[en] By solving the normalized dimensionless linear Schrödinger-like equation with harmonic potential analytically, we have studied the spatiotemporal Airy Gaussian (AiG) and Airy Gaussian vortex (AiGV) light bullets. The AiG light bullets are composed of the chirped Airy functions in temporal domain and the AiG functions in spatial domain, while AiGV light bullets are AiG light bullets carrying the vortex. By selecting the negative or positive linear chirp we can obtain decelerating or accelerating light bullets, respectively. Combing effects from harmonic potential with the negative quadratic chirp, we can study reversed light bullets in both spatial and temporal domains. (letter)

[en] We use a novel method to get in a suitable approximation, a closed form solution of the Lorenz system in terms of the Airy functions

[en] We consider the asymptotic expansion of the logarithmic derivative of the Airy function Ai'(z)/Ai(z), and also its reciprocal Ai(z)/Ai'(z), as |z| → ∞. We derive simple, closed-form solutions for the coefficients which appear in these expansions, which are of interest since they are encountered in a wide variety of problems. The solutions are presented as Mellin transforms of given functions; this fact, together with the methods employed, suggests further avenues for research.

[en] Integral representations are obtained for some quartic products of the Airy functions Ai(z) and Bi(z). These integral representations are particularly useful in obtaining the constants which appear in the asymptotic expansions of their integrals. Some of these results are also relevant to the connection problem for the second Painlevé transcendent.

[en] Vesicles, or closed fluctuating membranes, have been modelled in two dimensions by self-avoiding polygons, weighted with respect to their perimeter and enclosed area, with the simplest model given by area-weighted excursions (Dyck paths). These models generically show a tricritical phase transition between an inflated and a crumpled phase, with a scaling function given by the logarithmic derivative of the Airy function. Extending such a model, we find realisations of multicritical points of arbitrary order, with the associated multivariate scaling functions expressible in terms of generalised Airy integrals, as previously conjectured by John Cardy. This work therefore adds to the small list of models with a critical phase transition, for which exponents and the associated scaling functions are explicitly known. (paper)

[en] The decay mode solutions for the Kadomtsev-Petviashvili (KP) equation are derived by Hirota method (direct method). The decay mode solution is a new set of analytical solutions with Airy function. (general)

[en] The presence of widely used features, in particular, the autofocusing feature, of classical Airy beams leads to a thought of a benefit of their modification. There are several generalizations of Airy function both on base of the differential equation modification and variations in the integral notion. In the paper one more type of extended Airy functions is investigated theoretically and numerically. The type is constructed by generalization of the integral notion in a wide range of non-integer values of power. A principal attention in the paper is devoted to obtaining of approximate analytical expressions for extended Airy functions under an arbitrary exponent. On base of introduced functions the new type of autofocusing beams is formed. Extended Airy beams are formed if a diffractive optical element called the generalized lens is placed in an entrance plane of the optical scheme. The numerical examination of beams features is implemented by usage of the fractional Fourier transformation. (paper)

[en] Integral representations are derived for the Riesz fractional derivatives of the product of two functions, D_x^α(uv). Here u(x)=∫^∞_-∞Ai(x- y)f(y) dy and v(x)=∫^∞_-∞Ai(x-y)g(y) dy are the Airy transforms of the functions f(x) and g(x), respectively, and Ai(x) is the Airy function of the first kind. This derivation is based on the new Hankel transform-type formula for Ai(x-a)Ai(x-b), where a, binR. Estimates of D_x^α(uv) in L_∞(R) are obtained. They can be used for the study of small data scattering for the Korteweg-de Vries-type equations.

[en] In this short paper we derive a formula for the spatial persistence probability of the Airy₁ and the Airy₂ processes. We then determine numerically a persistence coefficient for the Airy₁ process and its dependence on the threshold. (paper)

[en] Several recent works have considered the pressure exerted on a wall by a model polymer. We extend this consideration to vesicles attached to a wall, and hence include osmotic pressure. We do this by considering a two-dimensional directed model, namely that of area-weighted Dyck paths. Not surprisingly, the pressure exerted by the vesicle on the wall depends on the osmotic pressure inside, especially its sign. Here, we discuss the scaling of this pressure in the different regimes, paying particular attention to the crossover between positive and negative osmotic pressure. In our directed model, there exists an underlying Airy function scaling form, from which we extract the dependence of the bulk pressure on small osmotic pressures. (paper)