arXiv subjects

S. Leble

Publications and source records attributed to S. Leble.

7 recordsLinked to original sources

arXiv2006

On convergence and stability of a numerical scheme of coupled nonlinear Schr\"{o}dinger equations

We consider numerical solution of Coupled Nonlinear Schr\"{o}dinger Equation. We prove stability and convergence in the $L_2$ space for an explicit scheme which estimations is used for implicit scheme and compare both method. As a test we compare numerical solutions of Manakov system with known analytical solitonic solutions and as example of general system - evolution of two impulses with different group velocity (model of pulses interaction in optic fibers). Last example, a rectangular pulse evolution, shows asymptotic behavior typical for Nonlinear Schr\"{o}dinger Equation asymptotics with the same initial condition.

math.NA↗

arXiv2004

A dressing of zero-range potentials and electron-molecule scattering problem at Ramsauer-Townsend minimum

A dressing technique is used to improve zero range potential (ZRP) model. We consider a Darboux transformation starting with a ZRP, the result of the "dressing" gives a potential with non-zero range that depends on a seed solution parameters. Concepts of the partial waves and partial phases for non-spherical potential are used in order to perform Darboux transformation. The problem of scattering on the regular X$_{\hbox{n}}$ and YX$_{\hbox{n}}$ structures is studied. The results of the low-energy electron-molecule scattering on the dressed ZRPs are illustrated by model calculation for the configuration and parameters of the silane ($\hbox{SiH}_4$) molecule. \center{Key words: low-energy scattering, multiple scattering, Ramsauer-Townsend minimum, silane, zero range potential.

quant-ph↗

arXiv2002

On multisolitonic decay behavior of internal gravity waves

We claim that changes of scales and fine-structure could increase from multisoliton behavior of internal waves dynamics and, further, in the so-called "wave mixing". We consider initial-boundary problems for Euler equations with a stratified background state that is valid for internal water waves. The solution of the problem we search in the waveguide mode representation for a current function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem for the vertical variable. The Cauchy problem is solved for initial conditions with realistic geometry and magnitude. We choose the geometry and dimensions of the McEwan experiment with the stratification of constant buoyancy frequency. The horizontal profile is defined by numerical solutions of a coupled Korteweg-de Vries system. The numerical scheme is proved to be convergent, stable and tested by means of explicit solutions for integrable case of the system. Together with the solution of the Sturm-Liouville problem it gives the complete internal waves profile.

math-ph↗

arXiv2002

Analytical and numerical solution of coupled KdV-MKdV system

The matrix 2x2 spectral differential equation of the second order is considered on x in ($-\infty,+\infty$). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter solution for the coupled KdV-MKdV system with reductions. We show effects of parameters on the resulting solutions (reality, singularity). A numerical method for general coupled KdV-MKdV system is introduced. The method is based on a difference scheme for Cauchy problems for arbitrary number of equations with constants coefficients. We analyze stability and prove the convergence of the scheme which is also tested by numerical simulation of the explicit solutions.

quant-ph↗

arXiv2002

On numerical integration of coupled Korteweg-de Vries System

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to the non-integrable case.

quant-ph↗

arXiv2002

Tollmien-Shlichting and sound waves interaction: general description and nonlinear resonances

The general hydro-thermodynamic system of equations in 2+1 dimensions with arbitrary equations of state (Taylor series approximation) is split to eigen modes: Tollmienn-Schlichting (TS) wave and two acoustic ones. A mode definition is realized via local relation equations, extracted from the linearization of the general system over boundary layer flow. Each such connection defines the subspace and the corresponding projector. So the division is performed locally and could help in initial (or boundary) problems formulation. The general nonlinearity account determines the specific form of interaction between acoustic and vortical boundary layer perturbation fields. After the next projecting to a subspace of Orr-Sommerfeld equation solution for the TS wave and the corresponding procedure for acoustics, the equations go to one-dimensional system that describes evolution along the basic stream. A new mechanism of nonlinear resonance excitation of the TS wave by sound is proposed and modelled via four-wave interaction. Subjectclass: Primary 35Q30 ; Secondary 76F70, 35Q72; Keywords: keywords fluid mechanics, boundary layer, projector to eigen modes, Tollmien-Schlichting waves, acoustic waves, nonlinear resonance, N-wave system.

physics.flu-dyn↗

arXiv2002

Zero-Range Potentials in Multi-Channel Diatomic Molecule Scattering

The method of zero-range potentials is generalized to account for the molecular electron excitation process. It is made by a matrix formulation in which a state vector components are associated with a scattering channel. The multi-center target is considered and the model is applied to the example of $e + H_2$ low energy scattering. The results of evaluation of cross-sections are compared with ones of the MCF and SMC methods.

quant-ph↗