Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation

a b s t r a c tWe study a numerical solution of the multi-dimensional time dependent Schrödinger equation using a split-operator technique for time stepping and a spectral approximation in the spatial coordinates. We are particularly interested in systems with near spherical symmetries. One expects these problems to be most efficiently computed in spherical coordinates as a coarse grain discretization should be sufficient in the angular directions. However, in this coordinate system the standard Fourier basis does not provide a good basis set in the radial direction. Here, we suggest an alternative basis set based on Chebyshev polynomials and a variable transformation.Furthermore, it is shown how the use of operator splitting produces a splitting error which introduces high frequency modes in the numerical solution in the case of the singular Coulomb potential. Incorporating the Coulomb potential into the radial Laplacian provides a much better splitting. Fortunately our new basis set allows this in some cases.Numerical experiments are presented which demonstrate the advantages and limitations of our technique. Details are demonstrated by 1D toy examples, while the superior efficiency is demonstrated by a 3D example.

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“…For a thorough review of splitting techniques see [10]. [11] applies high-order splitting to the non-linear 1D Schrödinger equation.…”

Section: Operator Splittingmentioning

confidence: 99%

Numerical solution of the 3D time dependent Schrödinger equation in spherical coordinates: Spectral basis and effects of split-operator technique

Sørevik¹,

Birkeland²,

Okša³

2009

Journal of Computational and Applied Mathematics

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“…In this method the time discretization and calculations of time partial derivatives are the same as in the previous subsection, but the spatial derivatives are calculated by Runge-Kutta algorithm. What is applied here is the fourth order Runge-Kutta algorithm, very popular for solving the differential equations [16,21,25,26].…”

Section: Iii2 the Fourth Order Runge-kutta Algorithmmentioning

confidence: 99%

“…where r is the relative amplitude of the two solitons and θ is the relative phase between them [2,11,21,22]. Analytical results [27,30] show that neighboring solitons either come closer or move apart because of the nonlinear inter- action between them.…”

Section: Iii2 the Fourth Order Runge-kutta Algorithmmentioning

confidence: 99%

Propagation of Ultrashort Pulses in a Nonlinear Medium

Van¹,

Thuan²,

Goldstein³

et al. 2010

CMST

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Abstract:In this paper, using a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [8] we study in the specific case of Kerr media. An obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrödinger Equation usually has a very complicated form and looking for its solutions is usually a "mission impossible". Theoretical methods to solve this equation are effective only for some special cases. As an example we describe the method of a developed elliptic Jacobi function expansion. Several numerical methods of finding approximate solutions are simultaneously used. We focus mainly on the following methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon: the collapse of solitons.

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“…The Fourier pseudospectral method has been verified as an accurate and effective technique for solving the envelope-equation in nonlinear optics [15], soliton physics [16], Bose-Einstein condensates [17], and plasma physics [18], therefore in the present article we apply a Fourier pseudospectral algorithm to the solve a 2D paraxial envelopeequation of laser interactions in plasmas. The article in organized as follow: In Section 2, we present the envelope-equation with its main mathematical approximations and physical assumptions.…”

Section: Introductionmentioning

confidence: 99%

Fourier Pseudospectral Solution for a 2D Nonlinear Paraxial Envelope Equation of Laser Interactions in Plasmas

Mahdy¹

2016

JAMP

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We apply a Fourier pseudospectral algorithm to solve a 2D nonlinear paraxial envelope-equation of laser interactions in plasmas. In this algorithm, we first use the second order Strang time-splitting method to split the envelope-equation into a number of equations, next we spatially discrete the filed quantity and its spatial derivatives in these equations in term of Fourier interpolation polynomials (FFT), finally we sequentially integrate the resultant equations by means of a discrete integration method in order to obtain the solution of the envelope-equation. We carry out several numerical tests to illustrate the efficiency and to determine accuracy of the algorithm. In addition, we conduct a number of numerical experiments to examine its performance. The numerical results have shown that the algorithm is highly efficient and sufficiently accurate to solve the 2D envelope-equation, furthermore, it yields an optimal performance in simulating fundamental phenomena in laser interactions in plasmas.

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Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation

Cited by 101 publications

References 19 publications

Numerical solution of the 3D time dependent Schrödinger equation in spherical coordinates: Spectral basis and effects of split-operator technique

Numerical solution of the 3D time dependent Schrödinger equation in spherical coordinates: Spectral basis and effects of split-operator technique

Propagation of Ultrashort Pulses in a Nonlinear Medium

Fourier Pseudospectral Solution for a 2D Nonlinear Paraxial Envelope Equation of Laser Interactions in Plasmas

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