The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics

Balac, Stéphane; Fernandez, Arnaud; Mahé, Fabrice; Méhats, Florian; Texier-Picard, Rozenn

doi:10.1051/m2an/2015060

Cited by 24 publications

(33 citation statements)

References 32 publications

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“…From (9), one can show [6] that the new unknown A ip k is solution to the following problem over each subinterval [z …”

Section: Overview Of the Interaction Picture Methodsmentioning

confidence: 99%

“…To be comprehensive we may bring to the attention of the reader that a numerical comparison of the IP method and the Symmetric Split-Step method, the most popular method for solving the GNLSE, has been carried out on benchmark problems in optics in [5] and a theoretical comparison of the 2 methods has been achieved in [6]. These investigations demonstrate the supremacy of the IP method over the Symmetric Split-Step method for solving the GNLSE.…”

Section: Introductionmentioning

confidence: 94%

“…The spip program solves the PDE problem (7) with N as given by (6) or (8) by means of the Interaction Picture (IP) method. The main idea of the IP method is a change of unknown in order to transform the GNLSE for the unknown A into a new equation where only remains an explicit reference to the partial derivation with respect to the space variable z and where the time variable t appears as a parameter.…”

Section: Introductionmentioning

confidence: 99%

“…Latter the IP method has been applied for solving the GNLSE in [5]. A comprehensive mathematical study of the Interaction Picture method for solving the GNLSE has been conducted in [6]. Recently an efficient embedded RK method based on Dormand and Prince RK4(3)-T formula [7] and specifically designed for the IP method has been proposed in [8] to provide a costless adaptive step-size control in the RK4-IP method.…”

Section: Introductionmentioning

confidence: 99%

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SPIP : A computer program implementing the Interaction Picture method for simulation of light-wave propagation in optical fibre

Balac

Fernandez

2016

Computer Physics Communications

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The computer program spip is aimed at solving the Generalized Non-Linear Schrödinger equation (GNLSE), involved in optics e.g. in the modelling of light-wave propagation in an optical fibre, by the Interaction Picture method, a new efficient alternative method to the Symmetric Split-Step method. In the spip program a dedicated costless adaptive step-size control based on the use of a 4 th order embedded Runge-Kutta method is implemented in order to speed up the resolution. linear optics as a model of wave propagation in fibre optics. Solution method: The GNLSE is solved by the Interaction Picture method coupled with an embedded Runge-Kutta scheme of order 4. The program includes a costless adaptive step-size control strategy taking advantage of the features of an embedded Runge-Kutta scheme designed for delivering a local error estimate at no extra-cost compared to the standard 4 th order Runge-Kutta scheme. Keywords Unusual features:Running time: highly dependent on the fibre length and accuracy required for the results. Typically between half a minute and several dozens of minutes.

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“…From (9), one can show [6] that the new unknown A ip k is solution to the following problem over each subinterval [z …”

Section: Overview Of the Interaction Picture Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

SPIP : A computer program implementing the Interaction Picture method for simulation of light-wave propagation in optical fibre

Balac

Fernandez

2016

Computer Physics Communications

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“…(1) in this paper is performed by means of the ''fourth-order Runge-Kutta method in the interaction picture method'' (RK4IP method) RK4IP can generally be interpreted as an exponential Runge-Kutta method applied to the solution of parabolic PDE [33][34][35][36][37]. The method has a 4th-order convergence, which makes it a good alternative to all known variants of the split step Fourier methods (SSFM).…”

Section: Numerical Solution Of the Basic Equationmentioning

confidence: 99%

Influence of higher-order effects on pulsating solutions, stationary solutions and moving fronts in the presence of linear and nonlinear gain/loss and spectral filtering

Uzunov

Arabadzhiev

Georgiev

2015

Optical Fiber Technology

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On the convergence of Lawson methods for semilinear stiff problems

2020

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Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

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The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics

Cited by 24 publications

References 32 publications

SPIP : A computer program implementing the Interaction Picture method for simulation of light-wave propagation in optical fibre

SPIP : A computer program implementing the Interaction Picture method for simulation of light-wave propagation in optical fibre

Influence of higher-order effects on pulsating solutions, stationary solutions and moving fronts in the presence of linear and nonlinear gain/loss and spectral filtering

On the convergence of Lawson methods for semilinear stiff problems

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