We study the effect of Two-Photon Absorption (TPA) nonlinear losses on Gaussian pulses, with power that exceeds the critical power for self-focusing, propagating in bulk kerr media. Experiments performed in fused silica and silicon highlight a spontaneous reshaping of the input pulse into a pulsed Bessel beam. A filament is formed in which sub-diffractive propagation is sustained by the Bessel-...

The fuzzy disc is a matrix approximation of the functions on a disc which preserves rotational symmetry. In this paper we introduce a basis for the algebra of functions on the fuzzy disc in terms of the eigenfunctions of a properly defined fuzzy Laplacian. In the commutative limit they tend to the eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of the first kind, thu...

In this note our aim is to present some Jordan-type inequalities for generalized Bessel functions in order to extend some recent results concerning generalized and sharp versions of the well-known Jordan’s inequality. Acknowledgements: Research partially supported by the Institute of Mathematics, University of Debrecen, Hungary. The author is grateful to Prof. Lokenath Debnath for a copy of pap...

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge...

In the paper, the authors verify the complete monotonicity of the difference e − ψ′(t) on (0,∞), compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of e, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind. These results show us some new properties and relat...

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermor...

The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution represented by Fourier-Bessel series in given domain. It established connection between data unique solvability problem.

Diffraction-free beam and optical invariant field are special cases of optical wave propagation that have caught general attention in the last decade. Temporal solitons were the first class of optical signals to be recognized as being invariant under propagation (Hasegawa and Tapert, 1973) and to be experimentally demonstrated (Mollenaure et al., 1980). But, they require a nonlinearity to prese...