ИСТИНА

We consider a Cauchy problem with localized initial data for a one-dimensional pseudo-differential equation describing a run-up on a shore of water waves with dispersion. We present asymptotic formulas for the solution both before and after the moment in time, when the wavefront collides with the shore. The formulas are efficient from the practical viewpoint, and appeal only to trajectories of the corresponding Hamiltonian system. In the regime before the collision, the solution is described with help of Airy function. After the collision, the solution is a sum of incoming and reflected waves, where the former is given in terms of WKB asymptotics, and the latter in terms of Airy function. In a neighborhood of the shore, the sum of two waves admits an asymptotics in terms of Bessel functions. The work is supported by the Russian Science Foundation (Project No. 16-11-10282).