Stability of Travelling Waves with Applications in NonlinearOptics

立方非线性薛定谔(NLS)方程是描述固体光纤激光器中脉冲串的产生或理想条件下远距离通信系统中脉冲传输的偏微分方程组。为了描述在更真实的物理场景下的情况，即具有损耗、频谱滤波等，必须考虑扩展模型方程，该扩展模型方程是NLS的扰动，并且取决于被建模的物理情况。例子包括相位匹配三次谐波产生的模型，具有高阶非线性的材料，以及被动锁模光纤激光器。最近的大量研究表明，这些扩展模型是非常稳健的。计划研究的中心数学问题是传播脉冲的稳定性，包括例如光纤中小增益和小损耗的影响，以及保证稳定脉冲的这些影响的正确平衡。对于控制方程的孤子的存在性和稳定性，一直缺乏严格的数学分析研究，并将继续发展。这项工作将利用最近发现的扩展模型方程的数学结构。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。光缆在长距离通信中越来越重要。在这种光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，等等。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，这些脉冲在沿光纤传输时保持稳定是至关重要的。在这项工作中，将发展严格的数学方法来研究这种稳定性。这些方法在介质的变化下也将是健壮的，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.