Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation

We pose the Cauchy problem with localized initial data that arises when passing from an explicit difference scheme for the wave equation to a pseudodifferential equation. The solution of the Cauchy problem for the difference scheme is compared with the asymptotics of the solution of the Cauchy problem for the pseudodifferential equation. We give a detailed study of the behavior of the asymptotic solution in the vicinity of the leading edge, where yet another version of the asymptotic solution is constructed based on vertical manifolds.