Asymptotic Solution of the Cauchy Problem with Localized Initial Data for a Wave Equation with Small Dispersion Effects
S. A. Sergeev · Differential Equations · 2022
Abstract An asymptotic solution of the Cauchy problem for a multidimensional wave equation with a localized initial function is constructed taking into account small dispersion effects. The three-dimensional case and the construction of the asymptotics for it are considered separately. With a special choice of the initial function in the three-dimensional case, the asymptotics is presented explicitly using the Airy and Scorer functions. As an example where dispersion effects arise, we consider the homogenization of a wave equation with a rapidly oscillating coefficient and construct an asymptotic solution of such an equation.