Nonlocal Interactions Research Articles

Advanced Analytical and Numerical Studies on Coupled Schrödinger Equations with Fractional Order Damping

This paper embarks on a thorough analytical and numerical exploration of coupled Schrödinger equations under the influence of fractional order damping mechanisms. By integrating fractional damping, which introduces memory effects and non-local dissipative interactions, into the coupled Schrödinger framework, we aim to dissect and understand the nuanced dynamics that govern these complex quantum systems. The research delves into the mathematical underpinnings, stability characteristics, and the dynamical behaviors that emerge from the intricate balance between quantum coupling and fractional damping effects. Through a blend of analytical rigor and sophisticated numerical simulations, this study unveils new insights into the complex interplay among quantum entanglement, dissipation, and non-linear dynamics, offering potential implications for quantum computing, optical systems, and beyond.

Exploring Fractional Quantum Mechanics: Stability Analysis and Wave Propagation in Coupled Schrödinger Equations

Fractional Quantum Mechanics (FQM) has emerged as a fascinating theoretical framework extending traditional quantum mechanics to describe physical systems with non-local or long-range interactions. In this paper, we delve into the realm of FQM, focusing on stability analysis and wave propagation in coupled Schrödinger equations. We begin with a comprehensive overview of FQM, elucidating its fundamental principles and mathematical formalism. Subsequently, we conduct stability analysis of coupled fractional Schrödinger equations, exploring the conditions under which these systems exhibit stable behavior. Furthermore, we investigate wave propagation phenomena within such systems, shedding light on the unique characteristics of fractional quantum waves. Our findings not only contribute to advancing the theoretical understanding of FQM but also offer insights into potential applications in diverse fields ranging from condensed matter physics to quantum information processing.

Comparison of energy-dependent and independent interactions-a case study

In the present text we construct velocity-dependent or equivalently energy-dependent potential (EDP) to an energy-independent nonlocal potential (EIP) of rank-1 by Taylor series method. The phase shifts for the nucleon-nucleon and alpha-nucleon systems are computed for these two potentials by exploiting the variable phase method and the Fredholm determinant, respectively. The velocity-dependent potential is found to be superior to central nonlocal interaction in generating the scattering phase shifts up to high energy region.

Bath-Engineering Magnetic Order in Quantum Spin Chains: An Analytic Mapping Approach.

Dissipative processes can drive different magnetic orders in quantum spin chains. Using a nonperturbative analytic mapping framework, we systematically show how to structure different magnetic orders in spin systems by controlling the locality of the attached baths. Our mapping approach reveals analytically the impact of spin-bath couplings, leading to the suppression of spin splittings, bath dressing and mixing of spin-spin interactions, and emergence of nonlocal ferromagnetic interactions between spins coupled to the same bath, which become long ranged for a global bath. Our general mapping method can be readily applied to a variety of spin models: we demonstrate (i)a bath-induced transition from antiferromagnetic (AFM) to ferromagnetic ordering in a Heisenberg spin chain, (ii)AFM to extended Neel phase ordering within a transverse-field Ising chain with pairwise couplings to baths, and (iii)a quantum phase transition in the fully connected Ising model. Our method is nonperturbative in the system-bath coupling. It holds for a variety of non-Markovian baths and it can be readily applied towards studying bath-engineered phases in frustrated or topological materials.

Fourier–Gegenbauer Pseudospectral Method for Solving Periodic Higher-Order Fractional Optimal Control Problems

Optimal control (OC) problems involve determining the best control strategy for a system to achieve the desired outcome. Fractional-order OC models have emerged in recent decades to represent complex systems that exhibit memory effects and nonlocal interactions. These models offer advantages over the traditional integer-order models. However, a key challenge with existing fractional calculus methods to define fractional derivatives, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov derivatives, is that they often fail to maintain the periodic nature of dynamical systems. This limitation hinders their application to real-world scenarios, in which periodicity is crucial. This study proposes a novel model for periodic higher-order fractional OC problems (PHFOCPs). We achieve this by employing recently developed periodic fractional derivatives that specifically preserve the periodicity within the model. This study outlines also a robust numerical solution method for this new class of problems. The numerical method is applicable for any positive fractional order [Formula: see text] and leverages several key techniques, including a useful change of variables, Fourier collocation, and Fourier and Gegenbauer quadratures, to retain numerical stability and high accuracy. We theoretically prove the exponential convergence of the proposed numerical method for sufficiently smooth periodic functions. The effectiveness of the proposed approach is further validated through various numerical simulations. In essence, this work presents a novel framework for the OC of periodic systems using fractional calculus, overcoming the limitations of classical methods and offering a powerful tool for various scientific and engineering applications.

Існування надзвукових періодичних біжучих хвиль в дискретних рівняннях типу Клейна-Ґордона з нелокальною взаємодією

The article is devoted to discrete Klein-Gordon type equations that describe infinite chains of nonlinear oscillators with nonlocal interactions. This implies that each oscillator interacts with several of its neighbors on both sides. The main result of the article concerns the existence of periodic traveling waves in such equations. Sufficient conditions for the existence of such waves were established using the variational method and the mountain pass theorem.

Solitary traveling waves in Fermi-Pasta-Ulam-type systems with nonlocal interaction on a 2D lattice

Solitary traveling waves in Fermi-Pasta-Ulam-type systems with nonlocal interaction on a 2D lattice

A computational approach to integrate three-dimensional peridynamics and two-dimensional higher-order classical elasticity theory for fracture analysis

AbstractAs a nonlocal alternative of classical continuum theory, peridynamics (PD) is mathematically compatible to discontinuities, making it particularly attractive for failure prediction. The PD theory on the other side can be computationally demanding due to its nonlocal interactions. A coupling between PD and refined higher-order finite element method (FEM) integrates their salient features. The present study proposes a computational approach to couple three-dimensional peridynamics with two-dimensional higher-order finite elements based on classical elasticity. The bond-based PD modeling is considered in a region where damage might appear while refined finite element modeling is used for the remaining region. The refined finite elements employed in this study are based on the 2D Carrera Unified Formulation (CUF), which provides 3D-like accuracy with optimized computational efficiency. The coupling between PD and FEM is achieved through the Lagrange multiplier method which permits physical consistency and compatibility at the interface domain. An adaptive convergence check algorithm is also proposed to achieve predetermined accuracy in the solution with minimum computational effort. Simulations of quasi-static tension tests, wedge splitting tests and L-plate cracking tests are carried out for verification. In-depth analysis shows that the present approach can reproduce the linear deformation, material degradation and crack propagation in an effective way.

CDMFT+HFD: An extension of dynamical mean field theory for nonlocal interactions applied to the single band extended Hubbard model

We examine the phase diagram of the extended Hubbard model on a square lattice, for both attractive and repulsive nearest-neighbor interactions, using CDMFT+HFD, a combination of Cluster Dynamical Mean Field theory (CDMFT) and a Hartree-Fock mean-field decoupling of the inter-cluster extended interaction. For attractive non-local interactions, this model exhibits a region of phase separation near half-filling, in the vicinity of which we find islands of d-wave superconductivity, decaying rapidly as a function of doping, with disconnected regions of extended s-wave order at smaller (higher) electron densities. On the other hand, when the extended interaction is repulsive, a Mott insulating state at half-filling is destabilized by hole doping, in the strong-coupling limit, in favor of d-wave superconductivity. At the particle-hole invariant chemical potential, we find a first-order phase transition from antiferromagnetism (AF) to d-wave superconductivity as a function of the attractive nearest-neighbor interaction, along with a deviation of the density from the half-filled limit. A repulsive extended interaction instead favors charge-density wave (CDW) order at half-filling.

Learning unbounded-domain spatiotemporal differential equations using adaptive spectral methods

AbstractRapidly developing machine learning methods have stimulated research interest in computationally reconstructing differential equations (DEs) from observational data, providing insight into the underlying mechanistic models. In this paper, we propose a new neural-ODE-based method that spectrally expands the spatial dependence of solutions to learn the spatiotemporal DEs they obey. Our spectral spatiotemporal DE learning method has the advantage of not explicitly relying on spatial discretization (e.g., meshes or grids), thus allowing reconstruction of DEs that may be defined on unbounded spatial domains and that may contain long-ranged, nonlocal spatial interactions. By combining spectral methods with the neural ODE framework, our proposed spectral DE method addresses the inverse-type problem of reconstructing spatiotemporal equations in unbounded domains. Even for bounded domain problems, our spectral approach is as accurate as some of the latest machine learning approaches for learning or numerically solving partial differential equations (PDEs). By developing a spectral framework for reconstructing both PDEs and partial integro-differential equations (PIDEs), we extend dynamical reconstruction approaches to a wider range of problems, including those in unbounded domains.

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods’ spectral accuracy and efficiency are validated by numerical results.

Bifurcation and Turing instability for a freshwater tussock sedge model with nonlocal interaction

Bifurcation and Turing instability for a freshwater tussock sedge model with nonlocal interaction

Romanovski–Jacobi spectral collocation schemes for distributed order differential problems

The article highlights the growing importance of computational mathematics in tackling intricate real-world phenomena governed by distributed order fractional dynamics. These dynamics, characterized by memory effects and non-local interactions, are pervasive in various fields such as physics, biology, finance, and engineering. Despite their prevalence, analytical solutions for distributed order fractional differential equations remain difficult to find, necessitating the development of robust numerical methods. The paper explores the effectiveness of Romanovski–Jacobi spectral collocation techniques in this context and provides a thorough analysis of how to use them to solve distributed order fractional differential equations. By leveraging these schemes, researchers gain valuable insights into efficiently analyzing and simulating fractional systems, thereby advancing our understanding of complex dynamics across diverse domains. It provides precise numerical findings by using finite expansion to evaluate residuals. The precision of the approach is illustrated by numerical simulations, especially in distributed order fractional differential equations. Additionally, we provide a few numerical tests to demonstrate the method’s ability to preserve the underlying problem’s non-smooth solution.

Cold isospin asymmetric baryonic rich matter in nonlocal NJL-like models

We study the features of low energy strong interactions for a system at zero temperature and finite baryon and isospin chemical potentials, in the framework of a Nambu–Jona-Lasinio-like model that includes nonlocal four-point interactions. We analyze the phase transitions corresponding to chiral symmetry restoration and pion condensation, comparing our results with those obtained from local NJL-like models and lattice QCD calculations. Published by the American Physical Society 2024

Efficient Perovskite Nanograin Light-Emitting Diodes in Green-to-Blue Gamut with Co-Additive Engineering.

Perovskite nanograins exceeding the Bohr exciton diameter show great potential for high-performance light-emitting diodes (LEDs) owing to their bandgap homogeneity, spatial charge confinement, and nonlocal interaction. However, it is challenging to directly synthesize proper nanograins along with reduced crystal defects on functional substrate, and the corresponding high-efficiency perovskite LEDs (PeLEDs) have rarely been reported. In this study, crystallization modulation for perovskites with an effective co-additive system, including lithium bromide, p-fluorophenethylammonium bromide, and 18-crown-6, is performed. Furthermore, it is demonstrated that the proposed co-additive system can synergistically retard perovskite crystallization and reduce crystal defects. Consequently, high-quality perovskite nanograin solids (≈22.8nm) are obtained with a high photoluminescence quantum yield (≈88%). These superior optical properties contribute to developing efficient green PeLEDs with a champion external quantum efficiency (EQE) of 28.4% and an average EQE of 27.1%. The co-additive system can be universally applied to mixed-halide perovskite nanograin LED, presenting a maximum EQE of 24.4%, 21.6%, 17.5%, and 11.1% for the blue device at 496, 488, 478, and 472nm, respectively, along with a narrow spectral linewidth (17-14nm) and stable color. These results supplement the research on high-efficiency perovskite nanograin LEDs for multicolor displays and lighting.

Underwater Scattering Exceptional Point by Metasurface with Fluid‐Solid Interaction

AbstractExceptional point (EP), a special degeneracy in non‐Hermitian systems, has exhibited various distinctive wave characteristics. However, the conventional synthesis of scattering EPs in acoustics is confined to rather limited methods for breaking Hermiticity, typically requiring intrinsic losses or open interfaces. Here, the concept of synthesizing scattering EPs is theoretically and experimentally demonstrated by leveraging a metasurface with fluid‐solid interaction (FSI) in water. The incorporation of FSI offers a novel mechanism to customize natural radiation losses. Simulations and experiments consistently confirm that synthesized EPs result in extremely asymmetric scattering patterns, even in the case of spatially impenetrable metasurfaces. This enhanced spatial symmetry breaking benefits from the FSI on the interface and the nonlocal interaction between unit cells. The proposed non‐Hermitian framework, involving the interplay of sound in fluids and solids, is expected to open up new possibilities for exploring unique non‐Hermitian physics and underwater acoustic applications.

A fast explicit time-splitting spectral scheme for the viscous Cahn–Hilliard equation with nonlocal diffusion operator

The viscous Cahn–Hilliard equation (VCH) is a parameter-dependent model encapsulated Cahn–Hilliard (CH) model and constrained Allen–Cahn (CAC) model. In this paper, we propose a fast explicit time-splitting spectral scheme for the VCH with nonlocal diffusion operator. The numerical scheme is constructed based on the second-order symmetric time-splitting method, which results in a splitting of the original problem into nonlocal linear part and nonlinear part, respectively. The nonlocal linear subproblem can be solved exactly by applying the Fourier spectral discretization in space and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via a second-order strong stability preserving Runge–Kutta method. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. Theoretical analysis shows that the stability of the scheme depends on both the viscosity coefficient and the range of nonlocal interactions. In addition, a rigorously convergence analysis of the scheme is established, and the convergence rate is proved to be second-order in time and spectral-order in space, respectively. Four numerical experiments are performed to demonstrate the effectiveness of the proposed scheme.

NONLINEARITY AND MEMORY EFFECTS: THE INTERPLAY BETWEEN THESE TWO CRUCIAL FACTORS IN THE HARRY DYM MODEL

This study investigates the nonlinear time-fractional Harry Dym ([Formula: see text]) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym ([Formula: see text]) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the [Formula: see text] equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He’s variational iteration ([Formula: see text]) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the [Formula: see text] equation’s behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the [Formula: see text] equation, this study paves the way for further investigations into its potential applications in diverse physical settings.

Alignments of triad phases in extreme one-dimensional Burgers flows.

We analyze the fine structure of nonlinear modal interactions in inviscid and viscous Burgers flows in 1D, which serve as toy models for the Euler and Navier-Stokes dynamics. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating in triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop diagnostic tools designed to probe the level of coherence among triadic interactions and apply them to Burgers flows corresponding to different initial conditions, including unimodal, extreme (in the sense of maximizing the growth of enstrophy in finite time), and generic. We find that in all cases triads involving energy-containing Fourier modes align their phases so as to maximize the energy flux toward small scales, and most of this flux is realized by only a handful of triads revealing a universal statistical distribution. We then identify individual triads making the largest contributions to the flux at different wave numbers and show that they represent a mixture of local and nonlocal interactions, with the latter becoming dominant at later times. These results point to the possibility of constructing a strongly reduced modal representation of Burgers flows that would require a much smaller number of degrees of freedom. Another interesting observation is that removing the spatial coherence from the extreme initial data (by randomizing the phases while retaining the magnitudes of the Fourier coefficients) does not profoundly change the nature of triadic interactions and synchronization as well as the resulting fluxes in these flows.

Spin–orbit-coupled Bose gas with Rydberg interactions confined in a two-dimensional optical lattice

We consider the ground state of Rydberg-dressed Bose gas with spin–orbit coupling and confined in a two-dimensional optical lattice. The effects of system’s parameters, especially the nonlocal Rydberg interaction and spin–orbit coupling, on the ground-state structure of such a system are explored in full parameter space. It is found that in the absence of spin–orbit coupling, the ground-state structure of the system changes from the initial optical lattice structure to a two-dimensional periodic one with the cell size decreasing with the strength of Rydberg interaction. In the presence of spin–orbit coupling, both the phase and the internal structure of unit cell show strong dependence on the strength of spin–orbit coupling. Finally, the strong interacting gas is also considered.