Ultra-high power (exceeding the self-focusing threshold by more than three orders of magnitude) light beams from ground-based laser systems may find applications in space-debris cleaning. The propagation of such powerful laser beams through the atmosphere reveals many novel interesting features compared to traditional light self-focusing. It is demonstrated here that for the relevant laser parameters, when the thickness of the atmosphere is much shorter than the focusing length (that is, of the orbit scale), the beam transit through the atmosphere in lowest order produces phase distortion only. This means that by using adaptive optics it may be possible to eliminate the impact of self-focusing in the atmosphere on the laser beam. The area of applicability of the proposed “thin window” model is broader than the specific physical problem considered here. For instance, it might find applications in femtosecond laser material processing.

We propose two finite-difference algorithms of fourth order of accuracy for solving the initial problem of the Zakharov-Shabat system. Both schemes have the exponential form and conserve quadratic invariant of Zakharov-Shabat system. The second scheme contains the spectral parameter in exponent only and allows to apply the fast computational algorithm.

An elliptic method based on the application of both Beltrami and diffusion equations in monitor metrics is developed for generating adaptive and/or magnetic vector field-aligned grids. Some results of numerical experiments for constructing two-and three-dimensional grids are demonstrated.The elliptic systems of inverted Beltrami and diffusion equations in monitor metrics allow one to generate both structured and unstructured grids in domains or on surfaces in a unified manner, regardless of their dimension and geometry features. In particular, these systems can be applied to produce grids in spatial blocks by means of the successive generation of grids on curvilinear edges, faces, and parallelepipeds, using the solution at a step i n as the Dirichlet boundary condition for the following step i·1 n. Thus, both the interior and the boundary grid points of a domain or a surface can be calculated by the similar elliptic solver. In the same way the grid properties required in practical applications such as nonsingularity, adaptivity, vector-field alignment, and smooth block grid matching are readily realized by the monitor metrics. These features are indispensable for developing comprehensive automatic grid-generating codes. The paper is concerned with some aspects related to the development of grid generation techniques and codes based on the solution of the inverted Beltrami and diffusion equations.

We propose a new high-precision algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and is a generalization of the second order Boffetta-Osborne scheme. It is allowed by our method to solve more effectively the Zakharov-Shabat spectral problem for continuous and discrete spectra. Keywords Zakharov-Shabat problem, inverse scattering transform, nonlinear Schrödinger equation, numerical methods The solution of the direct problem for the Zakharov-Shabat problem (ZSP) is the first step in the inverse scattering transform (IST) for solving the nonlinear Schrödinger equation (NLSE) [1]. The numerical implementation of the IST has gained great importance and attracted special attention since Hasegawa and Tappert [2] proposed to use soliton solutions as a bit of information for fiber optic data transmission.