Mathematical Sciences: Nonlinear Dispersive Waves

该项目将支持非线性色散系统领域的研究，特别是非线性薛定谔方程和扎哈罗夫方程。研究将集中在四个方面：(1)非线性散射和稳定性；(2)非线性薛定谔方程和Zakharov系统的奇点形成和成核；(3)长波传播数学描述中出现的方程低能解的色散；(4)哈密顿系统中的参数共振和耗散的影响。非线性波动现象出现在物理系统中，范围从量子水平到海洋中流体的大尺度运动。非线性波中的问题可以用各种技术来研究：解析、形式渐近和计算机模拟。

This project will support research in the area of nonlinear dispersive systems, in particular, the nonlinear Schrodinger equation and the Zakharov equation. The research will be focused in four areas: (1) nonlinear scattering and stability, (2) singularity formation and nucleation for the nonlinear Schrodinger equation and Zakharov systems, (3) dispersion of low energy solutions of equations arising in the mathematical description of the propagation of long waves, and (4) parametric resonance in Hamiltonian systems and the effect of dissipation. Nonlinear wave phenomena arise in physical systems ranging from the quantum level to the large scale motion of fluids in oceans. Problems in nonlinear waves can be studied using various techniques: analytical, formal asymptotics, and computer simulations.