Exact Explode-Decay Soliton Solutions of a Coupled Nonlinear Schroedinger Equation

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,>4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.

We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL () (x). First we summarize the results of a paper of Frenzen and Wong in whichn→∞ and α>−1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which α is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.

The propagation characteristics of Airy beams is investigated and fully\ndescribed under the traveling waves approach analogous to that used for\nnon-diffracting Bessel beams. This is possible when noticing that Airy\nfunctions are in fact Bessel functions of fractional order 1 . We show how\nphysical 3 principles impose restrictions such that the non-diffracting Airy\nbeams cannot be of infinite extent as has been argued, and introduce for the\nfirst time quantitative expressions for the maximum transverse and longitudinal\nextent of Airy beams. We show that under the appropriate physical conditions it\nis possible to obtain higher-order Airy beams.\n

We consider polynomials that are orthogonal on $[-1,1]$ with respect to a modified Jacobi weight $(1-x)^\alpha (1+x)^\beta h(x)$, with $\alpha,\beta>-1$ and $h$ real analytic and stricly positive on $[-1,1]$. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval $[-1,1]$, for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for Riemann--Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szegő function associated with the weight and for the local analysis around the endpoints $\pm 1$ we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.

We investigate the internal space of Bessel functions which is associated to thegroup Z of positive and negative integers defining their orders. As a result wepropose and prove a new unifying formula (to be added to the huge literatureon Bessel functions) generating Bessel functions of real orders out of integerorder ones. The unifying formula is expected to be of great use in appliedmathematics. Some applications of the formula are given for illustration

: The report contains tables of the first five roots of the following transcendental equations: (a) J0(alpha) Y0(alpha) = Y0(alpha) J0(alpha); (b) J1(alpha) Y1(alpha) = Y1(alpha) J1(alpha); (c) J0(alpha) Y1(alpha) = Y0(alpha) J1(alpha) where J0(alpha), Y0(alpha), J1(alpha), Y1(alpha) are Bessel functions of order 0 and 1 respectively. In these equations, alpha is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. Additional tables are included listing an auxiliary quantity gamma which is better suited to interpolation particularly when k is close to unity. (Author)

The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant conjugation parameter according to the transformation , where the conjugation parameter is set to unity () at the end of the evaluations. Factorization in normal order form yields composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an reversal conjugation rule . Setting provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.

Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the ‘small values regime’ and the ‘asymptotic’ one, and covering, in this way, an area of great physical interest. The examples that illustrate our approach are accessible to the undergraduate student.

Radiative effects are useful in physics and engineering, especially in high-temperature operations and space technology. The impact of thermal radiation is important in managing the heat transfer process in the polymer production sector. This work examines the impact of the Soret and Dufour effect on the mass and heat transfer in the fluid flow between two cylinders. The only thing that creates the flow is the inner cylinder extending beneath the motionless outer cylinder. Thermal radiation and nonuniform heat source/sink are also considered. Through the application of similarity transformations, the governing nonlinear differential equations for this flow dynamics are converted into nonlinear ordinary differential equations (ODEs). After that, these simplified equations are solved by applying the collocation method with Probabilists’ Hermite polynomial. The physical behavior of important parameters is examined in depth using graphical representations. The results of a comprehensive examination are presented in tabular data. Result reveals that, the heat, momentum, and mass transfer fields are all significantly impacted by the gap size. As the Dufour number and radiation parameter value grows, the thermal profile also rises. The concentration profile rises as the Soret number values increases, whereas the rise in the Schmidt number shows the reverse pattern.

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

TRAITEMENT AUTOMATIQUE DES SONDAGES ELECTRIQUES AUTOMATIC PROCESSING OF ELECTRICAL SOUNDINGS*

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

From Mie's classical theory of the action of small spherical particles on plane waves of light, the expression giving the loss of light due to absorption and scattering is reduced to the formula involving only Bessel functions of orders given by half integral values. The result is used for calculating the absorption by small carbon particles whose diameter is comparable with the wave-length of the incident light, particles that can be measured only by interference methods. When the diameter is less than 0.2 μ the coefficient of absorption decreases toward the red end of the spectrum. The reverse is true for 0.3 and 0.4 μ particles.