Discrete Nonlinear Research Articles

Dynamics of the Infinite Discrete Nonlinear Schrödinger Equation

The discrete nonlinear Schrödinger equation on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb Z}^d$$\\end{document}, d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 1$$\\end{document} is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the ℓ2(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2({\\mathbb Z}^d)$$\\end{document}-norm, the well-posedness of the corresponding Cauchy problem follows for square-summable initial data. In this paper, we prove that the well-posedness continues to hold for initial data that can grow towards infinity, namely anything that has at most a certain power law growth far away from the origin. The growth condition is loose enough to guarantee that, at least in dimension d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, initial data sampled from any reasonable equilibrium distribution of the defocusing DNLS satisfies it almost surely.

The Dynamic Event-Based Non-Fragile H∞ State Estimation for Discrete Nonlinear Systems with Dynamical Bias and Fading Measurement

The present study investigates non-fragile H∞ state estimation based on a dynamic event-triggered mechanism for a class of discrete time-varying nonlinear systems subject to dynamical bias and fading measurements. The dynamic deviation caused by unknown inputs is represented by a dynamic equation with bounded noise. Subsequently, the augmentation technique is employed and the dynamic event-triggered mechanism is introduced in the sensor-to-estimator channel to determine whether data should be transmitted or not, thereby conserving resources. Furthermore, an augmented state-dependent non-fragile state estimator is constructed considering gain perturbation of the estimator and fading measurements during network transmission. Sufficient conditions are provided based on Lyapunov stability and matrix analysis techniques to ensure exponential mean-square stability of the estimation error system while satisfying the H∞ disturbance fading level. The desired estimator gain matrix can be obtained by solving the linear matrix inequality (LMI). Finally, an example is presented to illustrate the effectiveness of the proposed method for designing estimators.

The Discrete Nonlinear Schrödinger Equation with Linear Gain and Nonlinear Loss: The Infinite Lattice with Nonzero Boundary Conditions and Its Finite-Dimensional Approximations

The Discrete Nonlinear Schrödinger Equation with Linear Gain and Nonlinear Loss: The Infinite Lattice with Nonzero Boundary Conditions and Its Finite-Dimensional Approximations

The Ground State Solutions of Discrete Nonlinear Schrödinger Equations with Hardy Weights

The Ground State Solutions of Discrete Nonlinear Schrödinger Equations with Hardy Weights

PROPAGATION OF SOLITON IN A ONE-DIMENSIONAL DISCRETE SYSTEM WITHIN SELF-ACTION MODE

Abstract. This study investigates the soliton propagation in a one-dimensional discrete system characterized by the Discrete Nonlinear Schrödinger Equation (DNLSE). The DNLSE is a fundamental model in wave phenomena, encompassing a broad spectrum of physical systems ranging from optics to fluid dynamics. The study uses both analytical and numerical computations to comprehensively observe the process. Through the analytical approach, i.e., the variation approximation (VA) method, essential parameters governing soliton evolutions, such as width, center-of-mass position, and linear and quadratic phase-front corrections are determined and graphically interpreted. These results are compared against direct numerical simulations of the main equation for validation. The results show that an increase in linear phase-front correction corresponds to an increase in both the soliton’s initial velocity and propagation distance. Additionally, direct numerical simulation reveals the increasing prominence of the discreteness effects with higher initial velocities.

Design of Discrete and Hybrid Nonlinear Control Systems

In this article the new method of discrete control systems design for nonlinear plants with differentiable nonlinearities is suggested. The increasing demands on the quality of control processes and the widespread use of computer technology provide ample opportunities for the design and implementation of digital control systems. However, discrete models of control plants are needed to solve this problem. In the case of linear plants, such models are created on the basis of z-transformation, Euler or Tustin formulas. In the case of nonlinear plants, these transformations are not applicable, so a large number of approximate discretization methods have been developed to date. Euler and Runge-Kutt transformations are used for these purposes most often, but they lead to satisfactory results only with very small period of discretization. In the case of automatic control systems, this requires the use of digital automation tools with very high speed, which is often economically impractical. Methods of discretization with a long period were most often developed on the basis of decomposition into series of the right-hand sides of the differential equations, transformed on Euler. Here, firstly, the problem of selecting the number of the series members, which to be retained arises, and secondly, already in the third or fourth order of the plant, the calculating ratios turn out to be extremely complex. The discretization method suggested below differs in that it is not the equations of nonlinear plants in the Cauchy form that are discretized, but the corresponding quasilinear model. In this case, a modified trapezoid method is used, and the discretization purpose is not the most accurate approximation of the original equations of the plant, but the stability of a closed nonlinear control system with rather big period. This system is designed using the algebraic polynomial-matrix method for designing of the nonlinear control systems. As a result, a hybrid nonlinear system with fairly simple algebraic calculation expressions is formed. The suggested approach makes it possible to create the control systems for nonlinear controlled plants using conventional computational automation tools.

Primary Frequency Control of Wind-solar-storage Power Station Considering Discrete Control Nonlinearity

With the gradual advancement of dual-carbon goals, the wind-solar-storage power station has become the mainstream trend in constructing new energy stations due to their wind energy and luminous complementary characteristics. However, the current research on primary frequency control of new energy stations is still focused on a single-machine strategy. The communication delay and control period caused by long control links are not considered. It is difficult to describe the influence of the difference in dynamic characteristics of wind-solar-storage power on primary frequency control performance. Therefore, this paper takes the multi-layer control link wind-solar-storage power station as the research object, establishes the nonlinear control model considering the influence of control delay and control period, and puts forward the wind-solar-storage power plant primary frequency control method, which is suitable for participating in the frequency control of power grid.

Statistical Solution for the Nonlocal Discrete Nonlinear Schrödinger Equation

Statistical Solution for the Nonlocal Discrete Nonlinear Schrödinger Equation

A new look at the Equivalent Linearization Technique to predict LCO in aeroelastic systems with discrete nonlinearities

Control surfaces nonlinearities can lead to limit cycle oscillations (LCO). Several methods have been proposed to predict LCO, such as Harmonic Balance-based methods (HB). Describing function (DF) is the HB with a single harmonic motion assumed, and the approach can be combined with a classic eigenvalue stability analysis via the Equivalent Linearization Technique (ELT) to predict LCO. On the other hand, High-Order Harmonic Balance (HOHB) methods consider higher number of harmonics, but they lead to a more complex nonlinear algebraic system of equations. This paper introduces a new look at the ELT combining both eigenvalue analysis and describing functions. Two applications are considered. The first one is a new DF written in a matrix form for the ELT to consider both first and third harmonics in LCOs due to freeplay. The second application is an iterative procedure to combine the eigenvalue analysis with a describing function to predict LCOs in systems with both freeplay and friction nonlinearities. Numerical simulations are performed for the aeroelastic typical section airfoil. The results show that the new DF improves the classic DF by providing a more accurate prediction of LCOs that accounts for third harmonic. Also, the iterative-ELT is shown as an excellent predictor of LCO for systems with both freeplay and friction.

A Hybrid Model Combining Discrete Wavelet Transform and Nonlinear Autoregressive Neural Network for Stock Price Prediction: An Application in the Egyptian Exchange

Forecasting stock prices is crucial for successful investment in financial markets. However, it is challenging due to the nonlinearity and high volatility caused by various factors influencing price movements. This paper proposes a hybrid model that integrates the discrete wavelet transform (DWT) with the nonlinear autoregressive neural network (NARNN) to predict stock prices. Following the division of stock prices into training and testing sets, the DWT decomposes the training set into low- and high-frequency components reducing the noise and lessening the data's nonlinearity. Then, the obtained components are used to train the NARNNs. To predict the future components, the model decomposes the preceding available prices at each time step and utilizes the latest eight points as input to the NARNNs. Eventually, NARNNs' outputs are combined to provide the final predicted prices. In previous works, the entire dataset is first decomposed and then partitioned into training and testing sets. This unrealistic approach causes the testing set to inherit information regarding stocks' future performance, leading to optimistic deceptive results. Twenty-four stocks from the Egyptian Exchange (EGX-30) are utilized to validate the proposed model's performance. The DWT-NARNN model is compared against other methods, and the empirical findings show that it performs the best.

Distributed Motion Planning for Multiple Quadrotors in Presence of Wind Gusts

This work demonstrates distributed motion planning for multi-rotor unmanned aerial vehicle in a windy outdoor environment. The motion planning is modeled as a receding horizon mixed integer nonlinear programming (RH-MINLP) problem. Each quadrotor solves an RH-MINLP to generate its time-optimal speed profile along a minimum snap spline path while satisfying constraints on kinematics, dynamics, communication connectivity, and collision avoidance. The presence of wind disturbances causes the motion planner to continuously regenerate new motion plans, thereby significantly increasing the computational time and possibly leading to safety violations. Control Barrier Functions (CBFs) are used for assist in collision avoidance in the face of wind disturbances while alleviating the need to recalculate the motion plans continually. The RH-MINLPs are solved using a novel combination of heuristic and optimal methods, namely Simulated Annealing and interior-point methods, respectively, to handle discrete variables and nonlinearities in real-time feasibly. The framework is validated in simulations featuring up to 50 quadrotors and Hardware-in-the-loop (HWIL) experiments, followed by outdoor field tests featuring up to 6 DJI M100 quadrotors. Results demonstrate (1) fast online motion planning for outdoor communication-centric multi-quadrotor operations and (2) the utility of CBFs in providing effective motion plans.

Special Issue Editorial: “Discrete and Continuous Memristive Nonlinear Systems and Symmetry”

Memristor, as the fourth basic electronic component, was first reported by Chua in 1971 [...]

A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.

Analisis Kestabilan Solusi Soliton pada Persamaan Schrodinger Nonlinier Diskrit Nonlokal

In this paper, the Nonlocal Discrete Nonlinear Schrodinger (DNLS) equation that interpolates the Nonlocal Ablowitz-Ladik DNLS and the Nonlocal Cubic DNLS equations and its stability are studied in detail. The solution of the Nonlocal SNLD equation is a soliton wave in the form of a Gaussian ansatz obtained using the method of Variational Approximation (VA). The stability of the solution is also analyzed using the VA. These semi-analytical results are then compared to numerical results. The soliton and its stability obtained via VA is concluded to be having a fairly good conformity with numerical results.

The closeness of localized structures between the Ablowitz–Ladik lattice and discrete nonlinear Schrödinger equations: Generalized AL and DNLS systems

The Ablowitz–Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localized solitons to rational solutions in the form of the spatiotemporally localized discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz–Ladik system and a wide class of Discrete Nonlinear Schrödinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in nonintegrable lattices for significantly large times. Nonintegrable systems exhibiting such behavior include a generalization of the Ablowitz–Ladik system with power-law nonlinearity and the discrete nonlinear Schrödinger equation with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates, in excellent agreement with the analytical results, the persistence of small amplitude Ablowitz–Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton.

Connecting Analog and Discrete Nonlinear Systems for Noise Generation

Abstract Nonlinear systems exhibit complex dynamic behaviour, including quasi-periodic and chaotic. The present contribution presents a composed analogue and discrete-time structure, based on second-order nonlinear building blocks with periodic oscillatory behaviour, that can be used for complex signal generation. The chosen feedback connection of the two modules aims at obtaining a more complex nonlinear dynamic behaviour than that of the building blocks. Performing a parameter scan, it is highlighted that the resulting nonlinear system has a quasi-periodic behaviour for large ranges of parameter values. The nonlinear system attractor projections are obtained by simulation and statistical numerical results are presented, both confirming the possible use of the designed system as a noise generator.

Strongly nonlinear wave dynamics of continuum phononic materials with periodic rough contacts.

We investigate strongly nonlinear wave dynamics of continuum phononic material with discrete nonlinearity. The studied phononic material is a layered medium such that the elastic layers are connected through contact interfaces with rough surfaces. These contacts exhibit nonlinearity by virtue of nonlinear mechanical deformation of roughness under compressive loads and strong nonlinearity stemming from their inability to support tensile loads. We study the evolution of propagating Gaussian tone bursts using time-domain finite element simulations. The elastodynamic effects of nonlinearly coupled layers enable strongly nonlinear energy transfer in the frequency domain by activating acoustic resonances of the layers. Further, the interplay of strong nonlinearity and dispersion in our phononic material forms stegotons, which are solitarylike localized traveling waves. These stegotons satisfy properties of solitary waves, yet exhibit local variations in their spatial profiles and amplitudes due to the presence of layers. We also elucidate the role of rough contact nonlinearity on the interrelationship between the stegoton parameters as well as on the generation of secondary stegotons from the collision of counterpropagating stegotons. The phononic material exhibits strong acoustic attenuation at frequencies close to (and fractional multiples of) layer resonances, whereas it causes energy propagation as stegotons for other frequencies. This study sheds light on the wave phenomena achievable in continuum periodic media with local nonlinearity, and opens opportunities for advanced wave control through discrete and local contact nonlinearity.

The closeness of the Ablowitz-Ladik lattice to the Discrete Nonlinear Schrödinger equation

While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schrödinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a “continuous dependence” on their initial data in the l2 and l∞ metrics. The most striking relevance of the analytical results is that small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schrödinger one. It is shown that the closeness results are also valid in higher dimensional lattices, as well as, for generalised nonlinearities. For illustration of the applicability of the approach, a brief numerical study is included, showing that when the 1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schrödinger system with cubic or saturable nonlinearity, it persists for long-times. Thereby, excellent agreement of the numerical findings with the theoretical predictions is obtained.

Equilibrium and nonequilibrium description of negative temperature states in a one-dimensional lattice using a wave kinetic approach.

We predict negative temperature states in the discrete nonlinear Schödinger (DNLS) equation as exact solutions of the associated wave kinetic equation. Within the wave kinetic approach, we define an entropy that results monotonic in time and reaches a stationary state, that is consistent with classical equilibrium statistical mechanics. We also perform a detailed analysis of the fluctuations of the actions at fixed wave numbers around their mean values. We give evidence that such fluctuations relax to their equilibrium behavior on a shorter timescale than the one needed for the spectrum to reach the equilibrium state. Numerical simulations of the DNLS equation are shown to be in agreement with our theoretical results. The key ingredient for observing negative temperatures in lattices characterized by two invariants is the boundedness of the dispersion relation.

Rogue Waves generation by Using higher order rational solutions of Discrete Nonlinear Schrödinger Equation

The higher-order discrete nonlinear Schrödinger equation has been demonstrated to be related to cubic-quintic nonlinearity. This executes the specified number of periodic solutions. The localized solution of the homogeneous cubic-quintic discrete nonlinear Schrödinger equation has been researched. By employing scaling transformations, it was possible to convert cubic-quintic discrete nonlinear Schrödinger equations to constant-coefficient higher-order rational solutions with discrete nonlinear Schrödinger equations (DNLSE). We investigated our solution in the periodic form of the gain/loss term and discovered some intriguing results. Even the solution is interconnected, and its features are extensively exploited in a variety of physical phenomena, including optical waveguides, B-E condensates in optical media, non-linearity in photonic crystals, Kerr-type nonlinearity and photo refracting medium.