Relevant bibliographies by topics / Schrödinger equation Numerical solutions

Academic literature on the topic 'Schrödinger equation Numerical solutions'

Author: Grafiati
Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "Schrödinger equation Numerical solutions"

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Паасонен, Виктор Иванович, and Михаил Петрович Федорук. "On the efficiency of high-order difference schemes for the Schro¨dinger equation." Вычислительные технологии, no. 6 (December 16, 2021): 68–81. http://dx.doi.org/10.25743/ict.2021.26.6.006.

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Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties
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Zlotnik, Alexander, and Olga Kireeva. "ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION." Mathematical Modelling and Analysis 26, no. 3 (September 10, 2021): 479–502. http://dx.doi.org/10.3846/mma.2021.13770.

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We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.
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Petridis, Athanasios N., Lawrence P. Staunton, Jon Vermedahl, and Marshall Luban. "Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times." Journal of Modern Physics 01, no. 02 (2010): 124–36. http://dx.doi.org/10.4236/jmp.2010.12018.

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Fermo, L., Mee Van, and S. Seatzu. "Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation." Publications de l'Institut Math?matique (Belgrade) 96, no. 110 (2014): 125–41. http://dx.doi.org/10.2298/pim1410125f.

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We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schr?dinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions.
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Kapoor, Mamta, Nehad Ali Shah, and Wajaree Weera. "Analytical solution of time-fractional Schr<i>ö</i>dinger equations via Shehu Adomian Decomposition Method." AIMS Mathematics 7, no. 10 (2022): 19562–96. http://dx.doi.org/10.3934/math.20221074.

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<abstract> <p>Present research deals with the time-fractional Schr<italic>ö</italic>dinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schr<italic>ö</italic>dinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.</p> </abstract>
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Ritchie, Burke, and Charles A. Weatherford. "Numerical solution of the time-dependent Schr�dinger equation for continuum states." International Journal of Quantum Chemistry 80, no. 4-5 (2000): 934–41. http://dx.doi.org/10.1002/1097-461x(2000)80:4/5<934::aid-qua42>3.0.co;2-o.

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Campoy, G., and A. Palma. "On the numerical solution of the Schr�dinger equation with a polynomial potential." International Journal of Quantum Chemistry 30, S20 (March 10, 1986): 33–43. http://dx.doi.org/10.1002/qua.560300706.

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Zeng Si-Liang, Zou Shi-Yang, Wang Jian-Guo, and Yan Jun. "Numerical solution of the three-dimensional time-dependent Schr?dinger equation and its application." Acta Physica Sinica 58, no. 12 (2009): 8180. http://dx.doi.org/10.7498/aps.58.8180.

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Vigo-Aguiar, Jes�s, and Higinio Ramos. "A variable-step Numerov method for the numerical solution of the Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 255–62. http://dx.doi.org/10.1007/s10910-004-1467-3.

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Anastassi, Z. A., and T. E. Simos. "Trigonometrically fitted Runge?Kutta methods for the numerical solution of the Schr�dinger equation." Journal of Mathematical Chemistry 37, no. 3 (April 2005): 281–93. http://dx.doi.org/10.1007/s10910-004-1470-8.

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Dissertations / Theses on the topic "Schrödinger equation Numerical solutions"

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Aydin, Ayhan. "Geometric Integrators For Coupled Nonlinear Schrodinger Equation." Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12605773/index.pdf.

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Multisymplectic integrators like Preissman and six-point schemes and a semi-explicit symplectic method are applied to the coupled nonlinear Schrö
dinger equations (CNLSE). Energy, momentum and additional conserved quantities are preserved by the multisymplectic integrators, which are shown using modified equations. The multisymplectic schemes are backward stable and non-dissipative. A semi-explicit method which is symplectic in the space variable and based on linear-nonlinear, even-odd splitting in time is derived. These methods are applied to the CNLSE with plane wave and soliton solutions for various combinations of the parameters of the equation. The numerical results confirm the excellent long time behavior of the conserved quantities and preservation of the shape of the soliton solutions in space and time.
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Conference papers on the topic "Schrödinger equation Numerical solutions"

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Manevitch, Leonid I., and Andrey I. Musienko. "Transversal Nonlinear Dynamics of Stretched Chain on Elastic Foundation." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84920.

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We present analytical and numerical study of short wavelength breathers in the system of asymmetric nonlinear oscillators coupled by stretched weightless beam. The study is focused on modulations of the nonlinear normal mode with shortest wavelength. A small parameter is introduced as ratio of distance between the particles in the chain and characteristic wavelength of modulation. It is shown that in main asymptotic approach the problem can be reduced to Nonlinear Schro¨dinger Equation (NSE). Breathers (envelope solitons) are localized oscillating solutions of this equation. We reveal the stability and high mobility of such solutions in their numerical study based on integrating of exact discrete equations of motion. We study breathers, localized on chain ends, and radiation of breathers from chain ends, induced by variable external force.
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Kuo, M. K., T. R. Lin, B. T. Liao, and C. H. Yu. "Optical Properties of InAs/GaAs Quantum Dots Grown by Epitaxy." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-59941.

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Strain effects on optical properties of self-assembled InAs/GaAs quantum dots grown by epitaxy are investigated. A new model based on the theory of linear elasticity is developed to analyze three-dimensional induced strain field. The model takes sequence of fabrication process of quantum-dot into account, where the mismatch of lattice constants between wetting layer and substrate is treated as initial strain first for the heterostructure system without capping material. The obtained total strain field is then treated as initial strain again for the whole heterostructure system with capping material. The computed strain from these two-steps analysis is then used as an input for the electronic band structure calculation. The numerical results show that strain field from this new model has significant difference from the usual model where the sequence of fabrication process is omitted. The strain-induced potential is incorporated into the three-dimensional steady state effective mass Schro¨dinger equation with the aid of the Pikus-Bir Hamiltonian and Luttinger-Kohn formalism. Both the strain field and the solutions of the steady state Schro¨dinger equations are found numerically by using of a commercial finite element package. The energy levels as well as the wave functions of both conduction and valence bands of quantum dots are calculated. Finally, energy of interband optical transitions is obtained in numerical experiments.
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Phibanchon, Sarun, and Michael A. Allen. "Numerical Solutions of the Nonlinear Schrödinger Equation with a Square Root Nonlinearity." In 2010 International Conference on Computational Science and Its Applications. IEEE, 2010. http://dx.doi.org/10.1109/iccsa.2010.68.

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Shemer, Lev, and Boris Dorfman. "Spatial vs. Temporal Evolution of Nonlinear Wave Groups: Experiments and Modeling Based on the Dysthe Equation." In ASME 2008 27th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2008. http://dx.doi.org/10.1115/omae2008-57652.

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The evolution along the tank of unidirectional nonlinear wave groups with narrow spectrum is studied both experimentally and numerically. Measurements of the instantaneous surface elevation within the tank are carried out using digital processing of video-recorded sequences of images of the contact line movement at the tank side wall. The accuracy of the video-derived results is verified by measurements performed by conventional resistance-type wave gauges. An experimental procedure is developed that enables processing of large volumes of video images and capturing the spatial structure of the instantaneous wave field in the whole tank. The experimentally obtained data are compared quantitatively with the solutions of the modified nonlinear Schro¨dinger (MNLS, or Dysthe) equation written in either temporal or spatial form. Results on the evolution along the tank of wave frequency spectra and on the temporal evolution of the wave number spectra are presented. It is demonstrated that accounting for the 2nd order bound (locked) waves is essential for getting a qualitative and quantitative agreement between the measured and the computed spectra.
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Fermo, Luisa, Cornelis van der Mee, and Sebastiano Seatzu. "Numerical solution of the direct scattering problem for the nonlinear Schrödinger equation." In 2015 Tyrrhenian International Workshop on Digital Communications (TIWDC). IEEE, 2015. http://dx.doi.org/10.1109/tiwdc.2015.7323323.

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Hu, Hanhong, and Ning Ma. "Numerical Simulation on Nonlinear Evolution of Rogue Waves on Currents Based on the NLS Equation." In ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2011. http://dx.doi.org/10.1115/omae2011-50079.

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In this paper, nonlinear instability and evolution of deep-water rogue waves on following and opposing currents were described by numerical simulation for laboratory investigation. The generation of rogue waves in a numerical tank by means of wave focusing technique had been studied. Here a spatial domain model of current modified nonlinear Schro¨dinger (NLSC) equations in one horizontal dimension (1D) was established for describing the deep-water wave trains in a prescribed stationary current field. The transient water waves (TWW) was adopted as the initial condition of the NLSC equation. The steady current was added to see the effect of wave-current interaction on the energy concentration of gravity waves. The influence of current as well as other terms in the NLSC equations on wave height, inclination, particle velocity and acceleration are shown. Meanwhile, the focusing time/position of TWW influenced by the current field is investigated, which is of course a very important factor in experimental research when we generate rogue waves in the laboratory.
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Krogstad, Harald E., Jingdong Liu, Herve´ Socquet-Juglard, Kristian B. Dysthe, and Karsten Trulsen. "Spatial Extreme Value Analysis of Nonlinear Simulations of Random Surface Waves." In ASME 2004 23rd International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2004. http://dx.doi.org/10.1115/omae2004-51336.

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The paper first recalls the Slepian Model Representation and a theorem of V. I. Piterbarg as generic tools for analyzing the spatial characteristic of ocean waves. We then consider numerical simulations of random surface gravity waves carried out in space and time by means of the modified nonlinear Schro¨dinger equation. It is shown that the extreme waves in the simulations are steeper and more asymmetric than predicted by the Gaussian theory. Moreover, the reconstructed wave fields shows extreme crest heights well in excess of what is expected from the Gaussian theory.
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Lin, T. R., M. K. Kuo, and K. B. Hong. "Strain Fields and Transition Energies in Multilayer InAs/GaAs Quantum Dots." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-68550.

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Strain fields and transition energies of vertically stacked semiconductor quantum dots are investigated. Strain fields, induced by lattice mismatches in heterostructures, in and around quantum dots are analyzed on the basis of the linear elasticity. The three-dimensional steady-state strained-effective-mass Schro¨dinger equation is then modified by incorporating the effects of strain fields into the carrier confinement potential and is analyzed by finite element methods numerically. The results include the energy levels and wavefunction spectra of InAs/GaAs quantum dots. The calculated optoelectronic transition energies agree well with the previous experimental photoluminescence data. Numerical results also suggest that transition energy decreases as the spacer thickness increases.
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Hu, Hanhong, Ning Ma, Xuefeng Wang, and Xiechong Gu. "Numerical Simulation of Rogue Waves Based on the Fourth-Order NLS Equation for Laboratory Experimental Investigation." In ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2010. http://dx.doi.org/10.1115/omae2010-20287.

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The main purposes of investigating the generation of the rogue waves in offshore engineering include: 1) prediction of its occurrence to protect the offshore structure from attacking; 2) the experimental investigation of rogue waves/structure interaction for the structure design. The latter one calls high requirement of wave generation and calculation. In this paper, we establish a spatial domain model of fourth order nonlinear Schro¨dinger (NLS) equation for describing deep-water wave trains in moving coordinate system. For the first purpose mentioned above, this paper presents the evolution of random wave trains in real sea state described by the Joint North Sea Wave Project (JONSWAP) power spectrum numerically, which is governed by the NLS equation. The parameters of the spectrum are evaluated to discuss their effect on the occurrence of rogue waves. For the second purpose to generate rogue waves in experimental tank efficiently, the transient wave is focused for its allowance of precise determination of concentration place/time. First we simulate the three-dimensional transient waves in the numerical tank modeling the deepwater basin with double-side multi-segmented wave-maker in Shanghai Jiao Tong University (SJTU) with linear superposing theory. To discuss its nonlinearity for the guidance of experiment, the transient wave is set as the initial condition of the NLS equation and the difference from the linear simulation is presented, which could be given as the suggestion to the preparation of experiment.
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