Theory of Partial Differential Equations
Various regimes
of soliton propagation in optical fibers and gratings
are modeled numerically by scalar, vector, and periodic
modules of the nonlinear Schrodinger (NLS) equation.
Linearization of the NLS equations results in a matrix
eigenvalue problem which typically has the unstable
eigenvalues leading to exponential growth of small
perturbations. Potential commercial use of solitons
in optical fibers, wave couplers, photonic devices
and semiconductor lasers can be achieved in the
regime where the soliton propagation is observed to be stable.
The challenging problem to detect soliton instabilities
in different nonlinear optical structures was attacked
simultaneously by many research groups during the last decade.
Up to the date, basic properties of the soliton spectrum
in the NLS equations have been understood and classified.
Many asymptotic and numerical approaches were developed
to explain different mechanisms for instabilities,
bifurcations, resonances, radiative decays and
oscillatory nonlinear dynamics of soliton signals.
It is now the time to complete the most general theory
of soliton instabilities and to implement the basic
principles and methods into powerful software packages.
The software is to be directed for solving
new modules of the NLS equations that arise in design
of new effective optical devices in fiber and photonic
optics communications.
