The cubic nonlinear Schroedinger (NLS) equation is a partial differential equation that describes pulse train generation in solid state or fiber lasers, or pulse propagation in long distance communication systems under ideal conditions. In order to describe the situation under more realistic physical scenarios, i.e., with losses, spectral filtering, etc., one must consider extended model equations which are perturbations of the NLS and which depend on the physical situation that is being modelled. Examples include models for phase-matched third-harmonic generation, materials with higher-order nonlinearities, and passively mode-locked fiber lasers. Numerous recent research efforts have indicated that these extended models are very robust. The central mathematical question of the planned research is the stability of propagating pulses, incorporating for example effects of small gains and losses in the optical fiber, and the correct balance of these effects which will guarantee stable pulses. A mathematically rigorous analytical study of the existence and stability of solitons for the governing equations has been lacking and will be developed. The work will exploit a recently discovered mathematical structure for the extended model equations. It makes use of the fact that these equations are very close to integrable equations that have been studied extensively.
Fiberoptic cables are of growing importance in long distance communications. The pulses which carry information in such cables can be described by mathematical equations that depend on the medium, among other things. The exact form of the equations is usually not known. In order for information to arrive undistorted at its destination, it is essential that these pulses remain stable as they travel along the fiber. In this work, rigorous mathematical methods to study this stability will be developed. The methods will also be robust under changes of the medium and will therefore have wide potential.
