Nonlinear grid mapping applied to an FDTD-based, multi-center 3D Schrödinger equation solver

Bigaouette, Nicolas; Ackad, Edward; Ramunno, Lora

doi:10.1016/j.cpc.2011.08.011

Cited by 15 publications

(20 citation statements)

References 32 publications

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“…Several methods for numerical treatment of time-dependent Schrödinger equations are known. If we enumerate some of them, they are the finite difference time domain (FDTD) method [26][27][28][29][30][31], the discretization method that takes advantage of the asymptotic behavior correspondence (ABC) [32,33], and the discrete local discontinuous Galerkin method [34]. In particular, the FDTD method has been widely applied to obtain numerical solutions of mechanical problems of dynamical systems including Maxwell-Schrödinger equations for electromagnetic fields [30,31].…”

Section: Resultsmentioning

confidence: 99%

Analyzing Quantum Time‐Dependent Singular Potential Systems in One Dimension

Choi¹

2016

Nonlinear Systems - Design, Analysis, Estimation and Control

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Quantum states of a particle subjected to time-dependent singular potentials in onedimension are investigated using invariant operator method and the Nikiforov-Uvarov method. We consider the case that the system is governed by two singular potentials which are the Coulomb potential and the inverse quadratic potential. An invariant operator that is a function of time has been constructed via a fundamental mechanics. This invariant operator is transformed to a simple one using a unitary operator, which is a time-independent invariant operator. By solving the Schrödinger equation in the transformed system, analytical forms of exact eigenvalues and eigenfunctions of the invariant operator are evaluated in a simple elegant manner with the help of the Nikiforov-Uvarov method. Eventually, the full wave functions in the original system (untransformed system) are obtained through an inverse unitary transformation from the wave functions in the transformed system. Quantum characteristics of the system associated with the wave functions are addressed in detail.

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Section: Resultsmentioning

confidence: 99%

Analyzing Quantum Time‐Dependent Singular Potential Systems in One Dimension

Choi¹

2016

Nonlinear Systems - Design, Analysis, Estimation and Control

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“…This stability criterion is very general as it is valid for every possible spatial scheme that uses the collocated scheme (18) or the equivalent staggered in time schemes ( 19)-( 20) or ( 21)- (22). For example, it would be possible to apply Theorem 1 to a Hamiltonian matrix with a Laplacian L constructed using secondorder forward or backward differences on a uniform grid or using the nonuniform grid mapping presented in [11]. In fact, the time step used in the latter was derived based on a uniform grid mapping.…”

Section: Stabilitymentioning

confidence: 99%

“…Instead of looking at the data in a few points, the method in [11] is applied, where the time-domain data at every point is multiplied by a random number between −0.5 and 0.5 and summed together. This is done to ensure that most eigenstates are found.…”

Section: A Particle-in-a-3-d-boxmentioning

confidence: 99%

“…To increase the accuracy in these areas, it is possible to use nonuniform tensor product grids, meaning that each spatial step can vary along its axis. In [11], such a scheme is proposed, where a mapping function is used to convert a nonuniform grid into a uniform grid. However, it does not allow for true higher-order accurate spatial schemes even when using extended stencils.…”

Section: Introductionmentioning

confidence: 99%

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Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation

Decleer

Londersele

Rogier

et al. 2021

Journal of Computational and Applied Mathematics

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“…In the ocean and nonlinear optics etc., a lot of mathematical models can be described by nonlinear partial differential equations, which are used to analyze and study rogue waves. One of the important known models is the nonlinear Schrödinger (NLS) equation, which can depict a large number of phenomena and dynamic processes in physics, chemistry, biology and computer science [12], [13]. Many research results have been obtained for the NLS equation [14]- [17].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics of Rogue Waves in a Fifth-Order Nonlinear Schrödinger Equation

2020

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In this paper, nonlinear dynamics of higher-order rogue waves are investigated for a fifthorder nonlinear Schrödinger equation, which can be used to depict the Heisenberg ferromagnetic spin chain. A generalized Darboux transformation is constructed based on the Lax pair. Higher-order rogue waves solutions are given in terms of a recursive formula. Using numerical simulation, the first-order to the thirdorder rogue waves are displayed on the basis of some free parameters, which play a crucial role in affecting the distribution of rogue waves. The results obtained should be useful in understanding the generation mechanism of rogue waves. INDEX TERMS Darboux transformation, fifth-order nonlinear Schrödinger equation, rogue waves.

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Nonlinear grid mapping applied to an FDTD-based, multi-center 3D Schrödinger equation solver

Cited by 15 publications

References 32 publications

Analyzing Quantum Time‐Dependent Singular Potential Systems in One Dimension

Analyzing Quantum Time‐Dependent Singular Potential Systems in One Dimension

Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation

Nonlinear Dynamics of Rogue Waves in a Fifth-Order Nonlinear Schrödinger Equation

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