Query: Numerical study of time-splitting spectral discretizations of nonlinear Schr¨odinger equations in the semi-classical regimes

Candidate papers:

abstract: An electrical meter has light emissive display elements and a meter circuit featuring electrically floating input terminals relative to device ground, and offers a potentially significant cost advantage compared to available alternatives in low accuracy measuring applications. A series string of the light emissive display elements are physically arranged along a line in order of increasing turn on current threshold of each element so that they are successively turned on along the line with increasing current to be measured. A control circuit included in the series string has first and second semiconductively complementary active semiconductor devices, such as an NPN and PNP transistor, each with base, emitter, and collector. The input sensing conductor, such as a base, of each active semiconductor is a respective floating input of the meter. The semiconductors have at least two output control conductors, such as collector and emitter, two corresponding control conductors being wired together (as emitter to emitter) and the other two being wired into the series string. The light emitting elements are suitably semiconductor diodes (LEDs) incandescent bulbs, neon bulbs, or other devices, with resistive shunting where necessary. Two such meter circuits are wired back to back with their light emissive elements arranged physically back to back to form an uncomplicated galvanometer device for measuring electrical currents of either positive or negative polarity.
author_names: Weizhu Bao

Shi Jin

Peter A Markowich
title: Numerical Study of Time Splitting Spectral Discretizations of Nonlinear Schrodinger Equations in the Semiclassical Regimes
venue: SIAM J. Sci. Comput.
year: 2003

abstract: Purpose The purpose of this paper is to consider the time splitting Fourier spectral (TSFS) method to solve the fractional coupled Klein Gordon Schrodinger (K G S) equations. A time splitting spectral approach is applied for discretizing the Schrodinger like equation and along with that, a pseudospectral discretization has been accurately utilized for the temporal derivatives in the Klein Gordon like equation. Furthermore, the time splitting scheme is proved to be unconditionally stable. Numerical experiments guarantee high accuracy of the TSFS scheme for the K G S equations. Here, the derivative of fractional order is taken in the Riesz sense. Design/methodology/approach The focus of this paper is to study the Riesz fractional coupled K G S equations using the TSFS method. This method is dependent on evaluating the solution to the given problem in small steps, and treating the nonlinear and linear steps separately. The nonlinear step is made in the time domain, while the linear step is made in the frequency domain, which necessitates the use of Fourier transform back and forth. It is a very effective, powerful and efficient method to solve the nonlinear differential equations, as in previous works (Bao et al. 2002; Bao and Yang, 2007; Muslu and Erbay, 2003; Borluk et al. 2007) the initial and boundary value problem is decomposed into linear and nonlinear subproblems. Summarizing the technique of the TSFS method, it can be stated that first the Schrodinger like equation is solved in two splitting steps. Then, the Klein Gordon like equation is solved by discretizing the spatial derivatives by means of the pseudospectral method. Findings The utilized method is found to be very efficient and accurate. Moreover, the time splitting spectral scheme is found to be unconditionally stable. By means of thorough study, it is found that the spectral method is time reversible, is gauge invariant and also conserves the total charge. Moreover, the results have been graphically presented to exhibit the accuracy of the proposed methods. Apart from that, the numerical solutions have been also compared with the exact solutions. Numerical experiments establish that the proposed technique manifests high accuracy and efficiency. Originality/value To the authors' best knowledge, the Riesz fractional coupled K G S equations have been for the first time solved by using the TSFS method.
author_names: Santanu Saha Ray
title: On the time splitting Fourier spectral method for the coupled Klein Gordon Schrodinger equations in the quantum field theory
year: 2017

abstract: We study stable blow up dynamics in the $L^2$ critical nonlinear Schr\"{o}dinger equation in high dimensions. First, we show that in dimensions $d=4$ to $d=12$ generic blow up behavior confirms the "log log" regime in our numerical simulations, including the log log rate and the convergence of the blow up profiles to the rescaled ground state; this matches the description of the stable blow up regime in the dimension $d =2$ (for the 2d cubic NLS equation) Next, we address the question of rigorous justification of the "log log" dynamics in higher dimensions $d \geq5) at least for the initial data with the mass slightly larger than the mass of the ground state, for which the spectral conjecture has yet to be proved, see [34] and [10] We give a numerically assisted proof of the spectral property for the dimensions from $d=5$ to $d=12$ and a modification of it in dimensions $2 \leq d \leq 12$ This, combined with previous results of Merle Rapha\"el, proves the "log log" stable blow up regime in dimensions $d \leq 10$ and radially stable for $d \leq 12$
author_names: Kai Yang

Svetlana Roudenko

Yanxiang Zhao
title: Blow up dynamics and spectral property in the $L^2$ critical nonlinear Schr\"odinger equation in high dimensions
year: 2017

abstract: This paper focuses on performance of several efficient and accurate numerical methods for the long wave short wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on: (i) the utilization of the first order or second order time splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives; (iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher order accurate, they are proved to preserve two invariants including the position density in L. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time step, mesh size and dispersion parameter e, and demonstrate the behavior of nonlinear dispersive waves in the semi classical limit regime. Mathematics subject classification: 65M06, 65M12
author_names: Tingchun Wang
title: Efficient and Accurate Numerical Methods for Long Wave Short Wave Interaction Equations in the Semiclassical Limit Regime
venue: Journal of Computational Mathematics
year: 2019

abstract: We numerically study supercontinuum (SC) generation (SCG) in a rare earth doped highly nonlinear photonic crystal fiber (HNL PCF) with anomalous dispersion (AD) in the sub picosecond pulse regime. We develop a semi classical numerical model based on the generalized Ginzburg Landau equation in order to take account of ultrafast interactions between gain ions and ultra broadband SC radiation encompassing sub 100 femtosecond solitons. Based on the numerical model, we analyze SCG characteristics of an active HNL PCF in comparison with a passive type counterpart, unveiling novel optical gain effects in a highly nonlinear optical fiber with AD. We rigorously investigate gain induced soliton dynamics, such as soliton cascade like behaviors, soliton quasi soliton collisions, and phase matched dispersive wave generation, which eventually contributes to enhancement of energy scaling of SC radiation without incurring considerable degradation of its spectral flatness. We also verify that such superior performance characteristics of an active HNL PCF make it suitable for the use as a boost amplifier for SC radiation. We think that the findings from this study will incite other subsequent studies on unveiling detailed nonlinear pulse dynamics in various gain embedded nonlinear optical media.
author_names: Kyoungyoon Park

Jeongkyun Na

Juhwan Kim

Yoonchan Jeong
title: Numerical Study on Supercontinuum Generation in an Active Highly Nonlinear Photonic Crystal Fiber With Anomalous Dispersion
venue: IEEE Journal of Quantum Electronics
year: 2020

abstract: Abstract The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time dependent Schrodinger equation in the semiclassical regime, where the Planck constant e is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time dependent Schrodinger equation with a linear potential. The local discretization error of the two time splitting methods constructed here is O max D t 3 D t 5 e while the well known Lie Trotter splitting scheme and the Strang splitting scheme are O D t 2 e and O D t 3 e respectively, where Dt is the time step size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrodinger equation with a quadratic potential.
author_names: Wansheng Wang

Jiao Tang
title: Efficient exponential splitting spectral methods for linear Schrodinger equation in the semiclassical regime
year: 2020

abstract: This paper presents and compares various numerical techniques for the long wave short wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time splitting approach combined with the Runge Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi classical limit.
author_names: Qianshun Chang

Yau Shu Wong

Chi-Kun Lin
title: Numerical computations for long wave short wave interaction equations in semi classical limit
venue: J. Comput. Phys.
year: 2008

abstract: As a basic principle, benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations. In this work, the capacity of locally adaptive space and time discretisations for the numerical solution of low dimensional nonlinear Schrodinger equations is investigated. The considered model equation is related to the time dependent Gross Pitaevskii equation arising in the description of Bose Einstein condensates in dilute gases. The performance of the Fourier pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher order exponential operator splitting methods with variable time stepsizes is studied. Numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs, respectively, is effective in different situations with an enhancement either in efficiency or reliability. As expected, adaptive time splitting schemes combined with fast Fourier transform techniques are favourable regarding accuracy and efficiency when applied to Gross Pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution. However, the numerical solution of nonlinear Schrodinger equations in the semi classical regime becomes a demanding task. Due to the highly oscillatory and nonlinear nature of the problem, the spatial mesh size and the time increments need to be of the size of the decisive parameter 0<e1, especially when it is desired to capture correctly the quantitative behaviour of the wave function itself. The required high resolution in space constricts the feasibility of numerical computations for both, the Fourier pseudo spectral and the finite element method. Nevertheless, for smaller parameter values locally adaptive time discretisations facilitate to determine the time stepsizes sufficiently small in order that the numerical approximation captures correctly the behaviour of the analytical solution. Further illustrations for Gross Pitaevskii equations with a focusing nonlinearity or a sharp Gaussian as initial condition, respectively, complement the numerical study.
author_names: Mechthild Thalhammer

Jochen Abhau
title: A numerical study of adaptive space and time discretisations for Gross Pitaevskii equations
venue: J. Comput. Phys.
year: 2012

abstract: In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schr\"odinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schr\"odinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical soliton which is close in energy norm to the continuous soliton. Such result is valid under a CFL condition between the time and space stepsizes. Furthermore we prove that if the initial datum is symmetric and close to the continuous soliton, then the associated numerical solution remains close to the orbit of the continuous soliton for very long times.
author_names: Dario Bambusi

Erwan Faou

Benoit Grebert
title: Existence and stability of solitons for fully discrete approximations of the nonlinear Schr\"odinger equation
year: 2012

abstract: In this paper we study time splitting spectral approximations for the linear Schrodinger equation in the semiclassical regime, where the Planck constant e is small. In this regime, the equation propagates oscillations with a wavelength of O (e) and finite difference approximations require the spatial mesh size h o(e) and the time step k o(e) in order to obtain physically correct observables. Much sharper mesh size constraints are necessary for a uniform L2 approximation of the wave function. The spectral time splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L2 approximation of the wave function for k o(e) and h O(e) Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g. k independent of e, and h O (e) are admissible for obtaining "correct" observables. Finally, we address the application to nonlinear Schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies.
author_names: Weizhu Bao

Shi Jin

Peter A Markowich
title: On time splitting spectral approximations for the Schrodinger equation in the semiclassical regime
year: 2002