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

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 ε, 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 slowly varying envelope approximation or generalizations thereof. Results in the literature state that the error of the slowly varying envelope approximation is of $\cal{O}(\varepsilon)$. ... mehrIn 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 $\cal{O}(\varepsilon^2)$ to $\cal{O}(\varepsilon^3)$. Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and non-oscillatory error terms, and on a priori bounds for certain “parts” of the approximations which are defined by suitable projections. As an important technical tool we use an advantageous transformation of the coefficient functions which appear in the approximations.