Phase-space asymptotic analysis of wave propagation in homogeneous dispersive and dissipative media

Arising naturally in the study of one-dimensional pulse propagation in homogeneous dispersive and/or dissipative media are certain classes of oscillatory and/or diffusion integrals which encompass the canonical diffraction catastrophe integrals due originally to Thorn and Arnold. This is especially evident within the framework of a phase-space asymptotic analysis. Depending on the order of approximation of the exact solution beyond the Liouville or "first-order quasiparticle" limit, one recognizes caustic-like structures smoothed over by hyperdiffusion. Asymptotic series for these structures, which essentially define new basic functions, have been derived, but will not be presented here. Only the salient features of these structures will be reviewed briefly, and they will be illustrated by means of several simple canonical problems. Also, their applicability to other physical areas (e.g., wave propagation in deterministically and/or randomly varying channels, diffraction, etc.) will be pointed out.