Abstract:
In this work we construct Gaussian beam approximations to solutions of the symmetric strongly hyperbolic system with highly oscillatory initial data. The evolution equations for each Gaussian beam component are derived. Under some regularity assumption of the data we obtain an optimal error estimate between the exact solution and the Gaussian beam superposition in terms of the high frequency parameter $\varepsilon$. The main result is that the relative local error measured in energy norm in the beam approximation decays as $\varepsilon^{\frac{1}{2}}$ independent of dimension and presence of caustics, for first order beams. This result is shown to be valid when the gradient of the initial phase may vanish on a set of measure zero. We provide applications of our results to acoustic and Maxwell equations. |