Vladimir I. Pulov

Symmetry properties of nonlinear pulse propagation in birefringent optical fibers with stimulated Raman scattering

A set of couled nonlinear equations describing propagation of two polarization light modes inside optical fibers with stimulated Raman scattering are considered. Using a Lie group technique, the most general Lie algebra is derived for the two cases: with and without orthogonal Raman gain. The optimal set of reduced systems of ordinary differential equations for the case without orthogonal Raman gain is obtained.

Lie group symmetry classification of solutions to coupled nonlinear Schrödinger equations

By applying the Lie group reduction method a full symmetry classification of one parameter group invariant solutions of two coupled nonlinear Schrodinger equations is presented. The physical situations under consideration include propagation of two polarization modes in weak and strong birefringent fibers, propagation of two waves at different carrier wavelengths, and nonlinear directional couplers.

Symmetries and Some Special Solutions of the Helfrich Model

The goal of this chapter is to present the results of the Lie group analysis in application to the Helfrich spontaneous curvature model. Special attention is paid to the translationally invariant solutions and the corresponding cylindrical equilibrium shapes. Graphs of closed diretrices of the obtained cylindrical surfaces in fixed and moving reference frame are presented.

Trajectories of the Plate-Ball Problem

The plate-ball problem concerns the shortest trajectories traced by a rolling sphere on a horizontal plane between the prescribed initial and final states meaning the positions and orientations of the sphere. Here we present an explicit parametric representation of these trajectories in terms of the Jacobian elliptic functions and elliptic integrals. MSC : 70E18, 70Q05, 81E15

Two light modes propagation in nonlinear media with negative cross-phase modulation via Lie group analysis

Two coupled nonlinear Schrodinger equations describing propagation of light pulses through nonlinear media with negative cross-phase modulation and different group-velocity dispersion regions for the two polarization modes are considered. Applying a Lie group technique, the most general Lie algebra and the corresponding Lie group of point symmetries permitted by the considered equations are derived. As a result, the optimal set of one-dimensional Lie algebras is obtained.

Group analysis of nonlinear pulse propagation in birefringent and multimode optical fibers

A set of two coupled nonlinear Schroedinger equations, describing nonlinear propagation in strongly birefringent and multimode optical fibers is analyzed by means of Lie group technique. As a result complete list of group- invariant exact solutions is obtained. A new exact solution is presented. The so-called symbiotic periodic and soliton solutions are included in this classification in natural ways. Laws of conservation are also derived.