A split-step Fourier method for the complex modified Korteweg-de Vries equation

Muslu, Gülçin M.; Erbay, H. A.

doi:10.1016/s0898-1221(03)80033-0

Cited by 74 publications

(52 citation statements)

References 17 publications

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“…Consistent with the linear operator, (8), periodic boundary conditions are assumed. Here, we implement boundary conditions for the temporal window by employing auxiliary points outside of the actual domain (also called "halo" or "ghost" by some authors)…”

Section: A Central Difference Methodsmentioning

confidence: 99%

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A Reliable Split-Step Fourier Method for the Propagation Equation of Ultra-Fast Pulses in Single-Mode Optical Fibers

Deiterding

Glowinski

Oliver

et al. 2013

J. Lightwave Technol.

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Abstract-The extension to the split-step Fourier method (SSFM) for Schrödinger-type pulse propagation equations that we propose in this article is designed with the accurate simulation of pulses in the femto-second regime in single-mode communication fibers in mind. We show that via an appropriate operator splitting scheme, Kerr nonlinearity and the self-steepening and stimulated Raman scattering terms can be combined into a single sub-step consisting of an inhomogeneous quasilinear first-order hyperbolic system for the real-valued quantities intensity and phase. First-and second-order accurate shock-capturing upwind schemes have been developed specifically for this nonlinear substep, which enables the accurate and oscillation-free simulation of signals under the influence of Raman scattering and extreme selfsteepening with the SSFM. Benchmark computations of ultra-fast Gaussian pulses in fibers with strong nonlinearity demonstrate the superior approximation properties of the proposed approach.

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Section: A Central Difference Methodsmentioning

confidence: 99%

“…Utilizing the Baker-Campbell-Hausdorff formula for expanding two non-commuting operators, (6) can be proven to be an O(h 2 ) approximation [8]. Comprehensive descriptions of the split-step approach for simulating pulse propagation in fibers are given for instance by Agrawal [1] and Hohage & Schmidt [3].…”

Section: A Splitting Algorithmsmentioning

confidence: 99%

A Reliable Split-Step Fourier Method for the Propagation Equation of Ultra-Fast Pulses in Single-Mode Optical Fibers

Deiterding

Glowinski

Oliver

et al. 2013

J. Lightwave Technol.

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show abstract

“…In some special case, such as q = 0 or p = 0 or θ = θ 0 , which respectively correspond to the 0, π/2 and θ 0 polarizations, the above coupled nonlinear equations reduce to a single MKDV equation. In this case, the CMKDV equation has the following analytical solution (Muslu and Erabay, 2003):…”

Section: Introductionmentioning

confidence: 99%

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Yan

Zheng

2017

International Journal of Applied Mathematics and Computer Science

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The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation.

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“…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]. The nonintegrable variant of Hirota equation in which the nonlinear term in (1) is replaced by |u| 2 u x , is solved numerically by [17,19].…”

Section: Introductionmentioning

confidence: 99%

“…The complex nonlinear partial differential equations have been solved in [17,18,19,20,21]. The nonintegrable variant of Hirota equation in which the nonlinear term in (1) is replaced by |u| 2 u x , is solved numerically by [17,19].…”

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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A split-step Fourier method for the complex modified Korteweg-de Vries equation

Cited by 74 publications

References 17 publications

A Reliable Split-Step Fourier Method for the Propagation Equation of Ultra-Fast Pulses in Single-Mode Optical Fibers

A Reliable Split-Step Fourier Method for the Propagation Equation of Ultra-Fast Pulses in Single-Mode Optical Fibers

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

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

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