怪波的数学理论及其在非线性系统中的应用

怪波作为一种极端非线性波，源于海洋学、非线性光学、玻色-爱因斯坦凝聚态、等离子体、大气科学、甚至金融学等领域。描述怪波的数学物理方程与可积和非可积系统有密切的联系。本项目首先研究连续非线性可积系统的非零背景的反散射理论和Riemann-Hilbert问题，分析怪波产生的机理和渐进行为，探讨连续怪波的产生与散射数据、谱参数及非零背景之间的影响，揭示怪波与呼吸子之间的内在关系。其次，研究离散非线性可积系统的怪波及动力学，探讨连续和离散怪波之间关联。然后，研究非局域非线性可积系统的极端波和怪波及稳定性，揭示局域和非局域极端波和怪波之间对应。最后，研究非线性非可积系统的调制不稳定性及近似怪波的动力学，探讨色散项和非线性耗散项对近似怪波的影响。以期揭示非线性可积和非可积系统的怪波产生的机理和数学性质，为怪波在非线性光学、海洋、金融经济等领域的应用提供更可靠的理论基础和依据。

Rogue waves, as a type of extreme waves, arise from ocean, nonlinear optics, Bose-Einstein condensates, plasmas physics, atmospherics, and even finance. The mathematical physics equations describing rogue waves have a close connection with integrable and non-integrable systems. This project firstly studies inverse scattering theory with non-zero background and Riemann-Hilbert problem of continuous integrable systems, analyzes the generation mechanism of rogue waves and asymptotic behavior, discuss the effects of rogue wave generation, spectral parameters, and non-zero background, even investigate the relations between rogue waves and breathers; Secondly, we focus on rogue waves and dynamics of discrete nonlinear integrable systems, and then investigate the relation between continuous and discrete rogue waves. Thirdly, we study extreme waves and rogue waves and dynamics of nonlocal nonlinear integrable systems, and then investigate the relation between local and nonlocal extreme and rogue waves. Finally, modulational instability and approximate rogue waves and dynamics, and explore the effects of dispersion and nonlinear dissipation on approximate rogue waves. The aim of this project is to the generation mechanism and mathematical properties of rogue waves of integrable and non-integrable systems, and to present more reliable theoretical basis for their applications in nonlinear optics, ocean, and finance.