Weak Nonlocality Research Articles

Jacobian elliptic wave trains in fourth-order dispersive nonlinear Schrödinger equation and the modulational instability

We analytically and numerically explore the existence of explicit exact Jacobian elliptic solutions of a nonlinear Schrödinger model with fourth-order dispersion and weak nonlocality. We delineate the parameter domain in which these highly dispersive doubly periodic waves balanced by the weak nonlocality and Kerr nonlinearity exist. Additionally, we have studied the modulational instability (MI) of these waves with the aid of linear stability analysis. We observe that in the presence of fourth-order dispersion, the MI gain increases with increasing the nonlocality parameter.

Extraction of new solitary wave solutions in a generalized nonlinear Schrödinger equation comprising weak nonlocality.

This article delves into examining exact soliton solutions within the context of the generalized nonlinear Schrödinger equation. It covers higher-order dispersion with higher order nonlinearity and a parameter associated with weak nonlocality. To tackle this equation, two reputable methods are harnessed: the sine-Gordon expansion method and the [Formula: see text]-expansion method. These methods are employed alongside suitable traveling wave transformation to yield novel, efficient single-wave soliton solutions for the governing model. To deepen our grasp of the equation's physical significance, we utilize Wolfram Mathematica 12, a computational tool, to produce both 3D and 2D visual depictions. These graphical representations shed light on diverse facets of the equation's dynamics, offering invaluable insights. Through the manipulation of parameter values, we achieve an array of solutions, encompassing kink-type, dark soliton, and solitary wave solutions. Our computational analysis affirms the effectiveness and versatility of our methods in tackling a wide spectrum of nonlinear challenges within the domains of mathematical science and engineering.

Optical dark, singular and bright soliton solutions with dual-mode fourth-order nonlinear Schrödinger equation involving different nonlinearities

This research article investigates the dynamic characteristics of solitary waves within a medium exhibiting weak nonlinearity in dual-mode. Firstly, the three models are converted to dual-mode. The primary focus is to investigate the behavior of solitary waves in a system described by a fourth-order nonlinear Schrödinger (NLS) equation, which incorporates Kerr nonlinearity, weak nonlocality, parabolic law nonlinearity and diffraction. By employing ansatz methods, the study successfully derives various localized waveforms, including bright, dark and singular solitons, based on the governing model. The examination extends to investigating the influence of nonlocality and power coefficients on dynamics of bright, dark and singular solitons. The findings presented in this article represent a crucial advancement in understanding the generation of solitary waves in nonlinear media. As the examined model finds application in various fields such as plasma physics, Bose-Einstein condensates and others, these findings not only enhance our theoretical understanding but also pave the way for potential practical applications and the advancement of innovative technologies.

A generalized nonlinear Schrödinger involving the weak nonlocality: Its Jacobi elliptic function solutions and modulational instability

The present paper investigates the features of specific waves in a weakly nonlocal nonlinear medium. More precisely, a generalized nonlinear Schrödinger equation (gNLSE) involving a high-order dispersion term, a high-order nonlinear term, and the effect of a term that accounts for the weak nonlocality is considered. By applying the Jacobi elliptic function (JEF) method, a wide range of Jacobi elliptic function solutions to the governing model is first derived. The characteristics of modulation instability (MI) and modulated wave (MW) topological and non-topological solitons are then analyzed by giving a series of two- and three-dimensional graphs. A major accomplishment of the study is the discovery of stable and unstable zones, as well as the control of the amplitude of topological and non-topological solitons.

A fourth-order nonlinear Schrödinger equation involving power law and weak nonlocality: Its solitary waves and modulational instability analysis

The present paper examines the dynamical features of solitary waves in a weakly nonlinear medium. More precisely, the propagation of solitary waves in a system is modeled by a fourth-order nonlinear Schrödinger equation involving diffraction, power law nonlinearity, and weak nonlocality. Several localized waves classified as bright and dark solitons to the governing model are derived using ansatz methods. It is shown how power and nonlocality coefficients affect the dynamics of bright and dark solitons. Furthermore, the modulational instability of continuous waves in the presence of such different effects is studied. The results of the current paper represent a significant advancement in exploring the propagation of solitary waves in a nonlinear medium.

Accessing and manipulating dispersive shock waves in a nonlinear and nonlocal Rydberg medium

Dispersive shock waves (DSWs) are fascinating wave phenomena occurring in media when nonlinearity overwhelms dispersion (or diffraction). Creating DSWs with low generation power and realizing their active controls is desirable but remains a longstanding challenge. Here, we propose a scheme to generate weak-light DSWs and realize their manipulations in an atomic gas involving strongly interacting Rydberg states under the condition of electromagnetically induced transparency (EIT). We show that for a two-dimensional (2D) Rydberg gas a weak nonlocality of optical Kerr nonlinearity can significantly change the edge speed of DSWs and induces a singular behavior of the edge speed and hence an instability of the DSWs. However, by increasing the degree of the Kerr nonlocality, the singular behavior of the edge speed and the instability of the DSWs can be suppressed. We also show that in a 3D Rydberg gas, DSWs can be created and propagate stably when the system works in the intermediate nonlocality regime. Due to the EIT effect and the giant nonlocal Kerr nonlinearity contributed by the Rydberg-Rydberg interaction, DSWs found here have extremely low generation power. In addition, an active control of DSWs can be realized; in particular, they can be stored and retrieved with high efficiency and fidelity through switching off and on a control laser field. The results reported here are useful not only for unveiling intriguing physics of DSWs but also for finding promising applications of nonlinear and nonlocal Rydberg media.

Chirped spatial solitons on a continuous-wave background in weak nonlocal media with polynomial law of nonlinearity

Spatial soliton propagation in a cubic-quintic-septimal optical material with a weakly nonlocal response is examined. We demonstrate that bright and gray solitons on a continuous-wave background exist for the envelope equation describing their dynamis in the physical media. In addition, optical kink and anti-kink type solitons are also identified. We find that these waveforms exhibit a frequency chirping property which is inversely proportional to the beam intensity. A noteworthy characteristic is that the chirped soliton solutions include no free parameters and their properties such as their amplitude, spatial width and propagation constant are determined solely by the material parameters. The numerical examples are structured for illustrating the propagation dynamics of these solitons in the optical system. The role of weak nonlocality on the modulation instability gain has also been discussed.

ПРИМЕРЫ ТОЧНЫХ РЕШЕНИЙ НЕЛОКАЛЬНОГО ВОЛНОВОГО УРАВНЕНИЯ С НЕЛИНЕЙНЫМИ ИСТОЧНИКАМИ

The scope of the study is a wave equation with a source in a medium with weak spatial nonlocality. The equation is distinguished by an additional term containing the function sought as a fourth order partial derivative of the spatial coordinate. The transformation of independent variables enables the construction of accurate partial solutions in the form of waves generated by a nonlinear source which depend on the desired function. The velocity regime of the wave (subsonic, sonic, supersonic) is characterized by the Mach number equal to the ratio of the velocity of the wave to the propagation ve-locity of small perturbations. A source function similar to the classical case for the double sine-Gordon equation is considered. A kink solution corresponds to two equilibrium states of the medium-source system. The relation between the source and the kink structure (the area of the solution, the sign of the kink obliquity, and the velocity of its movement) has been established. It is shown that in relation to the dimensionless parameter of nonlocality, the square of the Mach number is a monotonic increasing/decreasing function for the supersonic/subsonic velocity mode. In relation to one of the source parameters, the square of the Mach number is a non-monotonic function with a minimum/maximum in the supersonic/subsonic cases. The source functions corresponding to the extreme modes differ from each other by the inversion of the areas where these functions are positive and negative. For the sine-Gordon equation, the comparison of the classical and nonlocal processes are different not only in the areas of the solutions, but also in the velocity modes (subsonic/supersonic) of the motion of the kink. The cubic nonlinearity of the source gives solutions representing a weak discontinuity of the function sought or a solitary wave. A kink solution depends on the wave coordinate and is determined by a hyperbolic tangent. The paper provides a comparative analysis of the properties of the polynomial (third and fifth degree) functions of sources generating a wave in classical and nonlocal media.

Pure-quartic solitons in presence of weak nonlocality

In this paper, we study the characteristics of pure-quartic optical solitons in a nonlinear medium having a weakly nonlocal response. We propose an nonlinear Schrödinger equation incorporating pure fourth-order diffraction, Kerr nonlinearity and weak nonlocality to describe the transmission of solitons in the system. We obtain analytic pure-quartic soliton solutions to this model in a variety of shapes and forms, unlike the case of the model without taking into account the weak nonlocal contribution. Besides, we find that the obtained localized structures have both bright and dark type waveforms in the transmission system. Interestingly, the functional forms of these solitons are expressed in terms of sech and tanh functions which do not possess oscillating tails, markedly different from the conventional pure-quartic solitons in the absence of weak nonlocality. These structures exist for both positive and negative quartic diffraction coefficient whereas pure-quartic solitons in the absence of nonlocality occur only for negative fourth-order dispersion. The numerical examples are presented for illustrating the soliton evolution dynamics in the optical waveguide.

Type-II Dirac points and topological Fermi arcs in nonlocal metamaterials

Dirac semimetals are topological materials with degenerated Dirac points in their band structures. In particular, the Dirac points play a central role in the topologically protected phases. Recently, it was reported that the peculiar type-II Dirac points could be realized in photonics by means of photonic crystals (PCs). Nevertheless, the practical implementation is rather challenging. In this paper, we demonstrate that the nonlocal effect in the interior of a homogeneous photonic metamaterial (HPM) enables the existence of type-II Dirac points in the HPM's band structure. With theoretical formulations and numerical evaluations, we discuss the evolution of the HPM's band type and present the interesting transition between type-I and type-II Dirac degeneracy of the HPMs. Remarkably, a pair of the type-II Dirac points can be generated at two different frequencies, which is supported by the mode overlapping in the HPMs with weak nonlocality. The Fermi arc surface states connect the projections of the type-II Dirac points at the boundary between the vacuum and HPMs. The topological protection of the Fermi arc surface states is verified by the reflectionless transmission with sharp corners. Our paper provides a fundamental understanding of topological photonics by type-II Dirac points in HPMs.

Chaotic self-trapped beams in the nonlocal nonlinear system with the sine-oscillation response

The propagation of the unstable multipeak solitons in the nonlocal nonlinear system with the sine-oscillation response was examined in this paper. It was found that the beams are self-trapped while possessing the chaotic property under both the conditions of negative Kerr coefficient at strong nonlocality and positive Kerr coefficient at weak nonlocality. The self-trapping is represented by the invariant beam width and spectrum width. The chaotic property, denoted by the positive Lyapunov exponents, corresponds to the phenomenon that the profiles of the intensity change irregularly in the propagation direction. Compared with the known chaoticons, which were found only in the strongly nonlocal nonlinear system with the positively defined attenuating response for the positive Kerr coefficient, these beams have different characteristics in the interaction between beams and distribution of energy.

Chirped Bright and Kink Solitons in Nonlinear Optical Fibers with Weak Nonlocality and Cubic-Quantic-Septic Nonlinearity

This work focuses on chirped solitons in a higher-order nonlinear Schrödinger equation, including cubic-quintic-septic nonlinearity, weak nonlocal nonlinearity, self-frequency shift, and self-steepening effect. For the first time, analytical bright and kink solitons, as well as their corresponding chirping, are obtained. The influence of septic nonlinearity and weak nonlocality on the dynamical behaviors of those nonlinearly chirped solitons is thoroughly addressed. The findings of the study give an experimental basis for nonlinear-managed solitons in optical fibers.

Chirped periodic and localized waves in a weakly nonlocal media with cubic-quintic nonlinearity

We study the propagation of one-dimentional optical beams in a weakly nonlocal medium exhibiting cubic-quintic nonlinearity. A nonlinear equation governing the evolution of the beam intensity in the nonlocal medium allows us to examine whether the traveling-waves exist in such optical material. An efficient transformation is applied to obtain explicit solutions of the envelope model equation in the presence of all material parameters. We find that a variety of periodic waves accompanied with a nonlinear chirp do exist in the system in the presence of the weak nonlocality. Chirped localized intensity dips on a continuous-wave background as well as solitary waves of the bright and dark types are obtained in a long wave limit. A class of propagating chirped self-similar solitary beams is also identified in the material with the consideration of the inhomogeneities of media. The applications of the obtained self-similar structures are discussed by considering a periodic distributed amplification system.

Determining the Interphase Boundary Parameters in Heterogeneous Binary Alloys on the Basis of the Hypothesis of Weak Nonlocality

Based on our earlier hypothesis of weak nonlocality, we developed a method for calculating the temperature dependences of the width and specific energy of the interphase boundary for binary substitutional alloys, using the concentration dependence of the internal energy or the equilibrium phase diagram of the alloy. Numerically calculated temperature dependences of the solubility limits, specific energy, and thickness of the interphase boundary are presented for three thermodynamic models: the regular solution model (Nb–Cu alloy), the quasi-regular solution model, and the general case (Fe–Cu alloy and A–B alloy that differs from Fe–Cu in the sign of excess entropy).

Stability of quadrupole solitons in nonlocal media

We study quadruple-mode (quadrupole) solitons in nonlocal media with different degrees of nonlocality. The existence and stability of such quadrupole solitons is addressed systematically. The solution of quadrupole solitons is obtained analytically with variational approach. The propagation dynamics of quadrupole solitons is found numerically with the split-step method. The repulsion between the out-of-phase petals of quadrupole solitons can not be balanced by the weak nonlocality, leading to the formation of four separated scalar solitons. Completely stable quadrupole solitons can be achieved when the nonlocality is strong.

Gradient modification of Newtonian gravity

A second gradient generalization of Newtonian gravity is presented within the framework of gradient field theory. Weak nonlocality is introduced via first and second gradients of the gravitational field strength in the Lagrangian density. Gradient generalizations of the Poisson equation of Newtonian gravitation for the gravitational potential and of the generalized Gauss law for the gravitational field strength are presented. Such a gradient modification of Newtonian gravity provides a straightforward regularization of Newtonian gravity removing the classical Newtonian singularities. Finite gradient modifications of the gravitational potential energy and of the gravitational force law are constructed, with a possible connection to Yukawa interaction, and as suitable candidates for experimental tests of Newton's inverse-square law at short distances. In addition, nonlocal gravity of exponential type is investigated and its relation to gradient gravity theory is given.

Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and U ( 1 ) -gauge invariant. Second gradient electromagnetostatics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space and gives a regularization based on higher-order partial differential equations. From the group theoretical point of view, in second gradient electromagnetostatics the (isotropic) constitutive relations involve an invariant scalar differential operator of fourth order in addition to scalar constitutive parameters. We investigate the classical static problems of an electric point charge, and electric and magnetic dipoles in the framework of second gradient electromagnetostatics, and we show that all the electromagnetic fields (potential, field strength, interaction energy, interaction force) are singularity-free, unlike the corresponding solutions in the classical Maxwell electromagnetism and in the Bopp–Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality.

Electromagnetically induced grating in semiconductor quantum dot and metal nanoparticle hybrid system by considering nonlocality effects

The optical polarization from a hybrid system including a closely spaced spherical SQD (modeled as a three-level V-type system) and a metal nanoparticle which are considered classically and are connected by the dipole–dipole interaction mechanism is investigated. The interaction between the SQD and the MNP shows an interesting optical response. In the weak probe field regime and MNP nonlocality correction, the absorption spectrum of the hybrid system exhibits an EIT window with two absorption peaks and the plasmon-assisted quantum interference plays an important role in the position and amplitude of these peaks, which are intensely altered by including the nonlocal effects. The probe diffraction grating is created based on the excitons-induced transparency by applying a standing-wave coupling field. The results of this study are useful in numerous areas of all-optical communications.

Second gradient electrodynamics: Green functions, wave propagation, regularization and self-force

In this work, the theory of second gradient electrodynamics, which is an important example of generalized electrodynamics, is proposed and investigated. Second gradient electrodynamics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian density. Second gradient electrodynamics possesses a weak nonlocality in space and time. In the framework of second gradient electrodynamics, the retarded Green functions, first-order derivatives of the retarded Green functions, retarded potentials, retarded electromagnetic field strengths, generalized Liénard–Wiechert potentials and the corresponding electromagnetic field strengths are derived for three, two and one spatial dimensions. The behaviour of the electromagnetic fields is investigated on the light cone. In particular, the retarded Green functions and their first-order derivatives show oscillations around the classical solutions inside the forward light cone and it is shown that they are singularity-free and regular on the light cone in three, two and one spatial dimensions. In second gradient electrodynamics, the self-force and the energy release rate are calculated and the equation of motion of a charged point particle, which is an integro-differential equation where the infamous third-order time-derivative of the position does not appear, is determined.

Hypothesis of a Weak Nonlocality for Determining the Binodal of a Binary Alloy

This work proposes an approach for the calculation of equilibrium solubility limits in binary alloys, which is an alternative to approaches based on gradient expansions. The proposed approach rests on the description of alloys in terms of a weak nonlocality, which generalizes the classical local equilibrium hypothesis. This results in a theory that can be used to derive the explicit concentration dependence of all used parameters solely from knowledge of the concentration dependence of free energy density, which is impossible within the framework of existing approaches.