of nonlinear flow in saturated clays and radial consolidation

It was derived that micro-scale amount level of average pore radius of clay changed from 0.01 to 0.1 micron by an equivalent concept of flow in porous media. There is good agreement between the derived results and test ones. Results of experiments show that flow in micro-scale pore of saturated clays follows law of nonlinear flow. Theoretical analyses demonstrate that an interaction of solid-liquid interfaces varies inversely with permeability or porous radius. The interaction is an important reason why nonlinear flow in saturated clays occurs. An exact mathematical model was presented for nonlinear flow in micro-scale pore of saturated clays. Dimension and physical meanings of parameters of it are definite. A new law of nonlinear flow in saturated clays was established. It can describe characteristics of flow curve of the whole process of the nonlinear flow from low hydraulic gradient to high one. Darcy law is a special case of the new law. A math- ematical model was presented for consolidation of nonlinear flow in radius direction in saturated clays with constant rate based on the new law of nonlinear flow. Equations of average mass conservation and moving boundary, and formula of excess pore pressure distribution and average degree of consolidation for nonlinear flow in saturated clay were derived by using an idea of viscous boundary layer, a method of steady state in stead of transient state and a method of integral of an equation. Laws of excess pore pressure distribution and changes of average degree of consolidation with time were obtained. Re- sults show that velocity of moving boundary decreases because of the nonlinear flow in saturated clay. The results can provide geology engineering and geotechnical engineering of saturated clay with new scientific bases. Calculations of average degree of consolidation of the Darcy flow are a special case of that of the nonlinear flow.

Exact solitary wave solutions of a nonlinear Schrdinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice

In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable;which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.

Impacts of higher-order dispersions and saturable nonlinearities on modulation instability in negative-refractive metamaterials

On the basis of the standard linear stability analysis and Drude electromagnetic model,the impacts of higher-order dispersions and three kinds of typical saturable nonlinearities on modulation instability（MI） have been analyzed and calculated for negative-refractive metamaterials（MMs）.Our results show that the MI gain spectra consist of only one spectral region instead of one or two regions in ordinary materials,which may be close to or far from the zero point.Particularly,the spectrum far from the zero point has a high cut-off frequency but a narrow spectral width,which is obviously beneficial to the generation of high-repetition-rate pulse trains.Moreover,MI characteristics here will vary with the normalized angular frequency which can be modified by adjusting the structures of negative-refractive MMs,signifying the controllability of bistable solitons and MI based applications.The effects of saturable nonlinearities are similar to those in ordinary materials.

New exact solutions of nonlinear differential-difference equations with symbolic computation

In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schro¨dinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.

Spatial solitons in photonic lattices with large-scale defects

We address the existence, stability and propagation dynamics of solitons supported by large-scale defects surrounded by the harmonic photonic lattices imprinted in the defocusing saturable nonlinear medium. Several families of soliton solutions, including flat-topped, dipole-like, and multipole-like solitons, can be supported by the defected lattices with different heights of defects. The width of existence domain of solitons is determined solely by the saturable parameter. The existence domains of various types of solitons can be shifted by the variations of defect size, lattice depth and soliton order. Solitons in the model are stable in a wide parameter window, provided that the propagation constant exceeds a critical value, which is in sharp contrast to the case where the soliton trains is supported by periodic lattices imprinted in defocusing saturable nonlinear medium. We also find stable solitons in the semi-infinite gap which rarely occur in the defocusing media.

Surface defect gap solitons in two-dimensional optical lattices

We investigate the existence and stability of surface defect gap solitons at an interface between a defect in a two-dimensional optical lattice and a uniform saturable Kerr nonlinear medium. The surface defect embedded in the two-dimensional optical lattice gives rise to some unique properties. It is interestingly found that for the negative defect, stable surface defect gap solitons can exist both in the semi-infinite gap and in the first gap. The deeper the negative defect, the narrower the stable region in the semi-infinite gap will be. For a positive defect, the surface defect gap solitons exist only in the semi-infinite gap and the stable region localizes in a low power region.

Identification for the Low-Contrast Image Signal with Regularized Variational Term and Dynamical Saturating Nonlinearity

In recent years,image processing based on stochastic resonance(SR)has received more and more attention.In this paper,a new model combining dynamical saturating nonlinearity with regularized variational term for enhancement of low contrast image is proposed.The regularized variational term can be setting to total variation(TV),second order total generalized variation(TGV)and non-local means(NLM)in order to gradually suppress noise in the process of solving the model.In addition,the new model is tested on a mass of gray-scale images from standard test image and low contrast indoor color images from Low-Light dataset(LOL).By comparing the new model and other traditional image enhancement models,the results demonstrate the enhanced image not only obtain good perceptual quality but also get more excellent value of evaluation index compared with some previous methods.

Direct detection of the transient superresolution effect of nonlinear saturation absorption thin films

Using a strong nonlinear saturation absorption effect is one technique for breaking through the diffraction limit. In this technique, formation of a dynamic and reversible optical pinhole channel and transient superresolution is critical. In this work, a pump–probe transient detection and observation–experimental setup is constructed to explore the formation process directly. A Ge2Sb2Te5 thin film with strong nonlinear saturation absorption is investigated. The dynamic evolution of the optical pinhole channel is detected and imaged, and the transient superresolution spot is directly captured experimentally. Results verify that the superresolution effect originates from the generation of an optical pinhole channel and that the formation of the optical pinhole channel is dynamic and reversible. A good method is provided for direct detection and observation of the transient process of the superresolution effect of nonlinear thin films.

Transient Performance Improvement in Tracking Control for a Class of Nonlinear Systems with Input Saturation

This paper addresses the composite nonlinear feedback（CNF） control for a class of singleinput single-output nonlinear systems with input saturation to track a time varying reference target with good transient performance. The CNF control law consists of a tracking control law and a performance compensator. The tracking control law is designed to drive the output of the system to track the time varying reference target rapidly, while the performance compensator is used to reduce the overshoot caused by the tracking control law. The stability of the closed-loop system is established. The design procedure and the improvement of transient performance of the closed-loop system are illustrated with a numerical example and the controlled Van del Pol oscillator.

Wavelength-Dependent Transient Characteristics Caused by Gain Saturation in Highly Nonlinear Fiber-Based Raman Amplifiers

We have investigated the transient characteristics of discrete Raman Amplifiers and found that the response time caused by gain saturation is dependent upon the wavelength, which corresponds to the effective length of the pump light.

Nonlinear Saturation Amplitude in Classical Planar Richtmyer–Meshkov Instability

The classical planar Richtmyer–Meshkov instability(RMI) at a fluid interface supported by a constant pressure is investigated by a formal perturbation expansion up to the third order,and then according to definition of nonlinear saturation amplitude(NSA) in Rayleigh–Taylor instability(RTI),the NSA in planar RMI is obtained explicitly.It is found that the NSA in planar RMI is affected by the initial perturbation wavelength and the initial amplitude of the interface,while the effect of the initial amplitude of the interface on the NSA is less than that of the initial perturbation wavelength.Without marginal influence of the initial amplitude,the NSA increases linearly with wavelength.The NSA normalized by the wavelength in planar RMI is about 0.11,larger than that corresponding to RTI.

Localization of normalized solutions for saturable nonlinear Schr?dinger equations

In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.