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18 pages, 25291 KiB

Open AccessArticle

Theoretical and Computational Insights into a System of Time-Fractional Nonlinear Schrödinger Delay Equations

by Mai N. Elhamaky, Mohamed A. Abd Elgawad, Zhanwen Yang and Ahmed S. Rahby

Axioms 2025, 14(6), 432; https://doi.org/10.3390/axioms14060432 - 1 Jun 2025

Viewed by 408

This research focuses on the theoretical asymptotic stability and long-time decay of the zero solution for a system of time-fractional nonlinear Schrödinger delay equations (NSDEs) in the context of the Caputo fractional derivative. Using the fractional Halanay inequality, we demonstrate theoretically when the [...] Read more.

This research focuses on the theoretical asymptotic stability and long-time decay of the zero solution for a system of time-fractional nonlinear Schrödinger delay equations (NSDEs) in the context of the Caputo fractional derivative. Using the fractional Halanay inequality, we demonstrate theoretically when the considered system decays and behaves asymptotically, employing an energy function in the sense of the L₂ norm. Together with utilizing the finite difference method for the spatial variables, we investigate the long-time stability for the semi-discrete system. Furthermore, we operate the L1 scheme to approximate the Caputo fractional derivative and analyze the long-time stability of the fully discrete system through the discrete energy of the system. Moreover, we demonstrate that the proposed numerical technique energetically captures the long-time behavior of the original system of NSDEs. Finally, we provide numerical examples to validate the theoretical results. Full article

(This article belongs to the Section Mathematical Analysis)

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11 pages, 256 KiB

Open AccessArticle

Improved High-Order Difference Scheme for the Conservation of Mass and Energy in the Two-Dimensional Spatial Fractional Schrödinger Equation

by Junhong Tian and Hengfei Ding

Fractal Fract. 2025, 9(5), 280; https://doi.org/10.3390/fractalfract9050280 - 25 Apr 2025

Cited by 1 | Viewed by 330

Abstract

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and [...] Read more.

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results. Full article

(This article belongs to the Special Issue Exploration and Analysis of Higher-Order Numerical Methods for Fractional Differential Equations)

17 pages, 9977 KiB

Open AccessArticle

Statistical Properties of Correlated Semiclassical Bands in Tight-Binding Small-World Networks

by Natalya Almazova, Giorgos P. Tsironis and Efthimios Kaxiras

Entropy 2025, 27(4), 420; https://doi.org/10.3390/e27040420 - 12 Apr 2025

Viewed by 277

Abstract

Linear tight-binding models with long-range interactions and small-world geometry have a broad energy spectrum in the nearest neighbor coupling limit, while the spectrum becomes narrow in the fully connected limit due to the emergence of flat bands. A transition to a Wigner-like density [...] Read more.

Linear tight-binding models with long-range interactions and small-world geometry have a broad energy spectrum in the nearest neighbor coupling limit, while the spectrum becomes narrow in the fully connected limit due to the emergence of flat bands. A transition to a Wigner-like density of states appears at a low fraction of long-range bonds. Adding nonlinearity to the model introduces correlations among the stationary states, while multiple new states are generated as a result of the nonlinearity. In this work, we study the effect of band correlations on the local density of states for small-world networks as a function of the number of long-range bonds. We find that close to the nearest neighbor limit, the onset of correlations shifts the nonlinear density of states towards the band edge of the spectrum. Close to the opposite limit of the fully connected model, the band collapses in the band center, accompanied by a large increase in the new states induced by the nonlinearity. While in both limits the effect of correlations is to flatten the band, close to the mean field fully connected limit, the states are correlated and generally have distinct localized features. These effects may have implications for the dynamics of electrons in two-dimensional moiré structures and the onset of superconductivity in these systems. Full article

(This article belongs to the Special Issue New Challenges in Contemporary Statistical Physics)

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25 pages, 902 KiB

Open AccessArticle

Discrete Derivative Nonlinear Schrödinger Equations

by Dirk Hennig and Jesús Cuevas-Maraver

Mathematics 2025, 13(1), 105; https://doi.org/10.3390/math13010105 - 30 Dec 2024

Viewed by 745

Abstract

We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central difference schemes, respectively, and term the corresponding equations forward, backward, and central ddNLSs. [...] Read more.

We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central difference schemes, respectively, and term the corresponding equations forward, backward, and central ddNLSs. We show that in contrast to the dNLS, which is completely integrable and supports soliton solutions, the forward and backward ddNLSs can be either dissipative or expansive. As a consequence, solutions of the forward and backward ddNLSs behave drastically differently compared to those of the (integrable) dNLS. For the dissipative forward ddNLS, all solutions decay asymptotically to zero, whereas for the expansive forward ddNLS all solutions grow exponentially in time, features that are not present in the dynamics of the (integrable) dNLS. In comparison, the central ddNLS is characterized by conservative dynamics. Remarkably, for the central ddNLS the total momentum is conserved, allowing the existence of solitary travelling wave (TW) solutions. In fact, we prove the existence of solitary TWs, facilitating Schauder’s fixed-point theorem. For the damped forward expansive ddNLS we demonstrate that there exists such a balance of dissipation so that solitary stationary modes exist. Full article

(This article belongs to the Section E4: Mathematical Physics)

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10 pages, 725 KiB

Open AccessFeature PaperArticle

Coarse-Gridded Simulation of the Nonlinear Schrödinger Equation with Machine Learning

by Benjamin F. Akers and Kristina O. F. Williams

Mathematics 2024, 12(17), 2784; https://doi.org/10.3390/math12172784 - 9 Sep 2024

Cited by 1 | Viewed by 973

Abstract

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is [...] Read more.

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set. Full article

(This article belongs to the Special Issue Numerical Analysis in Computational Mathematics)

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16 pages, 1026 KiB

Open AccessArticle

Energy-Conserving Explicit Relaxed Runge–Kutta Methods for the Fractional Nonlinear Schrödinger Equation Based on Scalar Auxiliary Variable Approach

by Yizhuo Zhao, Yu Li, Jiaxin Zhu and Yang Cao

Axioms 2024, 13(9), 591; https://doi.org/10.3390/axioms13090591 - 30 Aug 2024

Viewed by 1141

Abstract

In this paper, we present a novel explicit structure-preserving numerical method for solving nonlinear space-fractional Schrödinger equations based on the concept of the scalar auxiliary variable approach. Firstly, we convert the equations into an equivalent system through the introduction of a scalar variable. [...] Read more.

In this paper, we present a novel explicit structure-preserving numerical method for solving nonlinear space-fractional Schrödinger equations based on the concept of the scalar auxiliary variable approach. Firstly, we convert the equations into an equivalent system through the introduction of a scalar variable. Subsequently, a semi-discrete energy-preserving scheme is developed by employing a fourth-order fractional difference operator to discretize the equivalent system in spatial direction, and obtain the fully discrete version by using an explicit relaxed Runge–Kutta method for temporal integration. The proposed method preserves the energy conservation property of the space-fractional nonlinear Schrödinger equation and achieves high accuracy. Numerical experiments are carried out to verify the structure-preserving qualities of the proposed method. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)

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15 pages, 1786 KiB

Open AccessArticle

Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations

by Xinyue Zhao, Yining Yang, Hong Li, Zhichao Fang and Yang Liu

Fractal Fract. 2024, 8(8), 480; https://doi.org/10.3390/fractalfract8080480 - 17 Aug 2024

Cited by 1 | Viewed by 1041

Abstract

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where [...] Read more.

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the

L 2 - 1_{σ}

formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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25 pages, 3439 KiB

Open AccessArticle

Split-Step Galerkin FE Method for Two-Dimensional Space-Fractional CNLS

by Xiaogang Zhu, Yaping Zhang and Yufeng Nie

Fractal Fract. 2024, 8(7), 402; https://doi.org/10.3390/fractalfract8070402 - 5 Jul 2024

Viewed by 1445

Abstract

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in [...] Read more.

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton’s iteration illustrate its effectiveness and advantages in computational efficiency. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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25 pages, 11760 KiB

Open AccessArticle

Regular, Beating and Dilogarithmic Breathers in Biased Photorefractive Crystals

by Carlos Alberto Betancur-Silvera, Aurea Espinosa-Cerón, Boris A. Malomed and Jorge Fujioka

Axioms 2024, 13(5), 338; https://doi.org/10.3390/axioms13050338 - 20 May 2024

Viewed by 1168

Abstract

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations [...] Read more.

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov–Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones. Full article

(This article belongs to the Special Issue Nonlinear Schrödinger Equations)

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14 pages, 891 KiB

Open AccessArticle

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

by Peiyao Wang, Shangwen Peng, Yihao Cao and Rongpei Zhang

Mathematics 2024, 12(7), 1110; https://doi.org/10.3390/math12071110 - 7 Apr 2024

Viewed by 1441

Abstract

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is [...] Read more.

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution. Full article

(This article belongs to the Section E4: Mathematical Physics)

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37 pages, 3477 KiB

Open AccessReview

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

by Boris A. Malomed

Entropy 2024, 26(2), 137; https://doi.org/10.3390/e26020137 - 2 Feb 2024

Cited by 3 | Viewed by 1874

Abstract

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete [...] Read more.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross–Pitaevskii (GP) equations with the Lee–Huang–Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole–dipole and quadrupole–quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (

χ^{(2)}

) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its

PT

-symmetric version, a system of DNLS equations for the spin–orbit-coupled (SOC) binary Bose–Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and

PT

-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others. Full article

(This article belongs to the Special Issue Recent Advances in the Theory of Nonlinear Lattices)

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15 pages, 645 KiB

Open AccessArticle

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

by Mengli Yao and Zhifeng Weng

Math. Comput. Appl. 2024, 29(1), 6; https://doi.org/10.3390/mca29010006 - 15 Jan 2024

Cited by 2 | Viewed by 2050

Abstract

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation [...] Read more.

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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13 pages, 2539 KiB

Open AccessArticle

Energy-Preserving AVF Methods for Riesz Space-Fractional Nonlinear KGZ and KGS Equations

by Jianqiang Sun, Siqi Yang and Lijuan Zhang

Fractal Fract. 2023, 7(10), 711; https://doi.org/10.3390/fractalfract7100711 - 27 Sep 2023

Cited by 1 | Viewed by 1440

Abstract

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained [...] Read more.

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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15 pages, 552 KiB

Open AccessArticle

An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation

by Zeting Liu, Baoli Yin and Yang Liu

Fractal Fract. 2023, 7(9), 654; https://doi.org/10.3390/fractalfract7090654 - 30 Aug 2023

Viewed by 1146

Abstract

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass [...] Read more.

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme. Full article

(This article belongs to the Section Numerical and Computational Methods)

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16 pages, 455 KiB

Open AccessArticle

The Convergence of Symmetric Discretization Models for Nonlinear Schrödinger Equation in Dark Solitons’ Motion

by Yazhuo Li, Qian Luo and Quandong Feng

Symmetry 2023, 15(6), 1229; https://doi.org/10.3390/sym15061229 - 9 Jun 2023

Cited by 1 | Viewed by 1083

Abstract

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the [...] Read more.

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schrödinger equation in dark solitons’ motion: the direct-discrete model and the Ablowitz–Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time–space discretization models: the Crank–Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schrödinger equation. Additionally, we prove that the convergence order of the two time–space discretization models is

O (h^{2} + τ^{2})

in discrete

L_{2}

-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results. Full article

(This article belongs to the Special Issue Advances in Nonlinear, Discrete, Continuous and Hamiltonian Systems II)

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