Asymptotics of the Solution of the Cauchy Problem for the Evolutionary Airy Equation at Large Times

Abstract

The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization of singularities to obtain an asymptotic Erdélyi series in inverse powers of the cubic root of the time variable with coefficients depending on the self-similar variable and the logarithm of time.