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Improved error bounds for approximations of high-frequency wave propagation in nonlinear dispersive media

Julian Baumstark, Tobias Jahnke

Abstract

High-frequency wave propagation is often modelled by nonlinear Friedrichs systems where both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, which causes oscillations with wavelengths proportional to $\varepsilon$ in time and space. A prominent example is the Maxwell--Lorentz system, which is a well-established model for the propagation of light in nonlinear media. In diffractive optics, such problems have to be solved on long time intervals with length proportional to $1/\varepsilon$. Approximating the solution of such a problem numerically with a standard method is hopeless, because traditional methods require an extremely fine resolution in time and space, which entails unacceptable computational costs. A possible alternative is to replace the original problem by a new system of PDEs which is more suitable for numerical computations but still yields a sufficiently accurate approximation. Such models are often based on the \emph{slowly varying envelope approximation} or generalizations thereof. Results in the literature state that the error of the slowly varying envelope approximation is of $\mathcal{O}(\varepsilon)$. In this work, however, we prove that the error is even proportional to $\varepsilon^2$, which is a substantial improvement, and which explains the error behavior observed in numerical experiments. For a higher-order generalization of the slowly varying envelope approximation we improve the error bound from $\mathcal{O}(\varepsilon^2)$ to $\mathcal{O}(\varepsilon^3)$. Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and non-oscillatory error terms, and on \textit{a priori} bounds for certain ``parts'' of the approximations which are defined by suitable projections.

Improved error bounds for approximations of high-frequency wave propagation in nonlinear dispersive media

Abstract

High-frequency wave propagation is often modelled by nonlinear Friedrichs systems where both the differential equation and the initial data contain the inverse of a small parameter , which causes oscillations with wavelengths proportional to in time and space. A prominent example is the Maxwell--Lorentz system, which is a well-established model for the propagation of light in nonlinear media. In diffractive optics, such problems have to be solved on long time intervals with length proportional to . Approximating the solution of such a problem numerically with a standard method is hopeless, because traditional methods require an extremely fine resolution in time and space, which entails unacceptable computational costs. A possible alternative is to replace the original problem by a new system of PDEs which is more suitable for numerical computations but still yields a sufficiently accurate approximation. Such models are often based on the \emph{slowly varying envelope approximation} or generalizations thereof. Results in the literature state that the error of the slowly varying envelope approximation is of . In this work, however, we prove that the error is even proportional to , which is a substantial improvement, and which explains the error behavior observed in numerical experiments. For a higher-order generalization of the slowly varying envelope approximation we improve the error bound from to . Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and non-oscillatory error terms, and on \textit{a priori} bounds for certain ``parts'' of the approximations which are defined by suitable projections.
Paper Structure (24 sections, 13 theorems, 198 equations, 2 figures)

This paper contains 24 sections, 13 theorems, 198 equations, 2 figures.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Notation.
  4. Analytical setting
  5. Wiener algebra and evolution equations in Fourier space.
  6. Local well-posedness.
  7. Eigendecompositions.
  8. Transformation of the coefficient functions.
  9. Projectors.
  10. Useful identities and inequalities.
  11. Why $\mathcal{P}_\varepsilon \widehat{u}_1$ is smooth and $\mathcal{P}_\varepsilon^\perp \widehat{u}_1$ is small in the slowly varying envelope approximation
  12. Convergence analysis for the slowly varying envelope approximation
  13. Improved error bound for the SVEA
  14. Step 1.
  15. Step 2.
  16. ...and 9 more sections

Key Result

Lemma 2.1

If $p_0, p_1 \in W$, then there is a $t_{\hbox{\tiny end}}>0$ such that for every $\varepsilon\in(0,1]$ the original problem PDE.uu with $p = p_0 + \varepsilon p_1$ has a unique mild solution $u \in C([0,t_{\hbox{\tiny end}}/\varepsilon),W)$ which is uniformly bounded, i.e. there is a constant $c>0$

Figures (2)

  • Figure 1: Accuracy of the SVEA for different values of $\varepsilon$. See text for details.
  • Figure 2: Accuracy of $\widetilde{u}^{(3)}$ for different values of $\varepsilon$. Parameters, data and discretizations are the same as in the numerical experiment described in Section \ref{['Subsec:NumEx.01']}.

Theorems & Definitions (18)

  • Lemma 2.1: Local well-posedness of \ref{['PDE.uu']}
  • Lemma 2.2: Local well-posedness of \ref{['PDE.mfe']}
  • Remark 2.4
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Remark 3.6
  • Remark 4.2
  • ...and 8 more