Mathematical Sciences: Nonlinear Wave Equations of Mathematical Physics

小克雷格9401377 该奖项支持三个领域的数学研究，这些领域与偏微分方程理论和数学物理有关。 重点主要是保守的发展方程。 例子包括非线性波动方程，薛定谔方程，Korteweg deVries方程和版本的Boussinesq系统。 这些问题都可以写成具有无限多个自由度的哈密顿系统。 与动力系统的类比可以追求的目标是描述解决方案的重要特征和相空间的主要结构，他们提出。 这些项目包括继续发展的理论Kolmogorov，阿诺德和莫泽系统的无限多个自由度，特别是上述非线性方程。 这项工作的目的是了解非线性波动方程的稳定运动，无论是在平衡点的邻域，或扰动的完全可积系统。 第二个项目涉及薛定谔方程的解的奇点的演化，在线性和非线性的情况下。 这需要在一个新的设置中，其中有一个相互依赖的空间和傅立叶域的属性的微局部分析技术。 最后的项目涉及工作的自由表面问题的水波，这是重要的数学流体动力学以及偏微分方程。 工作将解决稳定的自由表面问题和非平稳问题，在两个和三维设置。 偏微分方程是物理世界数学模型的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展的方法近似的解决方案和估计的准确性，这些近似。 ***

9401377 Craig This award support mathematical research in three areas which related the theory of partial differential equations and mathematical physics. The focus is principally on conservative evolution equations. Examples include nonlinear wave equations, Schrodinger equations, the Korteweg deVries equation and versions of the Boussinesq system. These problems can all be written as Hamiltonian systems, with infinitely many degrees of freedom. The analogy with dynamical systems can be pursued with the goal of describing the important features of solutions and the principal structures of the phase space in which they are posed. The projects include the continuation of development of the theory of Kolmogorov, Arnold and Moser to systems with infinitely many degrees of freedom, in particular, to the above nonlinear equations. The intent of this work is to understand the stable motions of nonlinear wave equations, either in neighborhoods of equilibrium points, or for perturbations of completely integrable systems. A second project concerns the evolution of singularities of solutions of the Schrodinger equation, in both linear and nonlinear cases. This requires microlocal analysis techniques in a novel setting in which there is an interdependence of properties in the spatial and Fourier domains. The final project involves work on the free surface problem of water wave, which is important in mathematical fluid dynamics as well as partial differential equations. Work will address steady free surface problems and nonstationary problems, in both two and three dimensional settings. Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops me thods for approximation of solutions and estimates on the accuracy of these approximations. ***