The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters

A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients

In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method

Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims

Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation

The Ginzburg-Landau equation has been applied widely in many fields. It describes the amplitude evolution of instability waves in a large variety of dissipative systems in fluid mechanics, which are close to criticality. In this chapter, we develop a local discontinuous Galerkin method to solve the nonlinear Ginzburg-Landau equation. The nonlinear Ginzburg-Landau problem has been expressed as a system of low-order differential equations. Moreover, we prove stability and optimal order of convergence OhN+1 for Ginzburg-Landau equation where h and N are the space step size and polynomial degree, respectively. The numerical experiments confirm the theoretical results of the method

The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters. © 2021, The Author(s).This study was supported financially by RFBR Grant (19-01-00019), the National Research Centre of Egypt (NRC) and Ghent university

A symmetric low-regularity integrator for the nonlinear Schr\"odinger equation

We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb{T}^d$ and on a smooth bounded domain $\Omega \subset \mathbb{R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e. under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes

Weak martingale solutions for the stochastic nonlinear Schrodinger equation driven by pure jump noise

We construct a martingale solution of the stochastic nonlinear Schrödinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in H^1 on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of càdlàg functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of θ and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.This research has been partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-0971-B-100 and Fundación Séneca -Agencia de Ciencia y Tecnología de la Región de Murcia grant number 20783/PI/18. Also, It has been supported by the National Research Program for Universities (NRPU), Higher Education Commission, Pakistan, No. 8103/Punjab/NRPU/R and D/HEC/2017