非线性波动方程的周期解及动力学性质研究

项目名称： 非线性波动方程的周期解及动力学性质研究

项目编号： No.10801060

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 冀书关

作者单位： 吉林大学

项目金额： 17万元

中文摘要： 本项目研究具有依赖于X（空间变量）系数的非线性波动方程，主要建立其时间周期解的存在性和正则性，并进一步尝试从无穷维动力系统的观点来发展与这种问题相适应的KAM理论。 波动方程是一类重要的偏微分方程，它通常描述自然界中各种各样的波动现象。特别地，周期波动是一种特殊而又十分重要的物理现象，在数学上它对应于波动方程的时间周期解。目前这是数学物理和应用数学领域中非常活跃的研究课题之一，从上世纪六十年代开始它一直受到国内外众多知名数学家的广泛关注，并取得了很多有意义的研究结果。但是现有的结果大多是处理较为理想的常系数情形。而具有依赖于X系数的情形是一种更为实际的模型，目前的结果还非常少。因此，我们的研究将会为人们进一步深入认识和理解波动现象提供必要的理论依据。

中文关键词： 波动方程；周期解；无穷维动力系统

英文摘要： In this project, we will study the nonlinear wave equation with x-dependent coefficients, the main aim is to establish the existence and regularity of its periodic solutions, and further try to develop the KAM theory adapted to this problem from the viewpoint of infinite-dimensional dynamical systems. The wave equation is an important partial differential equation in mathematical physics which generally describes all kinds of waves in nature. Particularly, it is well known that periodic wave motion is a special but very important physical phenomenon, which corresponds to the periodic solutions of wave equation in mathematics. At present, this is one of the most active research topics in mathematical physics and applied mathematics, and has attracted great attention of many mathematicians since 1960s. Until now, plenty of results on this problem have been obtained by using all kinds of methods and techniques. However, most of them dealt with the wave equation with constant coefficients, and very little is known for the wave equation with x-dependent coefficients which is a much better model for the description of wave motions in nonisotropic media. Therefore, our research will be able to provide some necessary theory basis for people's further understanding of the wave phenomena in nature.

英文关键词： wave equation; periodic solutions; infinite-dimensional dynamical systems