Higher-order Nonlinear Terms Research Articles

应用(<i>G'</i>/<i>G<sup>2</sup></i>)展开法求解含三阶色散项的薛定谔方程

在当今的信息社会中，信息量以指数级增长，传统的通信技术已经不能满足社会需要，所以新一代的高速率、大传输容量的高速光纤通信便成为了最理想方案。而在光纤通信领域中对超短脉冲传输的研究更有其实际的意义。经理论分析，光纤色散的高阶效应对超短脉冲传输的影响不可忽略，需用含三次或五次高阶非线性项的薛定谔方程来描述其传输规律。本文将应用(G'/G2)展开法求解含三阶色散项的非线性薛定谔方程的解析解，通过求解方程得到了方程的在取不同参数条件时的许多新解。相信本文对理解方程的物理意义及参数条件对孤子解的影响，对未来光纤孤子通信的研究具有参考价值。 In today’s information society, information is growing exponentially; the traditional communication technology has been unable to meet the needs of society, so the high speed optical fiber communication speed, a new generation of large transmission capacity has become the most ideal solution. But it has more practical significance on ultrashort pulse transmission research in the field of optical communication. By theoretical analysis, high order dispersion effect on the propagation of ultrashort pulse cannot be ignored, with three or five times with high order nonlinear term Schrodinger equations to describe the transmission rule. This paper applies the analytical method for solving nonlinear Schrodinger equations of the three order dispersion term expansion, by solving many new equations are obtained for different parameters in the condition of equation. To understand equations and parameters influence on soliton solutions and the future study on optical soliton, this physical meaning has reference value.

Nonlinear stabilization of high-energy and ultrashort pulses in passively modelocked lasers

The two most commonly used models for passively modelocked lasers with fast saturable absorbers are the Haus modelocking equation (HME) and the cubic-quintic modelocking equation (CQME). The HME corresponds to a special limit of the CQME in which only a cubic nonlinearity in the fast saturable absorber is kept in the model. Here, we use singular perturbation theory to demonstrate that the CQME has a stable high-energy solution for an arbitrarily small but non-zero quintic contribution to the fast saturable absorber. As a consequence, we find that the CQME predicts the existence of stable modelocked pulses when the cubic nonlinearity is orders of magnitude larger than the value at which the HME predicts that modelocked pulses become unstable. This intrinsically larger stability range is consistent with experiments. Our results suggest a possible path to obtain high-energy and ultrashort pulses by fine tuning the higher-order nonlinear terms in the fast saturable absorber.

Thermomagnetic instabilities in a vertical layer of ferrofluid: nonlinear analysis away from a critical point

A finite amplitude instability has been analysed to discover the exact mechanism leading to the appearance of stationary magnetoconvection patterns in a vertical layer of a non-conducting ferrofluid heated from the side and placed in an external magnetic field perpendicular to the walls. The physical results have been obtained using a version of a weakly nonlinear analysis that is based on the disturbance amplitude expansion. It enables a low-dimensional reduction of a full nonlinear problem in supercritical regimes away from a bifurcation point. The details of the reduction are given in comparison with traditional small-parameter expansions. It is also demonstrated that Squire’s transformation can be introduced for higher-order nonlinear terms thus reducing the full three-dimensional problem to its equivalent two-dimensional counterpart and enabling significant computational savings. The full three-dimensional instability patterns are subsequently recovered using the inverse transforms The analysed stationary thermomagnetic instability is shown to occur as a result of a supercritical pitchfork bifurcation.

Derivation of coupled envelope propagation equations in silicon-based photonic crystal waveguides

This paper presents a coupled-mode theoretical framework for deriving the propagation equations of the modal envelope functions in a silicon photonic crystal waveguide (PCW), taking into account the influence of linear coupling, Raman scattering, Kerr effect, two-photon absorption, free-carrier index change, and free-carrier absorption. Compared to previous work in the literature, we include the three-dimensional nature of the carrier density in our calculations, analyze the influence of carrier dynamics in the coupling of the modes, and show how higher-order nonlinear terms can be derived. The framework presented here is useful for analyzing a number of signal-processing schemes based on nonlinear PCWs.

Numerical investigation on the nonlinear flexural behaviour of wrapped glass/epoxy laminated composite panel and experimental validation

In this work, the geometrically nonlinear deflection responses of glass/epoxy composite flat/curved shell panel structure have been analysed theoretically with the help of three different displacement field kinematics and Green–Lagrange strain–displacement relation. In this analysis, the numerical models are developed based on two higher-order shear deformation mid-plane kinematics and one simulation model with the help of commercial finite element package (ANSYS). The present mathematical model is general in the sense that it includes all the nonlinear higher-order terms arising due to Green–Lagrange strain–displacement relation to capture the exact flexural strength of the laminated structure. The present nonlinear model is so generic that it can be easily extended for solving different kinds of geometrical configurations (spherical, cylindrical, elliptical, hyperboloid and plate). The equilibrium equation of the transversely loaded panel is achieved by minimizing total potential energy expression and discretized using the suitable finite element steps. The required deflection values are computed numerically via a homemade MATLAB code in conjunction with Picard’s iterative method. Consequently, the stability of the present numerical solutions has been established through the convergence test and validated by comparing the results with those available published results. In addition, the transverse deflections are obtained experimentally via three-point bend test and utilized for the comparison purpose to demonstrate the significance of the newly developed higher-order finite element model.

Supratransmission in a disordered nonlinear periodic structure

We study the interaction among dispersion, nonlinearity, and disorder effects in the context of wave transmission through a discrete periodic structure, subjected to continuous harmonic excitation in its stop band. We consider a damped nonlinear periodic structure of finite length with disorder. Disorder is introduced throughout the structure by small changes in the stiffness parameters drawn from a uniform statistical distribution. Dispersion effects forbid wave transmission within the stop band of the linear periodic structure. However, nonlinearity leads to supratransmission phenomenon, by which enhanced wave transmission occurs within the stop band of the periodic structure when forced at an amplitude exceeding a certain threshold. The frequency components of the transmitted waves lie within the pass band of the linear structure, where disorder is known to cause Anderson localization. There is therefore a competition between dispersion, nonlinearity, and disorder in the context of supratransmission. We show that supratransmission persists in the presence of disorder. The influence of disorder decreases in general as the forcing frequency moves away from the pass band edge, reminiscent of dispersion effects subsuming disorder effects in linear periodic structures. We compute the dependence of the supratransmission force threshold on nonlinearity and strength of coupling between units. We observe that nonlinear forces are confined to the driven unit for weakly coupled systems. This observation, together with the truncation of higher-order nonlinear terms, permits us to develop closed-form expressions for the supratransmission force threshold. In sum, in the frequency range studied here, disorder does not influence the supratransmission force threshold in the ensemble-average sense, but it does reduce the average transmitted wave energy.

Solitons and Other Solutions for the Generalized KdV–mKdV Equation with Higher-order Nonlinear Terms

Solitons and Other Solutions for the Generalized KdV–mKdV Equation with Higher-order Nonlinear Terms

Nonlinear Flexural Analysis of Laminated Composite Panel Under Hygro-Thermo-Mechanical Loading — A Micromechanical Approach

In this paper, the geometrically nonlinear transverse bending behavior of the shear deformable laminated composite spherical shell panel under hygro-thermo-mechanical loading is investigated. The laminated composite panel model has been developed mathematically based on the higher-order shear deformation theory and Green–Lagrange nonlinear strain kinematics. The material model is introduced through the micromechanical approach explicitly in terms of the matrix properties, the fiber properties, and the fiber volume fractions. Theoretical formulation is developed based on the hygrothermal dependent composite material properties to evaluate the corrugated behavior of the laminated structure. In addition to that, all the nonlinear higher-order terms arising in the strain displacement relation are included in the present formulation for accurate prediction of the flexural behavior. The desired nonlinear governing equations are obtained using the variational method and discretized with the help of suitable finite element steps. The desired responses are computed by solving the nonlinear equations numerically using the direct iterative method. The convergence behavior of the developed nonlinear numerical model has been checked and validated by comparing the responses with those available published literature. Finally, the effect of hygrothermal environment, geometrical and material parameters and the support conditions on the transverse bending behavior of the laminated composite curved shell panel have been highlighted by solving different numerical examples.

Nonlinear hygro-thermo-elastic vibration analysis of doubly curved composite shell panel using finite element micromechanical model

ABSTRACTNonlinear free vibration behavior of laminated composite curved panel under hygrothermal environment is investigated in this article. The mathematical model of the laminated panel is developed using Green–Lagrange-type geometrical nonlinearity in the framework of higher-order mid-plane kinematics. The corrugated composite properties are evaluated through the micromechanical model and all the nonlinear higher-order terms are included in the present model for the sake of generality. The equation of vibrated panel is obtained using Hamilton's principle and discretized with the help of the finite element steps. The solutions are computed numerically using the direct iterative method. The effect of parameters on the nonlinear vibration responses is examined thoroughly by solving the wide variety of numerical examples.

The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures

The main purpose of this article is to demonstrate that bursting oscillations can be observed not only in the slow–fast autonomous dynamical systems with multiple scales associated with time domain, but also in the non-autonomous dynamical systems with periodic excitations when an order gap exists between the exciting frequency and the natural frequency, implying multiple scales in frequency domain. Furthermore, we try to investigate the influence of different codimensional bifurcations between the quiescent states (QS) and repetitive spiking states (SP) and the nonlinear structures with different equilibrium branches on the bursting oscillations. By introducing an inductor as well as a periodically changed electrical current source in a traditional Chua’s circuit and taking suitable parameter values, a modified four-dimensional periodically excited oscillator with multiple scales in frequency domain is established. Bursting oscillations for two cases with nonlinear terms up to third and fifth order with codimension-1 and codimension-2 bifurcations have been explored, respectively. It is found that more equilibrium states may exist when higher order nonlinear terms are introduced in the vector field, which may cause multiple quiescent states, and accordingly, multiple forms of repetitive spiking oscillations in one bursting attractor, leading to more complicated bursting phenomena. Furthermore, instead of jumping from one stable equilibrium branch to settle down to another stable equilibrium branch when codimension-1 bifurcations (fold bifurcations) exist between QSs and SPs, codimension-2 bifurcation (fold-Hopf bifurcation) may cause QS approximately located on one stable equilibrium branch to jump to repetitive spiking oscillations surrounding another stable equilibrium branch of the generalized autonomous system.

Large amplitude bending behaviour of laminated composite curved panels

Purpose– The purpose of this paper is to analyse the nonlinear flexural behaviour of laminated curved panel under uniformly distributed load. The study has been extended to analyse different types of shell panels by employing the newly developed nonlinear mathematical model.Design/methodology/approach– The authors have developed a novel nonlinear mathematical model based on the higher order shear deformation theory for laminated curved panel by taking the geometric nonlinearity in Green-Lagrange sense. In addition to that all the nonlinear higher order terms are considered in the present formulation for more accurate prediction of the flexural behaviour of laminated panels. The sets of nonlinear governing equations are obtained using variational principle and discretised using nonlinear finite element steps. Finally, the nonlinear responses are computed through the direct iterative method for shell panels of various geometries (spherical/cylindrical/hyperboloid/elliptical).Findings– The importance of the present numerical model for small strain large deformation problems has been demonstrated through the convergence and the comparison studies. The results give insight into the laminated composite panel behaviour under mechanical loading and their deformation behaviour. The effects of different design parameters and the shell geometries on the flexural responses of the laminated curved structures are analysed in detailed. It is also observed that the present numerical model are realistic in nature as compared to other available mathematical model for the nonlinear analysis of the laminated structure.Originality/value– A novel nonlinear mathematical model is developed first time to address the severe geometrical nonlinearity for curved laminated structures. The outcome from this paper can be utilized for the design of the laminated structures under real life circumstances.

Weakly nonlinear stability analysis of subcritical electrohydrodynamic flow subject to strong unipolar injection

We analyse in detail the weakly nonlinear stability of electrohydrodynamic (EHD) flow of insulating fluids subject to strong unipolar injection, with and without cross-flow. We first consider the hydrostatic electroconvetion induced by a Coulomb force confined between two infinite flat electrodes, taking into account the charge diffusion effect. The effects of various non-dimensionalized parameters are examined in order to depict in detail and to understand better the subcritical bifurcation of hydrostatic electroconvetion. In addition, electrohydrodynamics with low- or high-$Re$cross-flow is also considered for investigating the combined effect of inertia and the electric field. It is found that the base cross-flow is modified by the electric effect and that, even when the inertia is dominating, the electric field can still strengthen effectively the subcritical characteristics of canonical channel flow. In this process, however, the electric field does not contribute directly to the subcriticality of the resultant flow and the intensified subcritical feature of such flow is thus entirely due to the modified hydrodynamic field as a result of the imposed electric field. This finding might be important for flow control strategies involving an electric field. Theoretically, the above results are obtained from a multiple-scale expansion method, which gives rise to the Ginzburg–Landau equation governing the amplitude of the first-order perturbation. The conclusions are deduced by probing the changes of value of the coefficients in this equation. In particular, the sign of the first Landau coefficient indicates the type of bifurcation, being subcritical or supercritical. Moreover, as a quintic-order Ginzburg–Landau equation is derived, the effects of higher-order nonlinear terms in EHD flow are also discussed.

Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field

The nonlinear free vibration behavior of functionally graded carbon nanotube reinforced composite flat panel is investigated using temperature dependent material properties for different grading. The carbon nanotube reinforced composite flat panel model has been developed mathematically using the higher-order shear deformation theory and Green-Lagrange nonlinearity. The present mathematical model has included all the nonlinear higher-order terms for the sake of generality. Further, the responses are computed numerically using the direct iteration method in conjunction with the finite element method. The validity and the convergence behavior of the present nonlinear model have been checked and the effect of different design parameters on the nonlinear vibration responses are discussed in detail.

Hygrothermal effects on the flexural strength of laminated composite cylindrical panels

In this article, the effect of hygrothermal conditions on the bending behaviour of laminated composite cylindrical panels has been investigated by using a nonlinear finite element micromechanical model. The panel model has been developed mathematically by introducing the geometrical nonlinearity in Green-Lagrange sense in the framework of the higher-order shear deformation theory. In order to achieve any general case, all the nonlinear higher order terms have been incorporated in the present formulation and the lamina material property variations due to hygrothermal effect are included via the micromechanical model. The governing equation is obtained using the variational principle and discretised using nonlinear finite element steps. The convergence behaviour of the present numerical model has been established and validated by comparing the responses with the existing results in the literature. The effects of thickness ratio, lamination scheme and support conditions under different sets of hygrothermal conditions are investigated in details.

Nonlinear Flexural Analysis of Shallow Carbon/Epoxy Laminated Composite Curved Panels: Experimental and Numerical Investigation

AbstractIn this work, the nonlinear flexural behavior of laminated carbon/epoxy composite panels is investigated numerically using a generalized nonlinear mathematical model based on two higher-order shear deformation midplane kinematics and Green-Lagrange type geometrical nonlinearity. The exact flexural behavior of the laminated panel is computed by considering all the nonlinear higher order terms in the present mathematical model. The nonlinear governing equations are obtained using variational principles and discretized through suitable finite-element steps. The desired nonlinear responses are computed numerically using the direct iterative method. The proposed nonlinear models have been validated by comparing the responses with those available in published literature and the experiment (three-point bend test) as well. In addition, the linear and nonlinear flexural responses of the laminated carbon/epoxy flat panel are also computed using ANSYS 13.0 simulation finite element analysis package. Finally,...

A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

Observed avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models.

Approximate Linearization of a Class of Underactuated System and its Application in the Ball and Beam System

In this paper, we consider the dynamical model of a class of underactuated systems. By combination of the partial fedback linearization and Yamada’s global linearization, we deduced a global approximate linearization method for underactuated systems. By using this method, the dynamical equations can be transformed into an state eqution that is expressed as a pseudolinear term with Brunovsky canonical form plus a high order nonlinear term, where the nonlinear term is high order on the equilibrium manifold of the system. By standard nonlinear feedback method, the system is transformed into the sum of a stable linear term and a high order nonlinear term. Take proper feedforward value as the input to reduce the influence of the nonlinear term to the system and thus the underactuated system can be regulated. This method is applied to the ball and beam system and simulation results show that the proposed approximate linearization method is effective for setpoint control.

Large amplitude free vibration analysis of laminated composite spherical shells embedded with piezoelectric layers

Numerical analysis of large amplitude free vibration behaviour of laminated composite spherical shell panel embedded with the piezoelectric layer is presented in this article. For the investigation purpose, a general nonlinear mathematical model has been developed using higher order shear deformation mid-plane kinematics and Green-Lagrange nonlinearity. In addition, all the nonlinear higher order terms are included in the present mathematical model to achieve any general case. The nonlinear governing equation of freely vibrated shell panel is obtained using Hamilton\'s principle and discretised using isoparametric finite element steps. The desired nonlinear solutions are computed numerically through a direct iterative method. The validity of present nonlinear model has been checked by comparing the responses to those available published literature. In order to examine the efficacy and applicability of the present developed model, few numerical examples are solved for different geometrical parameters (fibre orientation, thickness ratio, aspect ratio, curvature ratio, support conditions and amplitude ratio) with and/or without piezo embedded layers and discussed in details.

Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms.

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.