Highly accurate finite difference method for coupled nonlinear Schrödinger equation

Ismail, M.S.; Alamri, Sultan Z.

doi:10.1080/00207160410001661339

Cited by 82 publications

(54 citation statements)

References 7 publications

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“…For example, numerical simulations of the propagation and interactions of one-dimensional Langmuir solitons and their generation from random fluctuations by an external pump field were presented by Ismail and Taha [20,21,22,23,24].…”

Section: E55mentioning

confidence: 99%

“…Recently, various numerical methods such as finite difference methods [22], the finite element (fe) method [23], continuous Galerkin (cg) method [20], discontinuous Galerkin (dg) method [31], and discontinuous Petrov-Galerkin (dpg) with optimal test functions [37] have been widely used to solve this problem. However, to the best of our knowledge, only some did numerical work concerning these coupled equations using the finite difference CrankNicolson (cn) scheme and the standard finite element method, while the E56 intensive analysis of the precision of this method is very limited.…”

Section: E55mentioning

confidence: 99%

“…In addition to passing-through collision is an E53 important phenomena such that vector solitons also bounce off each other or trap each other. Solutions of this system have been studied intensively in recent years in numerical aspects and analytical [19,20,21,22,23,24,31,32,33,34,35,36,37]. Numerical examples were investigated by many authors in the one dimensional case [21,21,22,23,24,34].…”

Section: Introductionmentioning

confidence: 99%

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The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Rostamy¹

2013

ANZIAMJ

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We study the new streamline diffusion finite element method for treating the three dimensional coupled nonlinear Schrödinger equation. We derive stability estimates and optimal convergence rates. Moreover, an a priori error estimate is obtained and we compare the corresponding optimal convergence rate for popular numerical methods such as conservative finite difference, semi-implicit finite difference, semi-discrete finite element and the time-splitting spectral method. We justify the advantage of the streamline diffusion method versus the some numerical methods with some examples. Test problems are presented to verify the efficiency and accuracy of the method. The results reveal that the proposed scheme is very effective, convenient and quite accurate for such considered problems rather than other methods.

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Section: E55mentioning

confidence: 99%

Section: E55mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Rostamy¹

2013

ANZIAMJ

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“…Such as the Korteweg-de Vries (KdV) equation [3,4,5,6] and the nonlinear Schrodinger equation has been solved by [7,8]. Numerical solution of coupled partial differential equations, as an example, the coupled nonlinear Schrodinger equation admits soliton solution and it has many applications in communication, this system has been solved numerically by Ismail [9,10,11,12] and the coupled Korteweg-de Vries equation has been solved numerically [13,14,15,16]. The complex nonlinear partial differential equations have been solved in [17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Finite difference method with different high order approximations for solving complex equation

Raslan¹,

El-Danaf²,

Ali³

2017

ntmsci

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Abstract:In this paper, we take finite difference method with different high order approximations for solving Hirota Equation is presented. The stability analysis using Von-Neumann technique shows schemes are unconditionally stable. To test accuracy the error norms L 2 , L ∞ are computed. We compute local truncation error for different schemes. We make comparison between these approximations through the results that we are get it. These results show that the approximation of O(k 2 + h 4 ) introduced here is more accurate than others and easy to apply.

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“…In the literature there are several numerical approaches whose objective is to describe the propagation of solitons in dielectrical environments, most of which use the finite difference method (ISMAIL, 2004;WANG, 2005;CHEN, MALOMED, 2009), the finite element method (DAG, 1999;ISMAIL, 2008) and the split-step method (LIU, 2009 (REICH, 2000), among others. A review of the several numerical procedures applied to describe the propagation of solitons in optical fibers is found in Dehghan and Taleei (2010).…”

Section: Introductionmentioning

confidence: 99%

Sólitons em Fibras Óticas Ideais – Um Desenvolvimento Numérico.

et al. 2010

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ResumoThis work developed a numerical procedure for a system of partial differential equations (PDEs) describing the propagation of solitons in ideal optical fibers. The validation of the procedure was implemented from the numerical comparison between the known analytical solutions of the PDEs system and those obtained by using the numerical procedure developed. It was discovered that the procedure, based on the finite difference method and relaxation Gauss-Seidel method, was adequate in describing the propagation of soliton waves in ideals optical fibers. Key-words: Optical communication. Solitons. Finite differences. Relaxation Gauss-Seidel method. AbstractEste trabalho desenvolveu um procedimento numérico para um sistema de equações diferenciais parciais (EDP's) que descreve a propagação de sólitons em fibras óticas ideais. A validação do procedimento foi implementada a partir da comparação numérica entre as soluções analíticas conhecidas do sistema de EDP's e aquelas obtidas por meio do procedimento numérico desenvolvido. Verificou-se que o procedimento, baseado no método das diferenças finitas e no método de Gauss-Seidel com relaxação, mostrou-se adequado na descrição da propagação das ondas sólitons em fibras óticas ideais. Palavras-chave: Comunicação ótica. Sólitons. Diferenças finitas. Método de Gauss-Seidel com relaxação.

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Highly accurate finite difference method for coupled nonlinear Schrödinger equation

Cited by 82 publications

References 7 publications

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Finite difference method with different high order approximations for solving complex equation

Sólitons em Fibras Óticas Ideais – Um Desenvolvimento Numérico.

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