ИСТИНА

We consider a Cauchy problem with localized initial data for a one-dimensional pseudo-differential equation on a semi-axis. This equation describes water waves with dispersion running up on a shore of a basin. Using a modified construction of Maslov’s canonical operator on non-compact Lagrangian manifolds, we present asymptotic formulas for the solution both before and after the moment in time, when the wavefront reaches the shore. The formulas are efficient from the viewpoint of numerical realization, 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 a WKB asymptotics, and the latter is expressed in terms of Airy function. In a neighborhood of the shore, two waves admit an asymptotics in terms of Bessel functions.