An efficient block Gauss–Seidel iteration method for the space fractional coupled nonlinear Schrödinger equations

Dai, Pingfei; Wu, Qingbiao

doi:10.1016/j.aml.2021.107116

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…Solving nonlinear equations is a nontrivial problem for both the analytical and numerical methods. There are several numerical approaches to highly nonlinear problems, including the NLSE [42][43][44][45]. One of the practical drawbacks of numerical methods is that their efficiency drops for multidimensional problems, which leads to the lower achieved accuracy.…”

Section: Introductionmentioning

confidence: 99%

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Kulagin,

Shapovalov

2024

Mathematics

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The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

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Section: Introductionmentioning

confidence: 99%

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Kulagin,

Shapovalov

2024

Mathematics

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“…In [26], the Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. In [27], the authors propose an efficient block Gauss-Seidel iteration method for solving the complex linear equations arising from the space fractional coupled nonlinear Schrödinger equations. In [28], the authors have given a conservative Fourier spectral method, and used this conservative Fourier spectral method to solve space fractional Klein-Gordon-Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of fractal wave propagation of a multi-dimensional nonlinear fractional-in-space Schrödinger equation

Tang

Wang

2023

Phys. Scr.

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This paper studies a quantum particle traveling in a fractal space-time, which can be modelled by a fractional modiﬁcation of the Schrödinger equation with variable coeﬃcients. The Fourier spectral method is used to reveal the solution properties numerically, and the fractal properties are illustrated graphically by choosing diﬀerent coeﬃcients and diﬀerent fractional orders. Some novel isosurface plots of the dynamics of pattern formation in fractional Schrödinger equation with variable coeﬃcients are shown. Keywords: Fractal wave propagation; Multi-dimensional; Schrödinger equation; Fractal symmetry; Fourier spectral method.

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“…The block iteration approach was applied recently to solve the complex linear equations from the space fractional coupled nonlinear Schrodinger equations [5]. Apart from this, the block over-relaxation variants and their families have lately been utilized for quick computing [6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Enhancing Autonomous Guided Vehicles with Red-Black TOR Iterative Method

Dahalan,

Saudi,

Sulaiman

2023

Mathematics

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To address an autonomous guided vehicle problem, this article presents extended variants of the established block over-relaxation method known as the Block Modified Two-Parameter Over-relaxation (B-MTOR) method. The main challenge in handling autonomous-driven vehicles is to offer an efficient and reliable path-planning algorithm equipped with collision-free feature. This work intends to solve the path navigation with obstacle avoidance problem explicitly by using a numerical approach, where the mobile robot must project a route to outperform the efficiency of its travel from any initial position to the target location in the designated area. The solution builds on the potential field technique that uses Laplace’s equation to restrict the formation of potential functions across operating mobile robot regions. The existing block over-relaxation method and its variants evaluate the computation by obtaining four Laplacian potentials per computation in groups. These groups can also be viewed as groups of two points and single points if they’re close to the boundary. The proposed B-MTOR technique employs red-black ordering with four different weighted parameters. By carefully choosing the optimal parameter values, the suggested B-MTOR improved the computational execution of the algorithm. In red-black ordering, the computational molecules of red and black nodes are symmetrical. When the computation of red nodes is performed, the updated values of their four neighbouring black nodes are applied, and conversely. The performance of the newly proposed B-MTOR method is compared against the existing methods in terms of computational complexity and execution time. The simulation findings reveal that the red-black variants are superior to their corresponding regular variants, with the B-MTOR approach giving the best performance. The experiment also shows that, by applying a finite difference method, the mobile robot is capable of producing a collision-free path from any start to a given target point. In addition, the findings also verified that numerical techniques could provide an accelerated solution and have generated a smoother path than earlier work on the same issue.

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An efficient block Gauss–Seidel iteration method for the space fractional coupled nonlinear Schrödinger equations

Cited by 7 publications

References 11 publications

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Numerical simulation of fractal wave propagation of a multi-dimensional nonlinear fractional-in-space Schrödinger equation

Enhancing Autonomous Guided Vehicles with Red-Black TOR Iterative Method

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