1. Historical backgrounds: the Airy function in Physics 2 2. The area-Airy distributions: Brownian motion, linear probing hashing, additive parameters in grammars 4 2.1. Area under a Brownian excursion 4 2.2. On the analysis of linear probing hashing 4 2.3. Analytic variations on the Airy distribution 5 2.4. Hachage, arbres, chemins & graphes 6 3. Random matrices, Airy kernel and the Tracy–Wido...

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 present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting "incomplete Airy beam" is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its frin...

Using some properties of the Airy functions, we analyze the re ected Brownian Bridge area Wb conditioned on its local time b at the origin. We give a closed form expression of the Laplace transform of Wb, a recurrence equation for the moments, leading to an eÆcient computation algorithm and an asymptotic form for the density f(x; b) of Wb for x # 0.

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...

We simulate and analyze the propagation of truncated temporal Airy pulses in a single mode fiber in the presence of self-phase modulation and anomalous dispersion as a function of the launched Airy power and truncation coefficient. Soliton pulse shedding is observed, where the emergent soliton parameters depend on the launched Airy pulse characteristics. The Soliton temporal position shifts to ...

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...

The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem ut + uxxx + (u 4)x = 0, u(0) = u0 ∈ H s x(R), s > − 1 6 for the generalized Korteweg-deVries equation of order three, for short gKdV3. For real valued data u0 ∈ L 2 x(R) global wellposedness follows by the conservation of the L-norm. The main new tool is a bilinear estimate for solutions of the Airy-e...

We report experiments and numerical simulations on supercontinuum generation with femtosecond Airy pulses in a highly nonlinear optical fiber. The ability of the Airy waveform to regenerate its dominant intensity peak results in the generation of distinct spectral features. Airy pulses and other self-healing temporal waveforms may be useful for the generation of spectra with desired properties.

Owing to the nondiffracting, self-accelerating, and self-healing properties, Airy beams of different nature have become a subject of immense interest in the past decade. Their interesting properties have opened doors to many diverse applications. Consequently, the questions of how to properly design spatial manipulation of Airy beams or how to implement them in different setups have become impo...