Cauchy problem for the localized wave propagation in continuous model of the one-dimensional diatomic crystal

TL;DR

This paper analyzes the wave propagation in a one-dimensional diatomic crystal model using a continuous approximation, deriving asymptotic solutions with Airy functions based on small parameters like lattice step and initial perturbation size.

Contribution

It introduces a new asymptotic analysis of localized wave solutions in a continuous diatomic crystal model considering two small parameters.

Findings

Asymptotic solutions expressed via Airy functions

Different solution forms depending on perturbation size

Analytical formulas for wave behavior in diatomic crystals

Abstract

We study the continuous model of the localized wave propagation corresponding to the one-dimensional diatomic crystal lattice. From the mathematical point of view the problem can be described in terms of the Cauchy problem with localized initial data for a system of two pseudo-differential equations. We assume two small parameters in this formulation -- the lattice step and the size if the initial perturbation. We construct the asymptotic solution of the continuous Cauchy problem with respect to the size of perturbation. The ratio of the small parameters drastically affects the form of the solution. We consider two situations -- when the size of the perturbation is sufficiently large and when it is comparable with the lattice step. In each situations we provide analytical formulae for the asymptotic solution via Airy function.