Improved Error Bounds for Approximations of High-Frequency Wave Propagation in Nonlinear Dispersive Media

We analyze nonlinear Friedrichs systems where the differential equation and the initial data contain the inverse of a small parameter ε , which causes oscillations with wavelengths proportional to ε in time and space. In diffractive optics, such problems have to be solved on time intervals with length of O ( 1 / ε ) . Approximating the solution numerically with a standard method is hopeless, because traditional methods require an extremely fine resolution in time and space. A possible alternative is to replace the original problem by a new system of partial differential equation, 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. For applications in nonlinear optics, a rigorous analysis of the accuracy of such approximations is of utmost importance. We show that under a number of natural assumptions the error of the slowly varying envelope approximation is proportional to ε 2 . For a higher-order generalization, we improve the error bound from O ( ε 2 ) to O ( ε 3 ) . Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and nonoscillatory terms, and on a priori bounds for certain “parts” of the approximations, which are identified by suitable projections.