Mathematical Sciences: Nonlinear Evolutions Equations III

该奖项支持的研究将集中在非线性演化方程的各个方面：其解随时间演化的非线性偏微分方程。这项工作结合了微分方程的经典理论与谐波分析和数学物理。主要研究领域包括作为非线性波传播近似模型的非线性色散系统。我们将特别关注确定Sobolev指数的下界，它保证了广义Korteweg-deVries方程的局部适定性。类似的努力将应用于非线性薛定谔方程的研究。在这两种情况下，初步工作已经改进了对指数的已知估计。高阶方程也将被分析。这里方程的非线性部分将被限制为没有常数项的多项式。这类演化方程模拟弹性介质中的波动。局部结果已经建立了解决方案的平滑性，但长期行为还没有很好地理解。例如，五阶系统没有孤立波解。更多的工作计划在Zakharov-Schulman系统上进行，该系统描述了小振幅，高频波与声学型波的相互作用。其中线性部分是椭圆的，方程已被广泛研究，否则没有结果已知。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

Research supported by this award will focus on various aspects of nonlinear evolution equations: nonlinear partial differential equations whose solutions evolve in time. The work combines classical theory of differential equations with harmonic analysis and mathematical physics. The principal areas of study include nonlinear dispersive systems arising as approximate models in propagation of nonlinear waves. Particular interest will be paid to determining the lower bound of Sobolev exponents which guarantee local well-posedness for the generalized Korteweg-deVries equation. A similar effort will be applied to the study of nonlinear Schrodinger equations. In both cases preliminary work has already improved known estimates on the exponents. Higher order equations will also be analyzed. Here the nonlinear part of the equations will be restricted to polynomials having no constant term. This class of evolution equation models waves in elastic media. Local results have already been established for the smoothness of solutions, but long-time behavior is not well understood. For instance, fifth order systems do not have solitary wave solutions. Additional work is planned on the Zakharov-Schulman systems which describe interactions of small-amplitude, high frequencey wave with acoustic type waves. Where the linear part is elliptic, the equations have been extensively studied, otherwise no results are known. Partial differential equations form a basis for mathematical modeling 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 methods for approximation of solutions and estimates on the accuracy of these approximations.