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Decompositions of solutions of the wave equation in terms of localized solutions from the wide class of the same equation are constructed by means of the affine Poincare continuous wavelet analysis. Two particular decompositions are considered. One of them is represented in terms of nonstationary particle-like solutions, the Gaussian wave packets. The another one is a decomposition of a monochromatic field in terms of monochromatic Gaussian beams. A comparison with known results is given.

A formal asymptotic expansion of the solutions of the stationary Maxwell equations with periodic coefficients is constructed by the method of two-scale asymptotic expansions. The solutions oscillate with the period of the medium and depend on the slow change of the variables with the period. The frequency is assumed to be close to a stationary point of the dispersion function. The asymptotic results obtained in this way are applied to the solution of a boundary value problem for the Maxwell equations in a half-space.