Solving Cauchy Issues of Highly Nonlinear Elliptic Equations Using a Meshless Method

In this paper,we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data.We also add the average classification as an approximate solution to the nonlinear operator part,without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution.The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described by a new 3D homogenization function,which is right to find the numerical solutions with the errors smaller than the level of noise being put on the over-specified Neumann conditions on the bottom of the cuboid.Besides,note that the new homogenization functions method(HFM)does not require dealing with the regularization and highly nonlinear equations.The robustness and accuracy of the HFM are verified by comparing the recovered results of several numerical experiments to the exact solutions in the entire region,even though a very large level of noise 50%is imposed on the over specified Neumann conditions.The numerical errors of our scheme are in the order of O(10^(−1))-O(10^(−4)).

Cascaded four-wave mixing all-optical wavelength converter based on highly nonlinear fiber deposited graphene oxide

The all-optical wavelength conversion using cascaded four-wave mixing(FWM)phenomena with graphene oxide(GO)and a highly nonlinear fiber(HNLF)device is demonstrated experimentally.GO has a strong third-order nonlinear effect and is an excellent nonlinear optical material for nonlinear optical wavelength conversion.The group velocity matching of pump and signal,HNLF,and GO amplification promote the cascaded nonlinear frequency mixing process.From the experimental and analytical results,the maximum spacing between signal and pump is 21 nm,and specifically,the order of cascaded FWM light increases from order 1 to order 2 with increasing GO,and the first-order FWM conversion effi-ciency increases to a maximum of−14.5 dB.To the best of our knowledge,this is the first time that cascaded FWM-based all-optical wavelength conversion in HNLF-GO with wide wavelength selectivity is investigated with combined pump and signal light.Our findings not only provide an effective method for achieving all-optical wavelength conversion in cascaded FWM but also offer the possibility of fabricating high-performance nonlinear optical devices.

Ultra-broadband modulation instability gain characteristics in As_2S_3 and As_2Se_3 chalcogenide glass photonic crystal fiber

Based on the designed As2Se3 and As2S3 chalcogenide glass photonic crystal fiber（PCF） and the scalar nonlinear Schrdinger equation,the effects of pump power and wavelength on modulation instability（MI） gain are comprehensively studied in the abnormal dispersion regime of chalcogenide glass PCF.Owing to high Raman effect and high nonlinearity,ultra-broadband MI gain is obtained in chalcogenide glass PCF.By choosing the appropriate pump parameter,the MI gain bandwidth reaches 2738 nm for the As2Se3 glass PCF in the abnormal-dispersion region,while it is 1961 nm for the As2S3 glass PCF.

Nonlinear optics on silicon-rich nitride--a high nonlinear figure of merit CMOS platform[Invited]

CMOS platforms with a high nonlinear figure of merit are highly sought after for high photonic quantum efficiencies, enabling functionalities not possible from purely linear effects and ease of integration with CMOS electronics. Silicon-based platforms have been prolific amongst the suite of advanced nonlinear optical signal processes demonstrated to date. These include crystalline silicon, amorphous silicon, Hydex glass, and stoichiometric silicon nitride. Residing between stoichiometric silicon nitride and amorphous silicon in composition,silicon-rich nitride films of various formulations have emerged recently as promising nonlinear platforms for high nonlinear figure of merit nonlinear optics. Silicon-rich nitride films are compositionally engineered to create bandgaps that are sufficiently large to eliminate two-photon absorption at telecommunications wavelengths while enabling much larger nonlinear waveguide parameters(5 x–500 x) than those in stoichiometric silicon nitride. This paper reviews recent developments in the field of nonlinear optics using silicon-rich nitride platforms, as well as the outlook and future opportunities in this burgeoning field.

A LEAP FROG FINITE DIFFERENCE SCHEME FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS OF HIGHORDER

In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.

Few-layer MoS2-deposited microfiber as highly nonlinear photonic device for pulse shaping in a fiber laser [Invited]

Two-dimensional（2D） materials have emerged as attractive mediums for fabricating versatile optoelectronic devices. Recently, few-layer molybdenum disulfide（MoS2）, as a shining 2D material, has been discovered to possess both the saturable absorption effect and large nonlinear refractive index. Herein, taking advantage of the unique nonlinear optical properties of MoS2, we fabricated a highly nonlinear saturable absorption photonic device by depositing the few-layer MoS2 onto the microfiber. With the proposed MoS2 photonic device, apart from the conventional soliton patterns, the mode-locked pulses could be shaped into some new soliton patterns, namely,multiple soliton molecules, localized chaotic multipulses, and double-scale soliton clusters. Our findings indicate that the few-layer MoS2-deposited microfiber could operate as a promising highlynonlinear photonic device for the related nonlinear optics applications.

Flexible,robust and highly efficient broadband nonlinear optical materials based on graphene oxide impregnated polymer sheets

We report a simple solution-processed method for the fabrication of low-cost,flexible optical limiting materials based on graphene oxide(GO) impregnated polyvinyl alcohol(PVA) sheets.Such GO–PVA composite sheets display highly efficient broadband optical limiting activities for femtosecond laser pulses at 400,800,and 1400 nm with very low limiting thresholds.Femtosecond pump–probe measurement results revealed that nonlinear absorption played an important role for the observed optical limiting activities.High flexibility and efficient optical limiting activities of these materials allow these composite sheets to be attached to nonplanar optical sensors in order to protect them from light-induced damage.

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.

Canonical logic units using bidirectional four-wave mixing in highly nonlinear fiber

All-optical canonical logic units at 40 Gb/s using bidirectional four-wave mixing(FWM) in highly nonlinear fiber are proposed and experimentally demonstrated. Clear temporal waveforms and correct pattern streams are successfully observed in the experiment. This scheme can reduce the amount of nonlinear devices and enlarge the computing capacity compared with general ones. The numerical simulations are made to analyze the relationship between the FWM efficiency and the position of two interactional signals.

Competition mechanism of multiple four-wave mixing in highly nonlinear fiber： spatial instability and satellite characteristics

Competition mechanism in multiple four-wave mixing （MFWM） processes is demonstrated theoretically. Provided considering only two waves injected into a highly nonlinear fiber （HNLF）, there are three modes displaying comprehensive dynamic behaviors, such as fixed points, periodic motion, and chaotic motion. Especially, Mode C of MFWM is emphasized by analyzing its phase-space trajectory to demonstrate nonlinear wave- wave interactions. The study shows that, when the phase- space trajectory approaches or gets through a saddle point, a dramatic power depletion for the injected wave can be realized, with the representative point moving chaotically, but when phase-space trajectories are distributed around a center point, the power for the injected wave is retained almost invariable, with the representative point moving periodically. Finally, the evolvement of satellite wave over an optical fiber is investigated by comparing it with the interference pattern in Young＇s double-slit experiment.

Transient Response of High Dimensional Nonlinear Dynamic System for a Rotating Cantilever Twisted Plate

Dynamic transient responses of rotating twisted plate under the air-blast loading and step loading respectively considering the geometric nonlinear relationships are investigated using classical shallow shell theory.By applying energy principle,a novel high dimensional nonlinear dynamic system of the rotating cantilever twisted plate is derived for the first time.The use of variable mode functions by polynomial functions according to the twist angles and geometric of the plate makes it more accurate to describe the dynamic system than that using the classic cantilever beam functions and the free-free beam functions.The comparison researches are carried out between the present results and other literatures to validate present model,formulation and computer process.Equations of motion describing the transient high dimensional nonlinear dynamic response are reduced to a four degree of freedom dynamic system which expressed by out-plane displacement.The effects of twisted angle,stagger angle,rotation speed,load intensity and viscous damping on nonlinear dynamic transient responses of the twisted plate have been investigated.It’s important to note that although the homogeneous and isotropic material is applied here,it might be helpful for laminated composite,functionally graded material as long as the equivalent material parameters are obtained.

Analytic approximations of Von Krmn plate under arbitrary uniform pressure—equations in integral form

Analytic approximations of the Von Krmn's plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method(HAM), an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c_1 and c_2 in the frame of the HAM. Besides, it is found that the HAMbased iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c_1 =.θ and c_2 =-1, where θ denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c_1 =.θ and c_2 =-1. In addition, we prove that the HAM approach for the Von Krmn's plate equations in differential form is just a special case of the HAM for the Von Krmn's plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems,and its superiority over perturbation techniques.

Fiber soliton-form 3R regenerator and its performance analysis

A novel fiber optical 3R regenerator based on optical soliton-effect using highly nonlinear fiber is constructed and investigated for the needs of the high rate and long-haul optical communications. The propagation equation of the pulses in the proposed optical 3R regenerator with the control of optical modulator and filter is established. By the use of the variational approach, the evolution of the distorted optical pulses in the regenerator and the functions of reamplification, reshaping, and reUming are investigated. The relation between the construction parameters and the output performance of the regenerator is discussed. The stable operation condition of the regenerator is revealed.

Extensible Framework for Rao-Blackwellized Filtering

The Rao-Blackwellized Particle Filter （RBPF） is widely used for high dimensional nonlinear sys- tems, often with a linear Gaussian substructure. However, the RBPF is just a specific method in the class of Rao-Blackwellized Filtering （RBF）. This paper analyzes the recursive structure of the RBF from a more gen- eral perspective. The research starts from a general system model and studies the interconnected relation- ships between the two subspaces during the iterations. The results illustrate the working mechanisms of the RBF with an extensible framework for easily building Rao-Blackwellized algorithms with common nonlinear filters. Several examples are given to illustrate how to build new filters using this framework.