New Boundary-Value Problem Techniques for Nonlinear Wave Problems

PI和他的学生和合作者将进一步发展Athanassios Fokas用于边值问题的新方法，以研究两个不同的问题。首先，可积微分方程的周期性问题将使用这种新方法重新审视，目标是获得比当前方法更有效的（从渐近和数值的角度来看）答案。 第二，PI和他的学生将使用新的方法来分析研究非标准域上的不同问题的线性稳定性，例如具有指定边界条件的半直线或有限区间。此外，由于Ablowitz，Fokas和Muslimmani的水波问题的新公式将用于启动逆水波问题的解决方案：利用水波面测量恢复海底地形的问题。使用和扩展新方法研究在边界处具有规定输入的问题，PI和他的合作者和学生将研究描述广泛现象的问题。其中一些应用领域与海洋表面的波浪（海啸和流氓波）有关，而另一些则影响具有广泛社会影响的工程应用，如高速通信（非线性光学）或帮助我们理解当前的环境问题（烟雾形成和扩散）。最后，PI和他的研究小组将启动一种新的方法，通过对表面的遥感测量来确定近岸海洋底部的表面。这项工作有直接的商业和国防海军的后果。PI致力于培养下一代美国科学家，在当地范围内与学生合作，并在全球范围内编写一本描述所使用技术的教科书。

The PI and his students and collaborators will further develop the new methods of Athanassios Fokas for boundary value problems to investigate two different problems. First, the periodic problem for integrable differential equations will be revisited using this new approach, with the goal of obtaining a more efficient (from the point of view of asymptotics and numerics) answer than the current approach. Second, the PI and his students will use the new approach to analytically investigate the linear stability of different problems posed on non-standard domains such as the half-line or finite intervals with prescribed boundary conditions. In addition, the new formulation of the water wave problem due to Ablowitz, Fokas and Muslimmani will be used to initiate the solution of the inverse water wave problem: the problem of recovering the bottom topography using measurements of the water wave surface.Using and expanding on new methods for studying problems with prescribed input at their boundaries, the PI and his collaborators and students will investigate problems describing a wide range of phenomena. Some of these application areas are relevant for waves on the surface of the ocean (tsunamis and rogue waves), while others impact engineering applications with wide sociological impact such as high-speed communication (nonlinear optics) or help with our understanding of current environmental problems (smog formation and dispersal). Lastly, the PI and his research group will initiate a new approach to determine the surface at the bottom of the near-shore ocean from remote sensing measurements of the surface. This work has direct commercial and defense naval consequences. The PI is committed to the training of the next generation of US scientists, on a local scale by working with students, and on a more global scale by preparing a text book describing the techniques used.