Approximated exponential integrators for the stochastic Manakov equation

André Berg; David Cohen; Guillaume Dujardin

doi:10.3934/jcd.2023002

Article Contents

2023, Volume 10, Issue 2: 323-344. Doi: 10.3934/jcd.2023002

This issue Previous Article On the preservation of second integrals by Runge-Kutta methods Next Article Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

Approximated exponential integrators for the stochastic Manakov equation

1.
Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden
2.
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 41296 Gothenburg, Sweden
3.
Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France

^* Corresponding author: David Cohen
^* Corresponding author: David Cohen

Received: September 2021

Revised: March 2023

Early access: March 2023

Published: April 2023

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Abstract

This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is $ 1/2 $. This is then used to prove that the approximated exponential integrator has convergence order $ 1/2 $ in probability and almost sure order $ 1/2^{-} $, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the approximated exponential integrator as well as a modified version of it.

Keywords:

Mathematics Subject Classification: 65C30, 65C50, 65J08, 60H15, 60M15, 60-08, 35Q55.

Citation:

FullText(HTML)

Figure 1. $ 3d $ space-time evolution (Figure 1a) and colormap (Figure 1b) of the intensity of the first component (left) and the second component (right) of the numerical solution to the stochastic Manakov system (1) with initial value (8) computed using the approximated exponential integrator (3). The discretization parameters are $ h = 3/625 $ and $ \Delta x = 1/4 $

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Figure 2. Strong rates of convergence for the stochastic Manakov system (1) with initial value (8): Error $ {\mathbb E}\left[\left\lVert{X^N-X_{\text{ref}}(T)}\right\rVert_{ {\mathbb H}^1}^2\right] $ at the time $ T = 1 $ as a function of the time step $ h $ in loglog scale. The discretization parameters are $ h $ from $ 2^{-6} $ to $ 2^{-16} $ and $ \Delta x = 0.4 $

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Figure 3. Efficiency of the time integrators for the stochastic Manakov system (1) with initial value (8): Computational time as a function of the averaged final error $ {\mathbb E}\left[\left\lVert{X^N-X_{\text{ref}}(T)}\right\rVert_{ {\mathbb L}^2}^2\right] $ at $ T = 0.5 $ over $ 500 $ samples (loglog scale). The discretization parameters are $ h $ from $ 2^{-9} $ to $ 2^{-17} $ and $ \Delta x = 0.2 $

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Figure 4. Evolution of the (squared) $ {\mathbb L}^2 $-norm along the numerical solutions of the four numerical integrators applied to the stochastic Manakov system (1) with initial values (8). The discretization parameters are $ h = 0.006 $ and $ \Delta x = 0.25 $

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Figure 5. Left: Evolution of the (squared) $ {\mathbb L}^2 $-norm along the numerical solutions of both exponential schemes applied to the stochastic Manakov system (1) with initial values (8). The discretization parameters are $ h = 0.006 $ and $ \Delta x = 0.25 $. Right: Strong rates of convergence for the stochastic Manakov system (1) with initial value (8): Error $ {\mathbb E}\left[\left\lVert{X^N-X_{\text{ref}}(T)}\right\rVert_{ {\mathbb H}^1}^2\right] $ at the time $ T = 1 $ as a function of the time step $ h $ in loglog scale. The discretization parameters are $ h $ from $ 2^{-6} $ to $ 2^{-16} $ and $ \Delta x = 0.4 $

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Figure 6. Strong rate of convergence (left) and evolution of the (squared) $ {\mathbb L}^2 $-norm (right) of the approximated exponential scheme when applied to the inhomogeneous stochastic Manakov equation (9). The discretization parameters are $ h $ from $ 2^{-13} $ to $ 2^{-19} $ and $ \Delta x = 0.4 $ (left) and $ h = 0.006 $ and $ \Delta x = 0.25 $ (right)

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Figure 7. Weak rates of convergence of three time integrators when applied to the inhomogeneous stochastic Manakov equation (9). The test functions are $ \phi_1(X) = \left\lVert{X_1}\right\rVert_{L^2} $ (left) and $ \phi_2(X) = \exp(-\left\lVert{X_2}\right\rVert_{L^2}^2) $ (right). The discretization parameters are $ h $ from $ 2^{-6} $ to $ 2^{-13} $ and $ \Delta x = 0.4 $

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