Riesz fractional derivatives of a function, Dα xf(x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy fun...

We derive integral formulas, involving the Airy function, for moments of the time a two-sided Brownian motion with parabolic drift attains its maximum.

We have theoretically and numerically investigated the self-healing peculiarity of Airy beams truncated by different decay factors. The self-healing dynamic processes of truncated Airy beams are different from that of blocked Airy beams. The wing with large truncation of Airy beam can propagate much longer distance than that of small truncated wing. When one of the wings is complete truncated, ...

We investigate on the nonlinear dynamics of Airy beams in a regime where nonlinear losses due to multi-photon absorption are significant. We identify the nonlinear Airy beam (NAB) that preserves the amplitude of the inward Hänkel component as an attractor of the dynamics. This attractor governs also the dynamics of finite-power (apodized) Airy beams, irrespective of the location of the entrance...

In deriving uniform asymptotic expansions of a certain class of integrals one encounters the problem of expanding a function, that is analytic in some domain Ω of the complex plane, in two points. The first mention of the use of such expansions in asymptotics is given in [1], where Airy-type expansions are derived for integrals having two nearby (or coalescing) saddle points. This reference doe...

The Witten–Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the topological recursion applied to the Airy curve x = 1 2 y 2. In this paper, we consider the topological recursion applied to the irregular spectral curve xy2 = 2...

In general, there is an inverse relation between the degree of localization of a wave function of a certain class and its transform representation dictated by the scaling property of the Fourier transform. We report that in the case of finite energy Airy wave packets a simultaneous increase in their localization in the direct and transform domains can be obtained as the apodization parameter is...

In this Letter a new class of light beam arisen from the symmetrization of the spectral cubic phase of an Airy beam is presented. The symmetric Airy beam exhibits peculiar features. It propagates at initial stages with a single central lobe that autofocuses and then collapses immediately behind the autofocus. Then, the beam splits into two specular off-axis parabolic lobes like those correspond...