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.     

Multi-crack imaging using nonclassical nonlinear acoustic method

Solid materials with cracks exhibit the nonclassical nonlinear acoustical behavior. The micro-defects in solid materials can be detected by nonlinear elastic wave spectroscopy （NEWS） method with a time-reversal （TR） mirror. While defects lie in viscoelastic solid material with different distances from one another, the nonlinear and hysteretic stress-strain relation is established with Preisach-Mayergoyz （PM） model in crack zone. Pulse inversion （PI） and TR methods are used in numerical simulation and defect locations can be determined from images obtained by the maximum value. Since false-positive defects might appear and degrade the imaging when the defects are located quite closely, the maximum value imaging with a time window is introduced to analyze how defects affect each other and how the fake one occurs. Furthermore, NEWS-TR- NEWS method is put forward to improve NEWS-TR scheme, with another forward propagation （NEWS） added to the existing phases （NEWS and TR）. In the added phase, scanner locations are determined by locations of all defects imaged in previous phases, so that whether an imaged defect is real can be deduced. NEWS-TR-NEWS method is proved to be effective to distinguish real defects from the false-positive ones. Moreover, it is also helpful to detect the crack that is weaker than others during imaging procedure.     

Geometric discord for non-X states

The level surfaces of geometric discord for a class of two-qubit non-X states are investigated when the Bloch vectors are in arbitrary directions. The level surfaces of constant geometric discord are formed by three intersecting open tubes along three orthogc~nal directions. When Bloch vectors increase, the tubes along one or two directions shrink towards the center and may either totally disappear or the open tubes may become closed tubes when the Bloch vectors reach a critical value. In the generalized amplitude damping channel, the evolution of geometric discord shows double sudden changes when the parameter γ, increases. In the phase damping channel, the freezing phenomenon of geometric discord also exists.     

Turing pattern selection in a reaction-diffusion epidemic model

We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions.The value of this study is twofold.First,it establishes the amplitude equations for the excited modes,which determines the stability of amplitudes towards uniform and inhomogeneous perturbations.Second,it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication:on increasing the control parameter ν,the sequence "H 0 hexagons → H 0-hexagon-stripe mixtures → stripes → H π-hexagon-stripe mixtures → H π hexagons" is observed.This may enrich the pattern dynamics in a diffusive epidemic model.     

N-Soliton Solutions of General Nonlinear Schrodinger Equation with Derivative

The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative （DNLSE） and its multisoliton solutions are given by reduction.     

Memristance controlling approach based on modification of linear M–q curve

The memristor has broad application prospects in many fields, while in many cases, those fields require accurate impedance control. The nonlinear model is of great importance for realizing memristance control accurately, but the im- plementing complexity caused by iteration has limited the actual application of this model. Considering the approximate linear characteristics at the middle region of the memristance-charge （M-q） curve of the nonlinear model, this paper pro- poses a memristance controlling approach, which is achieved by linearizing the middle region of the M-q curve of the nonlinear memristor, and establishes the linear relationship between memristances M and input excitations so that it can realize impedance control precisely by only adjusting input signals briefly. First, it analyzes the feasibility for linearizing the middle part of the M-q curve of the memristor with a nonlinear model from the qualitative perspective. Then, the lin- earization equations of the middle region of the M-q curve is constructed by using the shift method, and under a sinusoidal excitation case, the analytical relation between the memristance M and the charge time t is derived through the Taylor series expansions. At last, the performance of the proposed approach is demonstrated, including the linearizing capability for the middle part of the M-q curve of the nonlinear model memristor, the controlling ability for memristance M, and the influence of input excitation on linearization errors.     

Nonlinear polarization rotation-induced pulse shaping in a stretched-pulse ytterbium-doped fiber laser

We report on controllable pulse shaping in a Yb-doped stretched-pulse fiber laser followed by a high-power chirped pulse amplifier. We demonstrate that the pulses after an extra-cavity grating pair change their intensity profile from Lorentz to Gaussian and then to sech2 shapes by adjusting the intra-cavity polarization through a quarter-wave plate inside the fiber laser cavity. The laser pulses with different pulse shapes exhibit pulse-to-pulse amplitude fluctuation of -- 1.02%, while the sech2-shaped pulse train is provided with a more stable free-running repetition rate as a result of the stronger self-phase modulation in the fiber laser cavity than Lorentz- and Gaussian-shaped pulse trains.     

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.     

Solitons in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in a Harmonic Trap

We obtain the integrable relation for the one-dimensional nonlinear Schrodinger equations which describes the dynamics of a Bos-Einstein Condensates with time-dependent scattering length in a harmonic potential. The exact one- and two-soliton solutions are constructed analytically by using the Hirota method. Then we further discuss the dynamics of the one soliton and the interactions between two solitons in currently experimental conditions.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f（ξ）, is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Soliton Solutions and Bilinear B/icklund Transformation for Generalized Nonlinear Schr6dinger Equation with Radial Symmetry

Investigated in this paper is the generalized nonlinear Schrodinger equation with radial symmetry. With the help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear method. B~cklund transformation in the bilinear form is presented, through which a new solution is constructed. Graphically, we have found that the solitons are symmetric about x = O, while the soliton pulse width and amplitude will change along with the distance and time during the propagation.     

Synchronization of the fractional-order generalized augmented Lii system and its circuit implementation

In this paper, the synchronization of the fractional-order generalized augmented Lti system is investigated. Based on the predictor--corrector method, we obtain phase portraits, bifurcation diagrams, Lyapunov exponent spectra, and Poincar6 maps of the fractional-order system and find that a four-wing chaotic attractor exists in the system when the system pa- rameters change within certain ranges. Further, by varying the system parameters, rich dynamical behaviors occur in the 2.7-order system. According to the stability theory of a fractional-order linear system, and adopting the linearization by feedback method, we have designed a nonlinear feedback controller in our theoretical analysis to implement the synchro- nization of the drive system with the response system. In addition, the synchronization is also shown by an electronic circuit implementation for the 2.7-order system. The obtained experiment results accord with the theoretical analyses, which further demonstrate the feasibility and effectiveness of the proposed synchronization scheme.     

Design on a highly birefringent and nonlinear photonic crystal fiber in the C waveband

In this paper, a novel photonic crystal fiber （PCF） with high birefringence and nonlinearity is designed. The charac- teristics of birefringence, dispersion and nonlinearity are studied by using the full-vector finite element method （FVFEM）. The numerical results show that the phase birefringence and nonlinear coefficient of PCF can be up to 4.51× 10-3 and 32.8972 w-l.km-1 at 1.55 μm, respectively. The proposed PCF could be found to have important applications in the polarization-dependent nonlinear optics such as the pulse compress and reshaping in the C waveband.     

Nonlinear Landau damping of high frequency waves in non-Maxwellian plasmas

Space plasmas often possess non-Maxwellian distribution functions which have a significant effect on the plasma waves.When a laser or electron beam passes through a dense plasma,hot low density electron populations can be generated to alter the wave damping/growth rate.In this paper,we present theoretical analysis of the nonlinear Landau damping for Langmuir waves in a plasma where two electron populations are found.The results show a marked difference between the Maxwellian and non-Maxwellian instantaneous damping rates when we employ a non-Maxwellian distribution function called the generalized(r,q)distribution function,which is the generalized form of the kappa and Maxwellian distribution functions.In the limiting case of r=0 and q→∞,it reduces to the classical Maxwellian distribution function,and when r=0 and q→κ+1,it reduces to the kappa distribution function.     

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrodinger Equations with Multiply Components

Considering the coupled nonlinear Schrodinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations. Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.     

Nonlinear effect of the structured light profilometry in the phase-shifting method and error correction

Digital structured light （SL） profilometry is increasingly used in three-dimensional （3D） measurement technology. However, the nonlinearity of the off-the-shelf projectors and cameras seriously reduces the measurement accuracy. In this paper, first, we review the nonlinear effects of the projector-camera system in the phase-shifting structured light depth measurement method. We show that high order harmonic wave components lead to phase error in the phase-shifting method. Then a practical method based on frequency domain filtering is proposed for nonlinear error reduction. By using this method, the nonlinear calibration of the SL system is not required. Moreover, both the nonlinear effects of the projector and the camera can be effectively reduced. The simulations and experiments have verified our nonlinear correction method.     

Numerical Solutions of a Class of Nonlinear Evolution Equations with Nonlinear Term of Any Order

In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.     

Analysis of detection limit to time-resolved coherent anti-Stokes Raman scattering nanoscopy

In the implementation of CARS nanoscopy, signal strength decreases with focal volume size decreasing. A crucial problem that remains to be solved is whether the reduced signal generated in the suppressed focal volume can be detected. Here reported is a theoretical analysis of detection limit （DL） to time-resolved CARS （T-CARS） nanoscopy based on our proposed additional probe-beam-induced phonon depletion （APIPD） method for the low concentration samples. In order to acquire a detailed shot-noise limited signal-to-noise （SNR） and the involved parameters to evaluate DL, the T-CARS process is described with full quantum theory to estimate the extreme power density levels of the pump and Stokes beams determined by saturation behavior of coherent phonons, which are both actually on the order of ~ 109 W/cm2. When the pump and Stokes intensities reach such values and the total intensity of the excitation beams arrives at a maximum tolerable by most biological samples in a certain suppressed focal volume （40-nm suppressed focal scale in APIPD method）, the DL correspondingly varies with exposure time, for example, DL values are 103 and 102 when exposure times are 20 ms and 200 ms respectively.     

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schrodinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.