Numerical and Analytical Investigations on Nonlocal Dispersive Wave Equations

近年来，非局部色散波动方程在电磁学、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、脑和意识等领域得到了广泛的应用。然而，它们的非定域性在数学分析和数值模拟方面都带来了相当大的挑战。该项目致力于解决与非局部色散波动方程的数学建模和数值模拟及其解的性质相关的基本问题。拟议的项目将弥合不同领域之间的差距，加强跨学科研究，并推动分数阶微分方程组在实践中的应用。由于非局域波动方程在物理、化学、生物和工程等领域有着广泛的应用，本课题的研究对推进相关领域的研究和技术具有很大的潜力。本研究的主要目的是建立非局域薛定谔波动方程的数学和数值处理方法，加深对具有远程相互作用的模型的理解，从而推动其在非局域问题中的应用。在这个项目中，我们将研究具有长程相互作用的离散的非线性薛定谔(DNLS)方程和具有分数拉普拉斯的分数阶非线性薛定谔(FNLS)方程。将离散模型和连续模型结合起来，为更深入地理解薛定谔波动方程的非局域性提供了新的机会。一方面，在模拟大格点的DNLS时，特别是在二维或三维晶格中，将开发精确的算法来提高计算效率和降低计算成本。另一方面，设计了高效、精确的分数拉普拉斯离散数值方法，并将其应用于研究fNLS的定态和动力学性质。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，并有助于分数阶常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.