Oscillatory integrals generated by the Ostrovsky equation

Abstract

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the KadomtsevPetviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. (2000). 35B40, 35C15, 35C20, 35Q53. Mathematics Subject Classification

Differential and integral relations involving fractional derivatives of Airy functions and applications

“…Integrals considered in the next statement may be used for studying the Ostrovsky equation (see [34]). …”

Fractional derivatives of products of Airy functions

“…Notice that oscillatory integrals of the type of (1.6) appear in the theory of electromagnetic wave propagation and diffraction (see [7,14]). In the paper [29] a special function representation was obtained for (1.6), namely…”