Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction,which may have potential applications in electric circuits.Nonlocal infinitely many conservation laws are constructed based on its Lax pair.Nonlocal discrete generalized(m,N?m)-fold Darboux transformation is extended and applied to solve this system.As an application of the method,we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized(1,N?1)-fold Darboux transformation,respectively.By using the asymptotic and graphic analysis,structures of one-,two-,three-and four-soliton solutions are shown and discussed graphically.We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures.It is shown that the soliton structures are quite different between discrete local and nonlocal systems.Results given in this paper may be helpful for understanding the electrical signals propagation.     

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.     

Effects of functionally graded materials on dynamics of molecular bond clusters

Unlike nonspecific adhesion of conventional hard materials in engineering commonly described by JKR and DMT type models,the molecular adhesion via specific receptor-ligand bonds is stochastic by nature and has the feature that its strength strongly depends on the medium stiffness surrounding the adhesion.In this paper,we demonstrate in a stochastic-elasticity framework that a type of materials with linearly graded elastic modulus can be designed to achieve "equal load sharing" and enhanced cooperative rebinding among interfacial molecular bonds.Upon modulus gradation,multiple molecular bonds can be elastically decoupled but statistically cooperative.In general,uniform molecular adhesion can be accomplished by two strategies:homogeneous materials with sufficient stiffness higher than a threshold or heterogeneous materials satisfying the criterion on modulus gradation.These results not only provide a theoretical principle for possible applications of functionally graded materials in quantitatively controlling cell-matrix adhesion,but also have general implications on adhesion between soft materials mediated by specific molecular binding.     

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.     

Correlating surface permeability with intracrystalline diffusivity in nanoporous solids

The rates of uptake and release of guest molecules in nanoporous solids are often strongly influenced or even controlled by transport resistances at the external surface ("surface barriers") rather than by intraparticle diffusion, which was assumed to be rate controlling in many of the earlier kinetic studies. By correlating the surface resistance with the intracrystalline diffusivity, we develop here a microkinetic model which closely reproduces the experimentally observed results for short-chain alkanes in Zn(tbip), a member of the novel metal-organic framework family of nanoporous materials. It seems likely that this mechanism, which is shown to provide a rational explanation of the commonly observed discrepancies between "macro" and "micro" measurements of intracrystalline diffusion, may be fairly general.     

Length scales at which classical elasticity breaks down for various materials

At what characteristic length scale does classical continuum elasticity cease to accurately describe small deformation mechanical behavior? The two dominant physical mechanisms that lead to size dependency of elastic behavior at the nanoscale are surface energy effects and nonlocal interactions. The latter arises due to the discrete structure of matter and the fluctuations in the interatomic forces that are smeared out within the phenomenological elastic modulus at coarser sizes. While surface energy effects have been well characterized in the literature, little is known about the length scales at which nonlocal effects manifest for different materials. Using a combination of empirical molecular dynamics and lattice dynamics (empirical and ab initio), we provide estimates of nonlocal elasticity length scales for various classes of materials: semiconductors, metals, amorphous solids, and polymers.     

Shock-induced thermal wave propagation and response analysis of a viscoelastic thin plate under transient heating loads

This paper is devoted to the thermal shock analysis for viscoelastic materials under transient heating loads. The governing coupled equations with time-delay parameter and nonlocal scale parameter are derived based on the generalized thermo-viscoelasticity theory. The problem of a thin plate composed of viscoelastic material, subjected to a sudden temperature rise at the boundary plane, is solved by employing Laplace transformation techniques. The transient responses, i.e. temperature, displacement, stresses, heat flux as well as strain, are obtained and discussed. The effects of time-delay and nonlocal scale parameter on the transient responses are analyzed and discussed. It can be observed that: the propagation of thermal wave is dynamically smoothed and changed with the variation of time-delay; while the displacement, strain, and stress can be rapidly reduced by nonlocal scale parameter, which can be viewed as an important indicator for predicting the stiffness softening behavior for viscoelastic materials.     

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the bright-dark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.     

Multi-place physics and multi-place nonlocal systems

Multi-place nonlocal systems have attracted attention from many scientists. In this paper, we mainly review the recent progresses on two-place nonlocal systems (Alice-Bob systems) and four-place nonlocal models. Multi-place systems can firstly be derived from many physical problems by using a multiple scaling method with a discrete symmetry group including parity, time reversal, charge conjugates, rotations, field reversal and exchange transformations. Multi-place nonlocal systems can also be derived from the symmetry reductions of coupled nonlinear systems via discrete symmetry reductions. On the other hand, to solve multi-place nonlocal systems, one can use the symmetry-antisymmetry separation approach related to a suitable discrete symmetry group, such that the separated systems are coupled local ones. By using the separation method, all the known powerful methods used in local systems can be applied to nonlocal cases. In this review article, we take two-place and four-place nonlocal nonlinear Schrödinger (NLS) systems and Kadomtsev-Petviashvili (KP) equations as simple examples to explain how to derive and solve them. Some types of novel physical and mathematical points related to the nonlocal systems are especially emphasized.     

Experimental falsification of Leggett's nonlocal variable model

Bell's theorem guarantees that no model based on local variables can reproduce quantum correlations. Also, some models based on nonlocal variables, if subject to apparently "reasonable" constraints, may fail to reproduce quantum physics. In this Letter, we introduce a family of inequalities, which use a finite number of measurement settings, and which therefore allow testing Leggett's nonlocal model versus quantum physics. Our experimental data falsify Leggett's model and are in agreement with quantum predictions.     

Nonlinear and nonlocal dynamics of spatially extended systems: Stationary states, bifurcations and stability

Spatially extended systems with nonlocal dynamics (e.g. ferromagnetic resonance or current instability) of the type

with uε ⁿ will be studied near the soft-mode instability (wave number k_c ≠ 0) of a stationary and uniform state. An amplitude equation is derived within the framework of a multiple-scale perturbation theory. A particular example of this class of nonlocal dynamics is also treated numerically. As the main result we find that in contrast to the well-known supercritical bifurcation into a stable periodic state, the uniform state can bifurcate supercritically into a stationary state of an amplitude-modulated fast oscillation in space.     

Bound nonlocality and activation

We investigate nonlocality distillation using measures of nonlocality based on the Elitzur-Popescu-Rohrlich decomposition. For a certain number of copies of a given nonlocal box, we define two quantities of interest: (i) the nonlocal cost and (ii) the distillable nonlocality. We find that there exist boxes whose distillable nonlocality is strictly smaller than their nonlocal cost. Thus nonlocality displays a form of irreversibility which we term "bound nonlocality." Finally, we show that nonlocal distillability can be activated.     

Generalized Noether identities for nonlocal transformations

Starting from the transformation properties of an action integral of a system under local and nonlocal transformations, we derive the generalized Noether identities for a variant system under those transformations. The applications of the theory to the Yang-Mills field with higher order Lagrangian is presented under the Coulomb gauge condition, a new conserved PBRS charge is found which differs from the BRS conserved charge, and another conserved charge connected with nonlocal transformation is also obtained.     

Should quantum mechanical description of physical reality be considered complete?

A brief and critical survey of wave-particle duality and nonlocality aspects of light is presented. A recent attempt to establish a reasonable framework for nonlocal realistic theories based on physically sound arguments and a proposed experiment to decide between such theories and the usual interpretation of quantum mechanical formalism are reviewed. It is shown that a nonlocal realistic approach may raise some new questions which could be answered by means of a program based on a sequence of experiments.     

Planelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces

This paper contains three types of results:

• the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,
• the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,
• the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.

In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

     

From point defects in graphene to two-dimensional amorphous carbon

While crystalline two-dimensional materials have become an experimental reality during the past few years, an amorphous 2D material has not been reported before. Here, using electron irradiation we create an sp2-hybridized one-atom-thick flat carbon membrane with a random arrangement of polygons, including four-membered carbon rings. We show how the transformation occurs step by step by nucleation and growth of low-energy multivacancy structures constructed of rotated hexagons and other polygons. Our observations, along with first-principles calculations, provide new insights to the bonding behavior of carbon and dynamics of defects in graphene. The created domains possess a band gap, which may open new possibilities for engineering graphene-based electronic devices.     

State and parameter estimation in nonlinear systems as an optimal tracking problem

In verifying and validating models of nonlinear processes it is important to incorporate information from observations in an efficient manner. Using the idea of synchronization of nonlinear dynamical systems, we present a framework for connecting a data signal with a model in a way that minimizes the required coupling yet allows the estimation of unknown parameters in the model. The need to evaluate unknown parameters in models of nonlinear physical, biophysical, and engineering systems occurs throughout the development of phenomenological or reduced models of dynamics. Our approach builds on existing work that uses synchronization as a tool for parameter estimation. We address some of the critical issues in that work and provide a practical framework for finding an accurate solution. In particular, we show the equivalence of this problem to that of tracking within an optimal control framework. This equivalence allows the application of powerful numerical methods that provide robust practical tools for model development and validation.     

Nonlocal Lagrange formalism in the thermodynamics of irreversible processes: variational procedures for kinetic equations

This paper is concerned with generalizations of the known local Lagrange formalism of first order. It will be applied to kinetic equations like the Fokker-Planck equation and the Boltzmann equation. In the latter case nonlocal methods are necessary from the very beginning. Nevertheless, in the framework of Fréchet's formalism the calculations are as easy as in the classical local case.Furthermore, a rather general entropy concept can be established within nonlocal Lagrange formalism for irreversible systems. As a main result of this paper we derive within our general concept the known entropy balances of the Boltzmann theory and the Fokker-Planck theory, respectively. It will be emphasized that our general concept may be applied to a very wide class of irreversible systems, in principle.     

N1-soliton solution for Schrödinger equation with competing weakly nonlocal and parabolic law nonlinearities

The nonlocal nonlinear Schrödinger equation (NNLSE) with competing weakly nonlocal nonlinearity and parabolic law nonlinearity is explored in the current work. A powerful integration tool, which is a modified form of the simple equation method, is used to construct the dark and singular 1-soliton solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical gadget for solving various types of NNLSEs.     

Dynamical Behavior of Solution in Integrable Nonlocal Lakshmanan-Porsezian-Daniel Equation

The integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation which has the higher-order terms (dispersions and nonlinear effects) is first introduced. We demonstrate the integrability of the nonlocal LPD equation, provide its Lax pair, and present its rational soliton solutions and self-potential function by using the degenerate Darboux transformation. From the numerical plots of solutions, the compression effects of the real refractive index profile and the gain-or-loss distribution produced by are discussed.