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

DOI10.1051/m2an/2015060zbMath1401.78014OpenAlexW1825496416MaRDI QIDQ2820338

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

Publication date: 15 September 2016

Published in: ESAIM: Mathematical Modelling and Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1051/m2an/2015060

zbMATH Keywords

nonlinear Schrödinger equationRunge-Kutta methodnonlinear opticsinteraction picture methodsymmetric split-step method

Mathematics Subject Classification ID

Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) NLS equations (nonlinear Schrödinger equations) (35Q55) Numerical methods for discrete and fast Fourier transforms (65T50) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Lasers, masers, optical bistability, nonlinear optics (78A60) Error bounds for initial value and initial-boundary value problems involving PDEs (65M15)

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Cites Work

• Unnamed Item
• Unnamed Item
• One-stage exponential integrators for nonlinear Schrödinger equations over long times
• Exponential time differencing for stiff systems
• Embedded Runge-Kutta scheme for step-size control in the interaction picture method
• Symmetric exponential integrators with an application to the cubic Schrödinger equation
• Exponential Runge-Kutta methods for the Schrödinger equation
• A family of embedded Runge-Kutta formulae
• A split-step Fourier method for the complex modified Korteweg-de Vries equation.
• Exponential time integration of solitary waves of cubic Schrödinger equation
• On the interaction picture
• Exponential integrators
• On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
• Solving Ordinary Differential Equations I
• Split-Step Methods for the Solution of the Nonlinear Schrödinger Equation
• The Fourier transform and the discrete Fourier transform
• Generalized Runge-Kutta Processes for Stiff Initial-value Problems†
• Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation
• Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations
• HIGH ORDER EMBEDDED RUNGE-KUTTA SCHEME FOR ADAPTIVE STEP-SIZE CONTROL IN THE INTERACTION PICTURE METHOD
• Fourth-Order Time-Stepping for Stiff PDEs
• Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants
• How well does the finite Fourier transform approximate the Fourier transform?
• Numerical Methods for Ordinary Differential Equations