Parabolic Cylinder Functions Research Articles

Fractional Wiener chaos: Part 1

In this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as Hα(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\alpha (x,y)$$\\end{document}, referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, Hα(Wt,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\alpha (W_t,t)$$\\end{document}, demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

On behavior analysis of solutions for the coupled higher-order WKI equation

A coupled higher-order WKI equation is proposed and its 4 × 4 matrix Lax pair is derived. On the basis of spectral analysis of the 4 × 4 matrix spectral problems related to the coupled higher-order WKI equation, the basic Riemann–Hilbert problem is established by using the inverse scattering transformation. Resorting to the nonlinear steepest descent method, the solution of the basic Riemann–Hilbert problem is transformed to the solution of the model Riemann–Hilbert problem after several contour deformation. Finally, we explicitly obtain the long-time asymptotics of the solution to the Cauchy problem of the coupled higher-order WKI equation in the solitonless sector region with the aid of the parabolic cylinder function.

The uniform asymptotic method “saddle point near an end point” revisited

We continue the program initiated in [López & all, 2009–2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method “saddle point near an end point”. We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions M(c,x/α+c+1,x) and U(c,αx+c+1,x) for large x, c fixed, uniformly valid for α∈(0,∞), are given as an illustration.

Computation of parabolic cylinder functions having complex argument

Numerical methods for the computation of the parabolic cylinder function U(a,z) for real a and complex z are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5×10−13 relative accuracy in double precision floating point arithmetic.

Approximation of Mathieu Functions by Parabolic Cylinder Functions

Approximation of Mathieu Functions by Parabolic Cylinder Functions

Fast reaction of soluble surfactant can remobilize a stagnant cap

Analytical solutions are derived showing that a stagnant cap of surfactant at the interface between two viscous fluids caused by a linear extensional flow can be remobilized by fast kinetic exchange of surfactant with one of the fluids. Using a complex variable formulation of this multiphysics problem at zero capillary number, zero Reynolds number and zero bulk Péclet number, and assuming a linear equation of state, it is shown that the system is governed by a forced complex Burgers equation at arbitrary surface Péclet number. Consequently, this nonlinear system is shown to be linearizable using a complex analogue of the Cole–Hopf transformation. Steady equilibria of the system at any finite value of the surface Péclet number are found explicitly in terms of parabolic cylinder functions. While surface diffusion is naturally expected to mollify sharp gradients associated with stagnant caps and to remobilize the interface, this work gives an analytical demonstration of the less intuitive result that fast kinetic exchange has a similar effect. Indeed, the analytical approach here imposes no limit on the surface Péclet number, which can be taken to be infinitely large so that surface diffusion is completely absent. Mathematically, the solution structure is then very rich allowing a theoretical investigation of this extreme case where it is seen that fast surfactant exchange with the bulk can alone remobilize a stagnant cap. Remarkably, it is also possible to track explicitly the time evolution of the system to these remobilized equilibria by finding time-evolving exact solutions.

Long‐time asymptotics to the defocusing generalized nonlinear Schrödinger equation with the decaying initial value problem

In this paper, the main work is to study the long‐time asymptotics of the defocusing generalized nonlinear Schrödinger equation with the decaying initial value. The Riemann‐Hilbert method and the nonlinear steepest descent method by Deift‐Zhou have made great contributions to obtain it. Starting from the Lax pair of the defocusing generalized nonlinear Schrödinger equation, the associated oscillatory Riemann‐Hilbert problem can be obtained. Then, via the stationary point, the steepest decent contours, and the trigonometric decomposition of jump matrix, we get the solvable Riemann‐Hilbert problem from the associated oscillatory Riemann‐Hilbert problem. Based on the decaying initial value in Schwartz space, the Weber equation, and the standard parabolic cylinder function, the expression of the solution for the generalized nonlinear Schrödinger equation can be given.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

Zigzag Edge States in Graphene in the Presence of In‐Plane Electric Field

Herein, the edge states in a finite‐width graphene ribbon and a semi‐infinite geometry subject to a perpendicular magnetic field and an in‐plane electric field, applied perpendicular to a zigzag edge, are explored. To accomplish this, a combination of analytic and numerical methods within the framework of low‐energy effective theory is employed. Both the gapless and gapped Dirac fermions in graphene are considered. It is found that a surface mode localized at the zigzag edge remains dispersionless even in the presence of electric field. This is shown analytically by employing Darwin's expansion of the parabolic cylinder functions of large order and argument.

On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: Without solitons

In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann–Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.

Global Asymptotics for Functions of Parabolic Cylinder and Solutions of the Schrödinger Equation with a Potential in the Form of a Nonsmooth Double Well

Global Asymptotics for Functions of Parabolic Cylinder and Solutions of the Schrödinger Equation with a Potential in the Form of a Nonsmooth Double Well

A Uniform Diffraction Theory for Wide-Angle Cones

In a recent article (Katsav and Heyman, 2020), we derived an approximate tip-diffraction coefficient for circular wide-angle cones. In the present work, we derive a uniform solution for the field in the geometrical optics (GO) transition zones. The uniform solution has the form of Weber parabolic cylinder function of order −3/2, but far from the transition zones, it reduces to the separate reflection and tip diffraction contributions. The derivation utilizes the approximate spectral integral solution for wide-angle cones derived by Katsav and Heyman (2020), whose form is much simpler than the exact spectral integral solution. The uniform solution is used therefore only inside the transition zone, whereas at large angles it is more accurate to switch to the nonuniform contributions of the reflection and of tip diffraction. The accuracy of the new solution is demonstrated via comparison to the exact conical harmonics solution.

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like κDα−122(2x) as x→+∞, where κ is some real constant and Dα−12(x) is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as x→−∞. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the σ-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the σ-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel.

Computation of the Complementary Landmark Generalized Parabolic Cylinder Function Distribution

Computation of the Complementary Landmark Generalized Parabolic Cylinder Function Distribution

Convolution of beta prime distribution

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae’s relationships for 3 F 2 ( 1 ) {}_3F_2(1) . Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán’s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill’s ratio is discussed in detail.

Charged particle motion in spherically symmetric distributions of magnetic monopoles

The classical equations of motion of a charged particle in a spherically symmetric distribution of magnetic monopoles can be transformed into a system of linear equations, thereby providing a type of integrability. In the case of a single monopole, the solution was given long ago by Poincaré. In the case of a uniform distribution of monopoles, the solution can be expressed in terms of parabolic cylinder functions (essentially the eigenfunctions of an inverted harmonic oscillator). This solution is relevant to recent studies of nonassociative star products, symplectic lifts of twisted Poisson structures, and fluids and plasmas of electric and magnetic charges.

A convergent and asymptotic Laplace method for integrals

Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F(z)≔∫0∞e−zf(t)g(t)dt for large values of the parameter z. They are useful tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where, by the use of inverse factorial series, a new asymptotic as well as convergent expansion of F(z), for the particular case f(t)=t, was derived. In this paper we go some steps further and investigate a modification of the Laplace’s method for F(z), with a general phase function f(t), to derive asymptotic expansions of F(z) that are also convergent, accompanied by error bounds. An analysis of the remainder of this new expansion shows that it is convergent under a mild condition for the functions f(t) and g(t), namely, these functions must be analytic in certain starlike complex regions that contain the positive axis [0,∞). In many practical situations (in many examples of special functions), the singularities of f(t) and g(t) are off this region and then this method provides asymptotic expansions that are also convergent. We illustrate this modification of the Laplace’s method with the parabolic cylinder function U(a,z), providing an asymptotic expansions of this function for large z that is also convergent.

The Hermitian symmetric space Fokas–Lenells equation: spectral analysis and long-time asymptotics

Abstract Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas–Lenells (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL equation is transformed into the solution of a Riemann–Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space FL equation with the aid of the parabolic cylinder function.

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

In this paper, the phase space representation for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. We have found the Husimi distribution function for the stationary states of the oscillator model under consideration for both cases without and with the applied external homogeneous field. The obtained function is expressed through the double sum of the parabolic cylinder function. Different special cases and the limit relations are discussed, too.