Mathematical Sciences: Nonlinear Waves

小行星9404528 我们将结合动力系统的规范形技术和非线性谐波分析来研究非线性色散波。我们将研究反应扩散方程组的平面波解的多维扰动，试图将简单的有效方程与完整的方程联系起来。我们将发展和分析随机微分方程的数值方法，特别注意边界。 非线性色散波包括水面上的波和原子晶格中的声波。反应扩散方程描述的是火焰。了解火焰的不稳定性是很重要的，例如在化学工程中。随机微分方程被用来模拟现代药物设计、股票价格和利率变动（在纽约市很重要）以及许多其他事情中的分子。这些领域迫切需要高效、准确的仿真技术。

9404528 Goodman We will study nonlinear dispersive waves by combining the normal forms technique of dynamical systems with nonlinear harmonic analysis. We will study multi-D perturbations of plane wave solutions of systems of reaction-diffusion equations, attempting to relate simple effective equations to the full equations. We will develop and analyze numerical methods for stochastic differential equations with special attention to boundaries. Nonlinear dispersive waves include waves on the surface of water and sound waves in a crystal lattice of atoms. Reaction diffusion equations describe, among other things, flames. Understanding of instabilities of flames is important, for example, in chemical engineering. Stochastic differential equations are used to model molecules in modern drug design, stock price and interest rate movements (important in New York City), and many other things. Efficient and accurate simulation technology is sorely needed in these areas.