Mathematical Modeling in Nonlinear Physics
Optical pulses are
used in fiber optics technology to code bit signals in data streams.
Optical pulses are transmitted through a long-haul
multi-channel telecommunication networks.
Multiple factors, such as polarization mode dispersion,
temporal instabilities, Gordon-Haus time jitting effects,
four-wave mixing and others, lead to degradation of the
bit-noise ratio between optical signals and accumulative
disturbances in the network. These factors become
dominant obstacles for the use of optical pulses in
transoceanic telecommunication lines that extend over
thousand miles. New revolutionary upgrading technologies,
dispersion and nonlinearity management, are designed
to reduce the degradation factors.
Nonlinear periodic
structures or simply photonic gratings are
small-sized devices used in photonic optics for all-optical
ultra-fast signal processing. Optical limiters, switches and
logic gates provide prospective basis for complex operations with
optical signals. New principles in design of such optical
gratings are based on nonlinear compensation of the
refractive index. This pioneer technology guarantees
stability of the photonic devices and their uniform behavior
suitable for signal processing.
Mathematical modeling in nonlinear physics is focused at
the three central problems:
- existence of stationary solutions in optical modules
- linear stability in the time evolution problem
- nonlinear transmission and interaction of optical signals
Research tools for solving these problems include
analysis of differential equations (exact integration,
asymptotic methods, numerical solutions), stability analysis
(eigenvalue problems, linear algebra methods),
and numerical modeling of partial differential equations
(split-step, Fourier, Runge-Kutta, finite differences).
