One-dimensional Discrete System Research Articles

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.

Image Encryption Schemes Based on a Class of Uniformly Distributed Chaotic Systems

This paper proposes a method to construct a one-dimensional discrete chaotic system. First, we define a generalized distance function to control the boundedness of the one-dimensional discrete system. Based on Marotto’s theorem, one-dimensional discrete systems are proven to be chaotic in the sense of Li–Yorke, and the corresponding chaos criterion theorem is proposed. The system can be distributed uniformly by adjusting the parameters. In this paper, we propose an image encryption scheme based on a uniformly distributed discrete chaotic system and DNA encoding. DNA encoding and decoding rules are determined by plain text. The experimental results demonstrate that our encryption algorithm has a large key space, high key sensitivity, and fast encryption speed and can resist differential and statistical attacks.

Energy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard-Jones (m, n) Interaction

It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the “step-bunching” phenomenon for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard-Jones (LJ)-type potential. It is characterized by two exponents m and n describing the singular and decaying behaviors of the interacting potential at small and large distances, and henceforth are called generalized LJ (m, n) potential. We provide a systematic analysis of the asymptotic properties of the step configurations and the value of the minimum energy, in particular their dependence on m and n and an additional parameter \(\alpha \) indicating the interaction range. Our results show that there is a phase transition between the bunching and non-bunching regimes. Moreover, some of our statements are applicable for any critical points of the energy, not necessarily minimizers. This work extends the technique and results of Luo et al. (SIAM Multiscale Model Simul 14(2):737–771, 2016) which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). As a by-product, our result also leads to the well-known fact that the classical LJ (6,12) potential does not demonstrate the step-bunching-type phenomenon.

Multiscale Mass-Spring Models of Carbon Nanotube Arrays Accounting for Mullins-like Behavior and Permanent Deformation

Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced to describe plasticity and damage in carbon nanotube (CNT) arrays. The energetics of the mechanical system are investigated analytically, the stress-strain law is derived, and the mechanical dissipation is computed, both for the discrete case as well as for the continuum limit. An information-passing approach is developed that permits the investigation of macroscopic portions of the material. As an application, the simulation of a cyclic compression experiment on real CNT foam is performed, considering both the material response during the primary loading path from the virgin state and the damaged response after preconditioning.

Pre-images of invariant sets of a discrete-time two-species competition model

In this paper, we explore the structure of pre-images of invariant sets of a discrete-time two-species competition model with singularity at the origin. We first show that this competition model is persistent with respect to the total population of two species, i.e. all initial conditions in are attracted to a compact set which is bounded away from the origin. Then we study the properties of pre-images of this system and give the explicit structure of all pre-images of invariant sets for this system under certain parameter range. These results are analogous to the one-dimensional discrete system. Our study is the first step to explore the structure of the basins of attractions of interior attractors of a general discrete-time two-species model (e.g. the locally asymptotically stable interior period-2 orbit). Finally, we discuss how our results give useful insights on the future study for coexistence of the species and list some open problems related to our system.

On the Existence Structure of One-dimensional Discrete Chaotic Systems

This paper investigates the existence and structure of one-dimensional discrete chaotic systems. It is expected to change the research perspective of discrete chaotic system, from being passively discovered to actively constructing. Firstly, some sufficient conditions guaranteeing a one-dimensional discrete system being chaotic are proposed and rigorously proved. Secondly, it presents a simple method for constructing the one-dimensional discrete chaotic system which can provide more models and inspecting tools for the study of the latter. Finally, by applying the method, some new discrete chaotic systems are constructed, and the numerical results with the bifurcation diagram and the Lyapunov exponent illustrate the chaos, which thus verifies the feasibility and effectiveness of the proposed method.

Ballistic Quantum Walk in a Discrete One-Dimensional System

A continuous-time quantum walk in a one-dimensional discrete system is considered. There are interactions \(\hbar v/\Delta x\) between adjacent sites, where Δ x denotes the lattice spacing. If the initial state is spread over N sites, the packet travels ballistically with velocity v for N ≫1.

Nonlinear Fano-Feshbach resonances

We study the wave scattering in an one-dimensional discrete system with two side-coupled defects. Each of the defects exhibits the Fano resonance as a resonant suppression of transmission, i.e., resonant reflection. We demonstrate that the interaction between two Fano resonances may give rise to a birth of a very narrow resonance. This effect may be understood in terms of the overlapping resonances, as suggested by Feshbach [Ann. Phys. 5, 357 (1958)]. We consider two cases, when the defects are coupled either locally or nonlocally to the discrete array. In the latter case, a sharp asymmetric resonance appears with a large quality factor. We demonstrate that by introducing a nonlinearity at side-coupled defects a closed loop in the nonlinear transmission coefficient may appear, which results in bistable response.

Phase Transitions in One-Dimensional Order—Disorder Systems: Application to Helix—Random-Coil Transition in Polymers

In this work a discussion is given of the present theory of the helix—random-coil transition in polypeptides. The seeming conflict between the description of this transition in terms of a one-dimensional model and the proofs that a phase transition cannot occur in a one-dimensional system is resolved. It is shown that the kind of interaction introduced in such models is singular and therefore falls outside the class for which the aforementioned proofs hold. A discussion is given of the mechanism of helix formation in the actual polymers, and it is shown how the description of the transition requires the use of such interactions. The properties which the singular interaction must have in order to cause a phase transition to occur in a one-dimensional discrete system are precisely stated and illustrated by simple examples. A general treatment of one-dimensional discrete systems with an arbitrary (but finite) range of interaction is given.