Long-time behavior of the solution of the Cauchy problem for the third-order Airy equation

Article type: Research Article

Authors: Zakharov, Sergei V.^{; *}

Affiliations: N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S. Kovalevskaja street, 620990, Ekaterinburg, Russia. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: The long-time asymptotic behavior of the solution of the Cauchy problem for the evolutionary third-order Airy equation describing wave propagation in dispersive physical media is derived by using the auxiliary parameter method. For the solution in the form of the convolution of the initial data and the Airy function, an asymptotic Erdélyi expansion in inverse powers of the cube root of the time variable with the coefficients depending on a self-similar variable and the logarithm of time is obtained. To refine the asymptotics, a family of special function classes for its coefficients is introduced. It is pointed out how the used method is connected with the geometrical optics approach and how the obtained result can be applied to nonlinear third-order PDEs.

Keywords: Third-order Airy equation, linearized KdV equation, Cauchy problem, Airy function, convolution, auxiliary parameter method, long-time asymptotics, Poincaré and Erdélyi expansions, self-similarity

DOI: 10.3233/ASY-191541

Journal: Asymptotic Analysis, vol. 116, no. 2, pp. 139-148, 2020