Generalized Airy Functions Research Articles

Transition of the semiclassical resonance widths across a tangential crossing energy-level

We consider a 1D 2×2 matrix-valued operator (1.1) with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order n, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order 2n. Below this level, they have no intersection, which suggests exponentially small widths of resonances (see e.g., [1,2]), while above this level, on the contrary, they intersect at two points, which implies a polynomial order of the widths as proved in [3]. We prove that the transition of the resonance widths near the crossing level is described in terms of a generalized Airy function. This result generalizes [4] to the tangential crossing and [3] to the crossing at a turning point. Our approach is based on the computation of the microlocal transfer matrix at the crossing point between the incoming and outgoing microlocal solutions.

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

Higher Order Airy and Painlevé Asymptotics for the mKdV Hierarchy

In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of $t^{-1/(2n+1)}$ with smooth coefficients of the variable $(-1)^{n+1}x((2n+1) t)^{-1/(2n+1)}$ in the self-similarity region for the solution of $n$-th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlev\'e II hierarchy, which generalize the classical Ablowitz-Segur solution for the Painelv\'{e} II equation and appear in a variety of random matrix and statistical physics models. We establish the connection formulas for this family of solutions. In the special case that the reflection coefficient vanishes at the origin, the solutions of Painlev\'e II hierarchy in the leading coefficient vanishes as well, the leading and subleading terms in the asymptotic expansion are instead given explicitly in terms of derivatives of the generalized Airy function.

The saddle-point method in ${\mathbb C}^N$ and the generalized Airy functions

The saddle-point method in ${\mathbb C}^N$ and the generalized Airy functions

Fermion propagator in an external potential and generalized Airy functions

We study the behavior of the fermion propagator in an external time-dependent potential in 0 + 1 dimension. We show that, when the potential has up to quadratic terms in time, the propagator can be expressed in terms of generalized Airy functions (or standard Airy functions depending on the exact time dependence). We study various properties of these new generalized functions which reduce to the standard Airy functions in a particular limit.

Bethe Ansatz and the Spectral Theory of Affine Lie algebra–Valued Connections II: The Non Simply–Laced Case

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $\Psi$-system among subdominant solutions. We then use the $\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

On the zeros of the Pearcey integral and a Rayleigh-type equation

In this work we find a sequence of functions at which the Pearcey function is identically zero. The sequence of functions can be expressed in terms of a second order non-linear ODE, which happens to be the Rayleigh-type. As a byproduct of these facts, we develop a methodology to find a class of functions which solve the moving boundary problem of the heat equation. To this end, we make use of generalized Airy functions, which in some particular cases fall within the category of functions with infinitely many real zeros, studied by P\'olya.

On the generalized mass transfer with a chemical reaction: Fractional derivative model

In this article using the inverse Laplace transform, we show analytical solutions for the generalized mass transfers with (and without) a chemical reaction. These transfers have been expressed as the Couette flow with the fractional derivative of the Caputo sense. Also, using the Hankel contour for the Bromwich's integral, the solutions are given in terms of the generalized Airy functions.

Diffraction by three-dimensional slit-shape curves: decomposition in terms of Airy and Pearcey functions

We analyze the diffraction field generated by coherent illumination of a three-dimensional transmittance characterized by a slit-shape curve. Generic features are obtained using the Frenet-Serret equations, which allow a decomposition of the optical field. The analysis is performed by describing the influence of the curvature and torsion on osculating, normal, and rectifying planes. We show that the diffracted field has a decomposition in three optical fields propagating along three optical axes that are mutually perpendicular. The decomposition is in terms of the Pearcey and Airy functions, and the generalized Airy function. Experimental results are shown.

On fractional calculus of [formula omitted] function

In this article, we consider the A2n+1(x) function (the generalized Airy function)A2n+1(x)=∫0∞coszx+(-1)n+1z2n+12n+1dzand find some properties of this function and its derivatives. These properties are included in integral representations for product of A2n+1(x) function, A2n+1(x) wavelets and integral and series identities. Moreover, the A2n+1(x) polynomials are derived and bi-orthogonality properties with their associated functions are obtained.

The higher-order heat-type equations via signed Lévy stable and generalized Airy functions

We study the higher-order heat-type equation with first time and Mth spatial partial derivatives, M = 2, 3, …. We demonstrate that its exact solutions for M even can be constructed with the help of signed Lévy stable functions. For M odd the same role is played by a special generalization of the Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spatial and temporary evolution of particular solutions for simple initial conditions.

The weighted [formula omitted] estimate on the solution of the Gellerstedt equation in the upper half space

In this note, we are concerned with the weighted W 2 , p estimate for the Gellerstedt equation, which is degenerate on the boundary ∂ R + 2 . Based on the estimates of the generalized Airy functions and Mihlin's multiplier theorem, we establish a weighted W 2 , p a priori estimate.

Cohomological Intersection Numbers for the Generalized Airy Functions at Veronese Points

We establish the exterior power structure for the polynomial twisted de Rham cohomology group associated with the generalized Airy functions at a point of extended Veronese variety. Using this structure, we obtain a natural basis of the twisted de Rham cohomology group from that of one dimensional case, which is considered as an analogue of the flat basis of the Jacobi ring of A-type simple singularity. We compute explicitly the cohomological intersection numbers for these basis and extend the result of Iwasaki and Matsumoto.

Generalized Airy Functions and the Cohomological Intersection Numbers

We compute explicitly the cohomological intersection numbers for the basis and extend the result of Iwasaki and Matsumoto. To this end, we establish the exterior power structure for the polynomial twisted de Rham cohomology group associated with the generalized Airy functions at a point of extended Veronese variety. Using this structure, we introduce a natural basis of the twisted de Rham cohomology group coming from that of the one-dimensional case, which is considered as an analogue of a flat basis of the Jacobi ring of A-type simple singularity.

Intersection matrix of a generalized Airy function in terms of skew-Schur polynomials

A duality is introduced between a pair of polynomial twisted de Rham cohomology groups associated with a generalized Airy function in several variables. Natural bases of the twisted de Rham groups are constructed in terms of Schur polynomials. Then the intersection matrix relative to these bases is calculated explicitly in terms of skew-Schur polynomials.

A novel solution of toroidal shells under axisymmetric loading

Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of them are expressed in terms of generalized Airy functions, instead of Bessel or Airy function for the homogeneous and Lommel function for the particular solutions, respectively, as in the existing work. In this paper, three particular solutions are given, one of which is just the solution obtained by Tumarkin(1959) and Clark(1963).

Electron field emission from a silicon subsurface based on a generalized Airy function approach

An analytical approach in calculating electron field emission current from a silicon subsurface layer is presented. By employing a nontriangular quantum well model for the silicon subsurface layer, wave functions of electrons in the silicon subsurface layer are given on the basis of generalized Airy functions. After tunneling through a vacuum barrier, the electrons from the silicon subsurface layer are represented by outgoing waves at infinity. Such a boundary condition results in a complex subband-state energy, which simultaneously provides a subband energy level and a tunneling probability per unit time. A negative differential resistance characteristic is shown, being consistent with the experimental result.

Calculation of Subband States in a Metal-Oxide-Semiconductor Inversion Layer with a Realistic Potential Profile

The ground and higher subband states in a metal-oxide-Si(100) inversion layer are calculated for a logarithmic potential model, in which the wave functions are represented by generalized Airy functions. It is found that the calculated subband energy levels for the Padmanabhan-Rothwarf logarithmic potential model are much lower than those self-consistently calculated. A modified Padmanabhan-Rothwarf logarithmic potential model gives the subband energy levels and the spatial extent of the electron density that agree well with those obtained from the self-consistent calculations.

Dispersion and birefringence in a synchrotron-emitting gas

The response tensor for a highly relativistic, magnetized electron gas is evaluated by performing the integrals over gyrophase and pitch angle using the method of stationary phase. Only the transverse components are considered explicitly. The anti-Hermitian part is expressed in terms of the Airy function $\mathrm{Ai}(z)$, and its derivative and integral, and is shown to reproduce the known formulas for synchrotron absorption. It is shown that the full tensor is obtained by the replacement $\mathrm{Ai}(z)\ensuremath{\rightarrow}\mathrm{Ai}(z)+i\mathrm{Gi}(z)$, where $\mathrm{Gi}(z)$ is a generalized Airy function. A simple form for the high-frequency response is derived for a power-law distribution of particles.

Semiclassical study of the laser transition

A semiclassical, nonlinear theory of a single-mode laser is developed so that the transition around threshold can be studied. The field is expressed in the frequency domain which allows us to emphasize the role of the Fabry-P\'erot cavity. A generalized Airy function is obtained for the laser line shape: it contains the source line shape and the empty cavity line shape. It describes the laser both above and below threshold, including the transition region where there is an abrupt increase of intensity and decrease of the laser linewidth. The general arguments are illustrated by detailed numerical calculations for the 3.39-\ensuremath{\mu}m line of the He-Ne laser. The intensity of the spontaneous emission source which plays a central role is computed from first principles.