Asymptotics of the convolution of the Airy function and a function of the power-like behavior

TL;DR

This paper derives the asymptotic behavior of a convolution involving the Airy function and a power-like function, relevant for wave propagation and solutions of the KdV equation.

Contribution

It provides new asymptotic formulas for convolutions of Airy functions with power-like functions, aiding analysis of wave equations.

Findings

Asymptotic formulas for the convolution integral are obtained.

Results can be applied to the asymptotic analysis of KdV solutions.

The work advances understanding of wave propagation in dispersive media.

Abstract

The asymptotic behavior of the convolution-integral of a special form of the Airy function and a function of the power-like behavior at infinity is obtained. The integral under consideration is the solution of the Cauchy problem for an evolutionary third-order partial differential equation used in the theory of wave propagation in physical media with dispersion. The obtained result can be applied to studying asymptotics of solutions of the KdV equation by the matching method.