Numerical and Analytical Investigations on Nonlocal Dispersive Wave Equations

非局部色散波方程在电磁学、声学、宇宙学、弹性力学、生物学、流体力学、粘弹性、地震学、水波、等离子体、量子力学、大脑和意识等领域有着广泛的应用，但其非局部性给数学分析和数值模拟带来了相当大的挑战。本计画旨在探讨非局部色散波方程之数学建模与数值模拟及其解之性质。该项目将弥合不同领域之间的差距，加强跨学科研究，并促进分数阶微分方程在实践中的应用。由于非局部波动方程在物理、化学、生物和工程等领域有着广泛的应用，本项目的研究对推动相关领域的研究和技术具有巨大的潜力，本研究的主要目的是建立非局部Schroedinger波动方程的数学和数值处理方法，加深对长程相互作用模型的理解，从而促进它们在非定域性问题中的应用。本计画将研究具有长程作用的离散非线性薛定谔（DNLS）方程和具有分数拉普拉斯算子的分数阶非线性薛定谔（fNLS）方程。结合离散和连续模型提供了一个新的机会，更深入地了解薛定谔波动方程的非局部性。一方面，精确的算法将被开发，以提高效率和减少计算成本，在模拟DNLS与大的晶格位置，特别是在二维或三维晶格。另一方面，有效和准确的数值方法离散的分数拉普拉斯算子将被设计和应用于研究的fNLS的定态和动力学的属性。对DNLS和fNLS的研究将加深对长程相互作用的建模和性质的理解，同时也有利于分数阶微分方程数值算法的发展。

Nonlocal dispersive wave equations have been recently applied in many areas such as electromagnetism, acoustics, cosmology, elasticity, biology, hydrodynamics, viscoelasticity, seismics, water wave, plasma, quantum mechanics, brain and consciousness, and so on. However, their nonlocality introduces considerable challenges in both mathematical analysis and numerical simulations. This project seeks to address fundamental issues related to mathematical modeling and numerical simulations of nonlocal dispersive wave equations as well as their solution properties. The proposed project will bridge the gap between different areas, enhance interdisciplinary research, and advance the application of fractional differential equations in practice. Since nonlocal wave equations have broad applications in physics, chemistry, biology and engineering, the research in this project has great potentials to advance the research and technology in relevant areas.The main objectives of this research are to build mathematical and numerical treatments for the nonlocal Schroedinger wave equations, and to provide a deeper understanding of the modeling with long-range interactions, so as to advance their application to problems with nonlocality. In this project, both the discrete nonlinear Schroedinger (DNLS) equation with long-range interactions and the fractional nonlinear Schroedinger (fNLS) equation with the fractional Laplacian will be investigated. Integrating the discrete and continuous models offers a new opportunity for a deeper understanding of the nonlocality of the Schroedinger wave equations. On the one hand, accurate algorithms will be developed to improve the efficiency and reduce the computational costs in simulating the DNLS with large lattice sites, especially in two- or three-dimensional lattices. On the other hand, efficient and accurate numerical methods for discretizing the fractional Laplacian will be designed and applied to study the properties of the stationary states and dynamics of fNLS. The study on the DNLS and fNLS will provide a deeper understanding on modeling and properties of long-range interactions, as well as is benefitting the development of numerical algorithms for fractional differential equations.