Stability of Travelling Waves with Applications in NonlinearOptics

三次非线性薛定谔（NLS）方程是描述理想条件下固体激光器中脉冲串产生或长距离通信系统中脉冲传输的偏微分方程。 为了描述更真实的物理场景下的情况，即，由于损耗、频谱滤波等，必须考虑扩展的模型方程，这些方程是NLS的扰动，并且依赖于被建模的物理情况。 例子包括相位匹配的三次谐波产生模型、高阶非线性材料和被动锁模光纤激光器。 许多最近的研究工作表明，这些扩展模型是非常强大的。 计划研究的中心数学问题是传播脉冲的稳定性，例如光纤中的小增益和损耗的影响，以及这些影响的正确平衡，这将保证稳定的脉冲。 一个数学上严格的分析研究的存在性和稳定性的控制方程一直缺乏和将开发。 这项工作将利用最近发现的数学结构的扩展模型方程。 它利用了这样一个事实，即这些方程非常接近已被广泛研究的可积方程。光纤电缆在长距离通信中越来越重要。在这种电缆中携带信息的脉冲可以用依赖于介质的数学方程来描述。 这些方程的确切形式通常是未知的。 为了使信息不失真地到达目的地，这些脉冲在沿着光纤传播时保持稳定是至关重要的。 在这项工作中，将开发严格的数学方法来研究这种稳定性。 该方法也将是稳健的介质下的变化，因此将具有广泛的潜力。

The cubic nonlinear Schroedinger (NLS) equation is a partial differential equation that describes pulse train generation in solid stateor 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 closeto 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 bymathematical equations that depend on the medium, among other things. The exactform 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 underchanges of the medium and will therefore have wide potential.