Fractional Nonlinear Schrodinger Equation Research Articles

The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory.

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

An Approximate Solution for the Non-Linear Fractional Schrodinger Equation with Harmonic Oscillator

An Approximate Solution for the Non-Linear Fractional Schrodinger Equation with Harmonic Oscillator

Novel optical solitons and other wave structures of solutions to the fractional order nonlinear Schrodinger equations

Novel optical solitons and other wave structures of solutions to the fractional order nonlinear Schrodinger equations

The study of new optical soliton solutions to the time-space fractional nonlinear dynamical model with novel mechanisms

In this work, we successfully employ two constructive mathematical tools such as the extended rational sine-cosine/sinh-cosh and novel Φ6-expansion model techniques, for obtaining different types of optical solitons and other solutions to the (2+1)-dimensional time-space fractional nonlinear Schrodinger (NLS) equation within the modified Riemann–Liouville derivatives. Solitary wave, dark, singular, and their combined solutions are retrieved. Moreover, hyperbolic, trigonometric, plane wave periodic wave, and Jacobi elliptic function solutions are also derived. The novelty of the obtained solutions is revealed by comparing them with the previous finding. By selecting suitable parametric values, numerical simulation and physical interpretations of the achieved outcomes are manifested with interesting figures presenting the meaning of the acquired results. The novel approaches are used through the symbolic computations impart a dynamical and potent mathematical implement for solving various benevolent nonlinear wave problems. The results suggest that these constructing techniques are concise, direct, efficient, and straightforward. The resulting solutions are novel, intriguing, and potentially helpful for understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including telecommunication, engineering, mathematical biology, mathematical physics, and nonlinear optics etc.

New optical solitons of fractional nonlinear Schrodinger equation with the oscillating nonlinear coefficient: A comparative study

In this current exploration, some new optical soliton structures of fractional nonlinear Schrodinger equation with the oscillating nonlinear coefficient are constructed with three different definitions of fractional operators beta, Riemann–Liouville, and M-Truncated derivatives. These structures are computed with the help of the new auxiliary equation method. This method gives the new analytical solutions of the considered model. The analysis is done by considering the different definitions of the derivatives like Beta, Riemann–Liouville (RL), and M-Truncated derivatives. The considered equation is converted to an ordinary differential equation (ODE) by the use of this complex transformation. The graphical explanation of some obtained results is also elaborated in detail. This work is new and does not exist in literature.

A Comparative Study for Fractional Nonlinear Schrodinger Equation in Optics

A Comparative Study for Fractional Nonlinear Schrodinger Equation in Optics

Fractionality-induced deformation in Airy and Hermit-Gaussian wavepackets traveling through the Gaussian wells

In this study, we investigate the transport properties of Airy and Hermit-Gaussian wavepackets traveling through a Gaussian potential well in the standard and fractional media. We use a split step Fourier method to solve the corresponding two-dimensional fractional nonlinear linear Schrodinger equation. We study the interference phenomenon, Huygens principle and trapping properties of the system in different situations. For this purpose, the position of the system has also selected inside the potential well to consider the wavepacket evolution. Then, the effect of the well and incident wavepackets parameters on the evolution the transport properties of employed wavepackets have properly been considered.

Diverse soliton structures for fractional nonlinear Schrodinger equation, KdV equation and WBBM equation adopting a new technique

Diverse soliton structures for fractional nonlinear Schrodinger equation, KdV equation and WBBM equation adopting a new technique

An effective technique for the conformable space-time fractional cubic-quartic nonlinear Schrodinger equation with different laws of nonlinearity

In the present study, we investigate the conformable space-time fractional cubic-quartic nonlinear Schrodinger equation with three different laws of nonlinearity namely, parabolic law, quadratic-cubic law, and weak non-local law.This model governs the propagation of solitons through nonlinear optical fibers. An effective approach namely, the exp(-Pi(xi))-expansion method is applied to construct some new exact solutions of the governing model. Consequently, the dark, singular, rational and periodic solitary wave solutions are successfully revealed. The comparisons with other results are also presented. In addition, the dynamical structures of obtained solutions are presented through 3D and 2D plots.

Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world. In particular, Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics. In this study, the (2 + 1)-dimensional time-fractional nonlinear Schrodinger equation and (1 + 1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures. This is shown by constructing appropriate analytic wave solutions. A successful implementation of the advised rational (1/ϕ′(ξ))-expansion method generates new outcomes of the considered equations, by comparing them with those already noted in the literature. On the basis of the conformable fractional derivative, a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme. Finally, the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures. The employed technique is competent, productive and concise enough, making it feasible for future studies.

Influence of fractional time order on W-shaped and Modulation Instability gain in fractional Nonlinear Schrödinger Equation

In this paper, we study the influence of fractional time derivative on W-shaped profile and Modulation Instability gain in fractional Nonlinear Schrodinger Equation (NLSE) which could be used to describe the propagation of pulses in random media, optical metamaterials and others nonlinear systems. We first imply the auxiliary equation method to set up bright, dark and W-shape optical solitons solutions under certain conditions. Thereafter, we graphically depict the obtained results which deeply show the potency of the fractional parameter order on the width and shape of the solitons. We then study the Modulation Instability (MI) gain spectra and we end up remarking that the MI gain sidebands and the MI gain shapes are mainly influenced by the fractional time derivative parameter compared to the previous works reported in nonlinear optic fibers (Wyller et al., 2002; Zhanga et al., 2017).

Wave Structures for Nonlinear Schrodinger Types Fractional Partial Differential Equations Arise in Physical Sciences

Nonlinear partial differential equations are mostly renowned for depicting the underlying behavior of nonlinear phenomena relating to the nature of the real world. In this paper, we discuss analytic solutions of fractional-order nonlinear Schrodinger types equations such as the space-time fractional nonlinear Schrodinger equation and the (2+1)-dimensional time-fractional Schrodinger equation. The considered equations are converted into ordinary differential equations with the help of wave variable transformation and then the recently established rational ( )-expansion method is employed to construct the exact solutions. The obtained solutions have appeared in the forms of a trigonometric function, hyperbolic function, and rational function which are compared with those of literature and claimed to be different. The graphical representations of the solutions are finally brought out for their physical appearances. The applied method is seemed to be efficient, concise, and productive which might be used for further research.
 Mathematics Subject Classifications: 35C08, 35R11

Accurate sets of solitary solutions for the quadratic–cubic fractional nonlinear Schrödinger equation

The analytical and semi-analytical solutions to the quadratic–cubic fractional nonlinear Schrödinger equation are discussed in this research article. The model’s fractional formula is transformed into an integer-order model by using a new fractional operator. The theoretical and computational approaches can now be applied to fractional models, thanks to this transition. The application of two separate computing schemes yields a large number of novel analytical strategies. The obtained solutions secure the original and boundary conditions, which are used to create semi-analytical solutions using the Adomian decomposition process, which is often used to verify the precision of the two computational methods. All the solutions obtained are used to describe the shifts in a physical structure over time in cases where the quantum effect is present, such as wave-particle duality. The precision of all analytical results is tested by re-entering them into the initial model using Mathematica software 12.

Analytical Solution Of Time Fractional Nonlinear Schrodinger Equation By Homotopy Analysis Method

In this research paper, we have applied Homotopy Analysis Approach to the “time fractional nonlinear Schrodinger equation” to find its analytical periodic and solitary wave solution. Presence of convergence control parameter in this method guarantee the solution of time fractional differential equation in the form of rapidly convergent series. Obtained analytical solution has been compared and found in good agreement. This work demonstrates reliability and potential of HAM to study the time fractional partial differential equation

Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation

The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of $$\min \{\kappa \theta ,2-\theta \}$$ order convergence in time and of spectral accuracy in space, where $$\theta $$ is the order of fractional derivative and $$\kappa $$ is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

New types of soliton solutions for space-time fractional cubic nonlinear Schrodinger equation

New definitions for traveling wave transformation and using of new conformable fractional derivative for converting fractional nonlinear evolution equations into the ordinary differential equations are presented in this study. For this aim we consider the time and space fractional derivatives cubic nonlinear Schrodinger equation. Then by using of the efficient and powerful method the exact traveling wave solutions of this equation are obtained. The new definition introduces a promising tool for solving many space-time fractional partial differential equations.

Wave Packets in the Fractional Nonlinear Schrödinger Equation with a Honeycomb Potential

In this article, we study wave dynamics in the fractional nonlinear Schrodinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay be...

Combined soliton solutions of a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation in the cubic–quintic nonlinear material

We concentrate on a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrodinger equation with cubic–quintic nonlinearities, and get some exact combined soliton solutions and periodic solutions based on the Jacobian elliptic function method and double function method. We discuss dynamical behaviors of bright soliton, dark soliton, periodic bright soliton train and periodic M-shaped soliton train, and find that shift distances along the x-direction of these solitons all add if the value of the fractional order increases.

English

We consider highly accurate schemes for nonlinear time fractional Schrodinger equations (NTFSEs). While an L1 strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid L1-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order 2 − γ in time, where 0 < γ < 1 is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.

Propagation dynamics of abruptly autofocusing circular Airy Gaussian vortex beams in the fractional Schrödinger equation

We have investigated the propagation dynamics of the circular Airy Gaussian vortex beams (CAGVBs) in a (2+1)-dimesional optical system discribed by fractional nonlinear Schr\"odinger equation (FNSE). By combining fractional diffraction with nonlinear effects, the abruptly autofocusing effect becomes weaker, the radius of the focusing beams becomes bigger and the autofocusing length will be shorter with increase of fractional diffraction L\'evy index. It has been found that the abruptly autofocusing effect becomes weaker and the abruptly autofocusing length becomes longer if distribution factor of CAGVBs increases for fixing the L\'evy index. The roles of the input power and the topological charge in determining the autofocusing properties are also discussed. Then, we have found the CAGVBs with outward acceleration and shown the autodefocusing properties. Finally, the off-axis CAGVBs with positive vortex pairs in the FNSE optical system have shown interesting features during propagation.