Airy Equation Research Articles

On the dynamics of the zeros of solutions of the Airy equation

We study the dependence of the zeros of solutions of the Airy equation on two parameters introduced in the equation. The parameters characterize the general solution of the equation. A system of infinitely many nonlinear evolution differential equations is obtained, displaying interesting properties.

Asymptotics of the Solution of the Cauchy Problem for the Evolutionary Airy Equation at Large Times

The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization of singularities to obtain an asymptotic Erdelyi series in inverse powers of the cubic root of the time variable with coefficients depending on the self-similar variable and the logarithm of time.

Non-commutative NLS-type hierarchies: Dressing & solutions

We consider the generalized matrix non-linear Schrödinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general n-th order matrix-differential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutative Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.

Asymptotics of the solution of the Cauchy problem for the evolutionary Airy equation at large times

Установлено асимптотическое поведение на больших временах решения задачи Коши для уравнения Эйри - эволюционного уравнения третьего порядка. Предполагается, что начальная функция локально интегрируема по Лебегу и имеет степенную асимптотику на бесконечности. С использованием метода вспомогательного параметра и регуляризации особенностей для решения в виде интеграла свертки с функцией Эйри получен асимптотический ряд Эрдейи по обратным степеням кубического корня из переменной времени с коэффициентами, зависящими от автомодельной переменной и логарифма времени.

The stein characterization of M-Wright distributions

ABSTRACTIn this paper, we use the Stein method to characterize the M-Wright distribution and its symmetrization. The Stein operator is associated with the general Airy equation and the corresponding Stein equation is nothing but a general inhomogeneous Airy equation.

Solvable problems in classical representation

Some solvable problems are considered in a classical representation: namely, the harmonic oscillator, the non-relativistic model of confinement of quarks and the time-dependent Feynman model. In a classical representation, the harmonic oscillator $$V(x)=\frac{1}{2}m\omega ^2x^2$$ transforms into an equation, which formally coincides with the radial Schrodinger equation for Sturmian states of a hydrogen atom with an energy $$ {\mathcal {E}} = -2 / \omega ^ 2 $$ , an effective charge $$ {\mathcal {Z}} = 2E_n / \omega ^ 2 $$ and an orbital quantum number $$ l = -1 / 2 $$ . The problem of V-type potential $$V(x)=F|x|$$ describing the confinement of quarks transforms again into the Airy equation. The classical solution of the time-dependent Feynman model $$V(x,t)=\frac{1}{2}m\omega ^2x^2+\alpha (t)x$$ after projection on the eigen-states in classical representation gives results for inelastic transitions which coincide with the exact results obtained by Feynman in a purely quantum approach. The analysis of harmonic oscillator, V-type potential and the Feynman model gives a new integral relation between the Laguerre polynomials and the square of the Hermite polynomials, the Airy function and the square of the Airy function, quadratic form of the Laguerre polynomials and quadratic form of the generalized Laguerre polynomials.

On the lack of compactness and existence of maximizers for some Airy-Strichartz inequalities

This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb R)$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.

Wrinkling Structures at the Rim of an Initially Stretched Circular Thin Plate Subjected to Transverse Pressure

Short-wavelength wrinkles that appear on an initially stretched thin elastic plate under transverse loading are examined. As the degree of loading is increased, wrinkles appear and their structure at the onset of buckling takes on one of three distinct forms depending on the size of the imposed stretching. With relatively little stretching, the wrinkles sit off the rim of the plate at a location which is not known a priori, but which is determined via a set of consistency conditions. These take the form of constraints on the solutions of certain coupled nonlinear differential equations that are solved numerically. As the degree of stretching grows, an asymptotic solution of the consistency conditions is possible, which heralds the structure that governs a second regime. Now the wrinkle sits next to the rim, where its detailed structure can be described by the solution of suitably scaled Airy equations. In each of these first two regimes the Foppl--von Karman bifurcation equations remain coupled, but as the initial stretching becomes stronger the governing equations separate. Further use of singular perturbation arguments allows us to identify the wavelength wrinkle which is likely to be preferred in practice.

Refinement of Strichartz Estimates for Airy Equation in Nondiagonal Case and its Application

In this paper, we give an improvement of nondiagonal Strichartz estimates for the Airy equation by using a Morrey-type space. As its applications, we prove the small data scattering and the existence of special nonscattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg--de Vries equation. Especially, the use of a refined nondiagonal estimate removes several technical restrictions on the previous work [S. Masaki and J. Segata, Existence of a Minimal Non-Scattering Solution to the Mass-Subcritical Generalized Korteweg-de Vries Equation, preprint, https://arxiv.org/abs/1602.05331] about the existence of the special non-scattering solution.

A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.

Numerical comparison of nonstandard schemes for the Airy equation

This paper considers the Airy ordinary differential equation (ODE) and different ways it can be discretized. We first consider a standard discretization using the central difference scheme. We then consider two difference schemes which were created using a nonstandard methodology. Finally, we compare the different schemes and how well they approximate solutions to the Airy ODE.

Using the argument of the Airy function as an analysis tool in ocean acoustics and oceanography

By assuming the index of refraction is linear between layers, it is possible to convert the depth dependent acoustic Helmholtz equation into an Airy equation. Decades ago this allowed the ocean acoustic pressure to be quickly calculated using tables of precalculated Airy functions with interpolation. High speed computers and the accurate parabolic equation model have rendered the Airy equation approach as a curiosity of history. However, the method can still be useful as an assessment tool. This has been demonstrated by using it to select similar ocean environmental parameters that have the same acoustic propagation characteristics. In this presentation, we apply the Airy equation method to evaluate the buoyancy frequency of similar ocean environments that support internal waves. A connection is made between the ocean acoustic propagation and the oceanography of the similar regions.

Universality of the topological string at large radius and NS-brane resurgence

We show that there is a natural universal limit of the topological string free energies at the large radius point. The new free energies keep a nonholomorphic dependence on the complex structure moduli space and their functional form is the same for all Calabi-Yau geometries, compact and noncompact alike. The asymptotic nature of the free energy expansion changes in this limit due to a milder factorial growth of its coefficients, and this implies a transseries extension with instanton effects in $\exp(- 1/g_s^2)$, of NS-brane type, rather than $\exp(-1/g_s)$, of D-brane type. We show a relation between the instanton action of NS-brane type and the volume of the Calabi-Yau manifold which points to a possible interpretation in terms of NS5-branes. A similar rescaling limit has been considered recently leading to an Airy equation for the partition function which is here used to explain the resurgent properties of the rescaled transseries.

On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical Lr-space

The purpose of this paper is to study local and global well-posedness of initial value problem for generalized Korteweg-de Vries (gKdV) equation in ^L^r. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for gKdV equation in the scale critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.

Airy Equation for the Topological String Partition Function in a Scaling Limit

We use the polynomial formulation of the holomorphic anomaly equations governing perturbative topological string theory to derive the free energies in a scaling limit to all orders in perturbation theory for any Calabi-Yau threefold. The partition function in this limit satisfies an Airy differential equation in a rescaled topological string coupling. One of the two solutions of this equation gives the perturbative expansion and the other solution provides geometric hints of the non-perturbative structure of topological string theory. Both solutions can be expanded naturally around strong coupling.

Global Well-Posedness for Higher-Order Schrödinger Equations in Weighted L2 Spaces

We obtain the global well-posedness for Schrödinger equations of higher orders in weighted L2 spaces. This is based on weighted L2 Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is also applied to the Airy equation which is the linear component of Korteweg-de Vries type equations.

Calculation of the finesse of an ideal Fabry-Perot resonator.

A new derivation of an exact finesse F for the description of an optical resonator is reported. The finesse is derived using the superposition principle of plane waves in an ideal Fabry-Perot resonator in combination with the standard definition of the quality factor Q which relates the energy loss of a resonator cycle to the energy stored in the resonator. The derived exact expression of the finesse is compared to equations found in the literature, and it is shown that they are based on approximations. The exact finesse is then used to convert an infinite series of Lorentz functions into the well-known Airy equation in the case of neglected absorption. The derived expression provides a framework for the discussion of the finesse in terms of decay times, free spectral ranges, and resonator line widths. The provided expression for the finesse is valid for any value of the intensity reflectivity and gives insight into the underlying physical principles of resonators.

On the complex zeros of Airy and Bessel functions and those of their derivatives

We study the distribution of zeros of general solutions of the Airy and Bessel equations in the complex plane. Our results characterize the patterns followed by the zeros for any solution, in such a way that if one zero is known it is possible to determine the location of the rest of zeros.

The Green function for the BFKL pomeron and the transition to DGLAP evolution

We consider the (process-independent) Green function for the BFKL equation in the next-to-leading order approximation, with running coupling, and explain how, within the semi-classical approximation, it is related to Green function of the Airy equation. The unique Green function is obtained from a combination of its required ultraviolet behaviour compatible with asymptotic freedom and an infrared limit phase imposed by the non-perturbative sector of QCD. We show that at sufficiently large gluon transverse momenta the corresponding gluon density matches that of the DGLAP analysis, whereas for relatively small values of the gluon transverse momentum the gluon distribution is sensitive to the Regge poles, whose positions are determined both by the non-pertubative QCD dynamics and physics at large transverse momenta.