Nonlocal Nonlinearity Research Articles

Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

On Impulsive Nonlocal Nonlinear Fuzzy Integro-Differential Equations in Banach Space

The aim of this article is to investigate the existence, uniqueness and other qualitative properties of the solution of first-order nonlocal impulsive nonlinear fuzzy integro-differential equations in Banach space by using the concept of fuzzy numbers whose values are normal, upper semicontinuous, compact, and convex. The result is attained by utilizing a modified version of the Banach contraction principle. We offer an example as an application of the results.

Quiescent optical solitons with Kudryashov’s generalized quintuple-power law and nonlocal nonlinearity having nonlinear chromatic dispersion with generalized temporal evolution by enhanced direct algebraic method and sub-ODE approach

Revisiting the study of quiescent optical solitons with quintuple-power-law self-phase modulation and nonlinear chromatic dispersion is the focus of the current paper. The soliton solutions to the model are revealed through the intermediary Jacobi’s elliptic functions using the enhanced direct algebraic method. The intermediary Weierstrass’ elliptic functions are used by the sub-ODE approach to reveal such quiescent soliton solutions.

The effect of Lévy index coefficient on modulational instability and rogue wave excitation in nonlocal media with cubic–quintic nonlinearities

This paper explores the modulational instability (MI) of a plane wave and its behavior in the nonlinear Schrödinger equation (NLSE) with a fractional diffraction term quantified by its Lévy index coefficient and nonlocal cubic–quintic nonlinearities. First, we analyze the stability of the plane wave solution and examine how nonlocal nonlinearities and the Lévy index coefficient affect the MI gain. We observe that the stability in the fractional NLSE exhibits new features that differ from those in the standard NLSE. Specifically, when dealing with competing cubic and quintic nonlinearities, the interaction between nonlocality and the Lévy index coefficient α can eliminate MI for low values of α, unlike the classical NLSE with α=2, where we find the plane wave to be unstable. Besides the linear stability analysis, numerical simulations are performed to understand further the plane wave dynamics from its nonlinear stage in this model. The results reveal the generation of periodic chains of localized peaks. Guided by analytical predictions and using the plane wave solution subject to Gaussian perturbation, we numerically investigate the possibility of exciting rogue waves in the parameter spaces where MI exists. We find that the different combinations of signs of the cubic and quintic nonlinearities (focusing and defocusing) and the fractional diffraction term significantly impact the formation of rogue waves. These results may pave the way for the theoretical and experimental study of nonlinear phenomena in physical models with fractional derivatives and nonlocal nonlinearities.

Orbital angular momentum and rotational properties of off-axis vortex beams in two-dimensional media with anisotropic nonlocal nonlinearity.

By introducing anisotropy into nonlinear propagations, off-axis vortex beams exhibit significantly different characteristics compared to the isotropic case. The orbital angular momentum (OAM) is non-conservative and can periodically change between positive and negative values. Accordingly, the rotation of phase singularity can transit between clockwise and counterclockwise directions. Furthermore, the phase singularity can move to infinity when the OAM approaches zero. By using the Ehrenfest theorem, the motion of the beam center is obtained. Its trajectory can be circular and parabolic or follow other complex shapes, depending closely on the anisotropy of the nonlinearity. The rotational velocity of the beam center can be modulated by the nonlinearity anisotropy and can far exceed the initial value during its propagation. These results may find potential applications in beam shaping and optical manipulation.

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Modulational instability and collapse of internal gravity waves in the atmosphere.

Nonlinear two-dimensional (IGWs) in the atmospheres of the Earth and the Sun are studied. The resulting two-dimensional nonlinear equationhas the form of a generalized nonlinear Schrödinger equationwith nonlocal nonlinearity, that is, when the nonlinear response depends on the wave intensity at some spatial domain. The modulation instability of IGWs is predicted, and specific cases for the Earth's atmosphere are considered. In a number of particular cases, the instability thresholds and instability growth rates are analytically found. Despite the nonlocal nonlinearity, we demonstrate the possibility of critical collapse of IGWs due to the scale homogeneity of the nonlinear term in spatial variables.

Hirota Bilinear Approach to Multi-Component Nonlocal Nonlinear Schrödinger Equations

Hirota Bilinear Approach to Multi-Component Nonlocal Nonlinear Schrödinger Equations

Orbital and Spin Parts of Angular Momentum Flux Density of Monochromatic Radiation in Nonabsorbing Media with Nonlocal Nonlinear Optical Response

Orbital and Spin Parts of Angular Momentum Flux Density of Monochromatic Radiation in Nonabsorbing Media with Nonlocal Nonlinear Optical Response

Two distinct algorithms for conformable time-fractional nonlinear Schrödinger equations with Kudryashov’s generalized non-local nonlinearity and arbitrary refractive index

Two distinct algorithms for conformable time-fractional nonlinear Schrödinger equations with Kudryashov’s generalized non-local nonlinearity and arbitrary refractive index

Shock-wave generation and propagation in dissipative and nonlocal nonlinear Rydberg media

We investigate the generation of optical shock waves in strongly interacting Rydberg atomic gases with a spatially homogeneous dissipative potential. The Rydberg atom interaction induces an optical nonlocal nonlinearity. We focus on local nonlinear (Rb≪R0) and nonlocal nonlinear (Rb∼R0) regimes, where Rb and R0 are the characteristic length of the Rydberg nonlinearity and beam width, respectively. In the local regime, we show spatial width and contrast of the shock wave change monotonically when increasing strength of the dissipative potential and optical intensity. In the nonlocal regime, the characteristic quantity of the shock wave depends on Rb/R0 and dissipative potential nontrivially and on the intensity monotonically. We find that formation of shock waves dominantly takes place when Rb is smaller than R0, while the propagation dynamics is largely linear when Rb is comparable to or larger than R0. Our results reveal nontrivial roles played by dissipation and nonlocality in the generation of shock waves, and provide a route to manipulate their profiles and stability. Our study furthermore opens new avenues to explore non-Hermitian physics and nonlinear wave generation and propagation by controlling dissipation and nonlocality in the Rydberg media. Published by the American Physical Society 2024

Dynamical optical soliton solutions and behavior for the nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index together with the dual-form of nonlocal nonlinearity

Dynamical optical soliton solutions and behavior for the nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index together with the dual-form of nonlocal nonlinearity

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

Features of dispersion properties of a waveguide with a modified Kerr weak nonlocal nonlinearity coated with a metal thin film

Features of dispersion properties of a waveguide with a modified Kerr weak nonlocal nonlinearity coated with a metal thin film

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

Effect of weak nonlocal nonlinearity on generalized sixth-order dispersion modulational instability in optical media

Effect of weak nonlocal nonlinearity on generalized sixth-order dispersion modulational instability in optical media

Optical solutions with Kudryashov’s arbitrary type of generalized non-local nonlinearity and refractive index via the new Kudryashov approach

Optical solutions with Kudryashov’s arbitrary type of generalized non-local nonlinearity and refractive index via the new Kudryashov approach

Quasi-Periodic Solutions to the Nonlocal Nonlinear Schrödinger Equations

Quasi-Periodic Solutions to the Nonlocal Nonlinear Schrödinger Equations

Retrieval of solitons and other wave solutions for stochastic nonlinear Schrödinger equation with non-local nonlinearity using the improved modified extended tanh-function method

Retrieval of solitons and other wave solutions for stochastic nonlinear Schrödinger equation with non-local nonlinearity using the improved modified extended tanh-function method