Broadband third-order optical nonlinearities of layered franckeite towards mid-infrared regime

The study of nonlinear optical responses in the mid-infrared(mid-IR)regime is essential for advancing ultrafast mid-IR laser applications.However,nonlinear optical effects under mid-IR excitation are rarely reported due to the lack of suitable nonlinear optical materials.The natural van derWaals heterostructure franckeite,known for its narrow bandgap and stability in air,shows great potential for developing mid-IR nonlinear optical devices.We have experimentally demonstrated that layered franckeite exhibits a broadband wavelength-dependent nonlinear optical response in the mid-IR spectral region.Franckeite nanosheets were prepared using a liquid-phase exfoliation method,and their nonlinear optical response was characterized in the spectral range of 3000 nm to 5000 nm.The franckeite nanosheets exhibit broadband wavelengthdependent third-order nonlinearities,with nonlinear absorption and refraction coefficients estimated to be about 10^(-7)cm/W and 10^(-11)cm^(2)/W,respectively.Additionally,a passively Q-switched fluoride fiber laser operating around a wavelength of 2800 nm was achieved,delivering nanosecond pulses with a signal-to-noise ratio of 43.6 dB,based on the nonlinear response of franckeite.These findings indicate that layered franckeite possesses broadband nonlinear optical characteristics in the mid-IR region,potentially enabling new possibilities for mid-IR photonic devices.

Output Feedback Stabilization of High-Order Nonlinear Time-Delay Systems With Low-Order and High-Order Nonlinearities

Dear Editor,to This letter deals with the output feedback stabilization of a class of high-order nonlinear time-delay systems with more general low-order and high-order nonlinearities.By constructing reduced-order observer,based on homogeneous domination theory together with the adding a power integrator method,an output feedback controller is developed guarantee the equilibrium of the closed system globally uniformly asymptotically stable.

MINIMIZERS OF L^(2)-SUBCRITICAL VARIATIONAL PROBLEMS WITH SPATIALLY DECAYING NONLINEARITIES IN BOUNDED DOMAINS

This paper is concerned with the minimizers of L^(2)-subcritical constraint variar tional problems with spatially decaying nonlinearities in a bounded domain Ω of R~N(N≥1).We prove that the problem admits minimizers for any M> 0.Moreover,the limiting behavior of minimizers as M→∞ is also analyzed rigorously.

Numerical Exploration of Asymmetrical Impact Dynamics: Unveiling Nonlinearities in Collision Problems and Resilience of Reinforced Concrete Structures

This study provides a comprehensive analysis of collision and impact problems’ numerical solutions, focusing ongeometric, contact, and material nonlinearities, all essential in solving large deformation problems during a collision.The initial discussion revolves around the stress and strain of large deformation during a collision, followedby explanations of the fundamental finite element solution method for addressing such issues. The hourglassmode’s control methods, such as single-point reduced integration and contact-collision algorithms are detailedand implemented within the finite element framework. The paper further investigates the dynamic responseand failure modes of Reinforced Concrete (RC) members under asymmetrical impact using a 3D discrete modelin ABAQUS that treats steel bars and concrete connections as bond slips. The model’s validity was confirmedthrough comparisons with the node-sharing algorithm and system energy relations. Experimental parameterswere varied, including the rigid hammer’s mass and initial velocity, concrete strength, and longitudinal and stirrupreinforcement ratios. Findings indicated that increased hammer mass and velocity escalated RC member damage,while increased reinforcement ratios improved impact resistance. Contrarily, increased concrete strength did notsignificantly reduce lateral displacement when considering strain rate effects. The study also explores materialnonlinearity, examining different materials’ responses to collision-induced forces and stresses, demonstratedthrough an elastic rod impact case study. The paper proposes a damage criterion based on the residual axialload-bearing capacity for assessing damage under the asymmetrical impact, showing a correlation betweendamage degree hammer mass and initial velocity. The results, validated through comparison with theoreticaland analytical solutions, verify the ABAQUS program’s accuracy and reliability in analyzing impact problems,offering valuable insights into collision and impact problems’ nonlinearities and practical strategies for enhancingRC structures’ resilience under dynamic stress.

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L2-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.

Stability of Standing Waves for the Nonlinear Schrödinger Equation with Mixed Power-Type and Hartree-Type Nonlinearities

This paper studies the existence of stable standing waves for the nonlinear Schrödinger equation with Hartree-type nonlinearity i∂tψ+Δψ+| ψ |pψ+(| x |−γ∗| ψ |2)ψ=0, (t,x)∈[ 0,T )×ℝN.Where ψ=ψ(t,x)is a complex valued function of (t,x)∈ℝ+×ℝN. The parameters N≥3, 0p4Nand 0γmin{ 4,N }. By using the variational methods and concentration compactness principle, we prove the orbital stability of standing waves.

Maximum Correntropy Kalman Filtering for Non-Gaussian Systems With State Saturations and Stochastic Nonlinearities

This paper tackles the maximum correntropy Kalman filtering problem for discrete time-varying non-Gaussian systems subject to state saturations and stochastic nonlinearities. The stochastic nonlinearities, which take the form of statemultiplicative noises, are introduced in systems to describe the phenomenon of nonlinear disturbances. To resist non-Gaussian noises, we consider a new performance index called maximum correntropy criterion(MCC) which describes the similarity between two stochastic variables. To enhance the “robustness” of the kernel parameter selection on the resultant filtering performance, the Cauchy kernel function is adopted to calculate the corresponding correntropy. The goal of this paper is to design a Kalman-type filter for the underlying systems via maximizing the correntropy between the system state and its estimate. By taking advantage of an upper bound on the one-step prediction error covariance, a modified MCC-based performance index is constructed. Subsequently, with the assistance of a fixed-point theorem, the filter gain is obtained by maximizing the proposed cost function. In addition, a sufficient condition is deduced to ensure the uniqueness of the fixed point. Finally, the validity of the filtering method is tested by simulating a numerical example.

Topology Optimization of Compliant Mechanisms with Geometrical Nonlinearities Using the Ground Structure Approach

The majority of topology optimization of compliant mechanisms uses linear finite element models to find the structure responses.Because the displacements of compliant mechanisms are intrinsically large,the topological design can not provide quantitatively accurate result.Thus,topological design of these mechanisms considering geometrical nonlinearities is essential.A new methodology for geometrical nonlinear topology optimization of compliant mechanisms under displacement loading is presented.Frame elements are chosen to represent the design domain because they are capable of capturing the bending modes.Geometrically nonlinear structural response is obtained by using the co-rotational total Lagrange finite element formulation,and the equilibrium is solved by using the incremental scheme combined with Newton-Raphson iteration.The multi-objective function is developed by the minimum strain energy and maximum geometric advantage to design the mechanism which meets both stiffness and flexibility requirements, respectively.The adjoint method and the direct differentiation method are applied to obtain the sensitivities of the objective functions. The method of moving asymptotes（MMA） is employed as optimizer.The numerical example is simulated to show that the optimal mechanism based on geometrically nonlinear formulation not only has more flexibility and stiffness than that based on linear formulation,but also has better stress distribution than the one.It is necessary to design compliant mechanisms using geometrically nonlinear topology optimization.Compared with linear formulation,the formulation for geometrically nonlinear topology optimization of compliant mechanisms can give the compliant mechanism that has better mechanical performance.A new method is provided for topological design of large displacement compliant mechanisms.

MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS

In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.

Observers for Descriptor Systems with Slope-restricted Nonlinearities

This paper addresses the observer design problem for a class of nonlinear descriptor systems whose nonlinear terms are slope-restricted. The full-order observer is formulated as a nonlinear descriptor system. A linear matrix inequality （LMI） condition is derived to construct the full-order observer. The existence and uniqueness of the solution to the obtained observer system are guaranteed. Furthermore, under the same LMI condition and a common assumption, a reduced-order observer is designed. Finally, the design methods are reduced to a strict LMI problem and illustrated by a numerical example.

1∶2 INTERNAL RESONANCE OF COUPLED DYNAMIC SYSTEM WITH QUADRATIC AND CUBIC NONLINEARITIES

The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.

Analysis of nonlinearities and effects in direct drive electro-hydraulic position servo system

The direct drive electro-hydraulic servo system is a new approach hydraulic system. It is much smaller and easier controlled than traditional systems and is a perfect energy saver. This paper will briefly introduce the popular nonlinearities in the electro-hydraulic system and analyse the effect of nonlinearities in direct drive electro-hydraulic position servo system by means of simulation research. Some valuable conclusions are given.

Adaptive Control of MIMO Mechanical Systems with Unknown Actuator Nonlinearities Based on the Nussbaum Gain Approach

This paper investigates MIMO mechanical systems with unknown actuator nonlinearities. A novel Nussbaum analysis tool for MIMO systems is established such that unknown time-varying control coefficients are tackled. In contrast to existing literatures on continuous-time systems, the newly-developed Nussbaum tool focuses on extending the traditional Nussbaum result from one dimensional case to the multiple one. Specifically, not only the multiple unknown input coefficients are extended to the time-varying, but also the limitation of the prior knowledge of coefficients' upper and lower bounds is removed. Furthermore, an adaptive robust controller associated with the proposed tool is presented. The asymptotic tracking of MIMO mechanical systems is guaranteed with the help of the Lyapunov Theorem. Finally, a simulation example is provided to examine the validity of the proposed scheme. © 2014 Chinese Association of Automation.

Solutions to the equations describing materials with competing quadratic and cubic nonlinearities

The Lie group theoretical method is used to study the equations describing materials with competing quadratic and cubic nonlinearities. The equations shave some of the nice properties of soliton equations. From the elliptic functions expansion method, we obtain large families of analytical solutions, in special cases, we have the periodic, kink and solitary solutions of the equations. Furthermore, we investigate the stability of these solutions under the perturbation of amplitude noises by numerical simulation.

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear SchrSdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.

QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical.The potential V is bounded from below and above by positive constants.Because we are considering a critical term which interacts with higher eigenvalues for the linear problem,we need to apply a linking theorem.Notice that the lack of compactness,which comes from critical problems and the fact that we are working in the whole space,are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems.The main feature in this article is to restore some compact results which are essential in variational methods.Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting.This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

The Physical Transient Spectrum for a Multi-Photon V-Type Three-Level Atom Interacting with a Squeezed Coherent Field in the Presence of Nonlinearities

We study the interaction of a multi-photon three-level atom with a single mode field in a cavity, taking explicitly into account the existence of forms of nonlinearities of both the field and the intensity-dependent atom-field coupling. The analytical forms of the emission spectrum is calculated using the dressed states of the system. The effects of photon multiplicities, mean photon number, detuning, Kerr-like medium and the intensity-dependent coupling functional on the emission spectrum are analyzed.

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities

In this paper, we studied the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic system. We prove the existence of at least four positive solutions for a semilinear elliptic system involving concave and convex nonlinearities by using the Nehari manifold and the center mass function.

Adaptive RBF neural network control of robot with actuator nonlinearities

In this paper, an adaptive neural network control scheme for robot manipulators with actuator nonlinearities is presented. The control scheme consists of an adaptive neural network controller and an actuator nonlinearities compensator. Since the actuator nonlinearities are usually included in the robot driving motor, a compensator using radial basis function (RBF) network is proposed to estimate the actuator nonlinearities and eliminate their effects. Subsequently, an adaptive neural network controller that neither requires the evaluation of inverse dynamical model nor the time-consuming training process is given. In addition, GL matrix and its product operator are introduced to help prove the stability of the closed control system. Considering the adaptive neural network controller and the RBF network compensator as the whole control scheme, the closed-loop system is proved to be uniformly ultimately bounded (UUB). The whole scheme provides a general procedure to control the robot manipulators with actuator nonlinearities. Simulation results verify the effectiveness of the designed scheme and the theoretical discussion.

SOLUTIONS TO ELLIPTIC SYSTEMS INVOLVING DOUBLY CRITICAL NONLINEARITIES AND HARDY-TYPE POTENTIALS

In this article, an elliptic system is investigated, which involves Hardy-type potentials, critical Sobolev-type nonlinearities, and critical Hardy-Sobolev-type nonlinearities. By a variational global-compactness argument, the Palais-Smale sequences of related approximation problems is analyzed and the existence of infinitely many solutions to the system is established.