Computational and Analytical Challenges in Nonlinear Dispersive Wave Equations

非线性色散波动方程模拟流体力学、等离子体物理、湍流和光学中的各种现象。对这些方程进行数值分析和模拟，使我们能够对相应的应用作出预测。例如，解决应用中出现的以下问题是可取的：用于传输和存储数据的光学设备中的信号是否稳定，或者能量如何在湍流系统中传递？这个项目将从两个方面增进我们的知识。首先，它将在数学上严格地证明方程的性质，其中一个例子就是光脉冲的稳定性。其次，它将开发和证明在计算机上模拟方程的数值算法。系统的算法分析保证了计算机模拟真实地反映了方程。最后，对这些方程的分析和模拟将决定这些方程作为物理应用模型的有效性。这项研究的另一个好处是，它开发的数值算法将适用于广泛的方程和物理问题。本项目研究的方程为非线性Schrödinger型，包括微分非线性Schrödinger方程和Gross-Pitaevskii方程。这项工作将解决的主要分析问题包括奇异解和非线性界态的存在性和稳定性。这些问题将通过渐近分析、变分方法和谱理论来解决。奇点形成也将通过模拟研究，使用自适应网格划分方法，如迭代网格重新分配。模拟时间相关问题的数值算法将通过算子分裂方法实现，分别处理方程的非线性和线性色散部分。这一数值分析的目的是构建适合长时间模拟的高效节能方法。本文将以半离散形式研究非线性约束状态的计算算法，使其不受空间离散化的影响。约束状态算法将有助于初始条件的构建，然后可以使用时间相关算法进行研究。

Nonlinear dispersive wave equations model a variety of phenomena in fluid mechanics, plasma physics, turbulence, and optics. Analyzing and simulating such equations numerically allows us to make predictions pertaining to the corresponding application. For example, it would be desirable to address the following questions arising in applications: Will signals in optical devices, used for the transmission and storage of data, be stable, or how is energy transferred in turbulent systems? This project will advance our knowledge in two ways. First, it will seek to prove, in a mathematically rigorous sense, properties of the equations, an example of which is the stability of optical pulses. Second, it will develop and justify numerical algorithms for simulating the equations on computers. Systematic analysis of the algorithms ensures that the computer simulations truly reflect the equations. Finally, both the analysis and simulation of the equations will determine the regime of validity of these equations as models for the physical applications. A supplemental benefit of this research is that the numerical algorithms it develops will be applicable to a broad class of equations and physical problems. The equations under investigation in this project are of the nonlinear Schrödinger type, including the derivative nonlinear Schrödinger equation and the Gross-Pitaevskii equation. Major analytical questions this work will address include the existence and stability of singular solutions and nonlinear bound states. These problems will be addressed through asymptotic analysis, variational methods, and spectral theory. Singularity formation will also be investigated through simulation, using adaptive meshing methodology, such as iterative grid redistribution. Numerical algorithms for simulating the time dependent problems will be implemented by operator splitting methods, treating the nonlinear and linear dispersive parts of the equation separately. This numerical analysis will be pursued with the intention of constructing efficient energy preserving methods suitable for long time simulation. An algorithm for computing nonlinear bound states will be studied in a semi-discrete form, so as to be insensitive to spatial discretization. The bound state algorithms will assist in the construction of initial conditions, which can then be studied using the time dependent algorithms.