Dynamics of Nonlinear Wave Equations

项目主要研究当前非线性波动方程的问题，包括解的渐近行为、奇点的发展和不稳定性。我们打算研究处理最小能量驻波附近的动力学问题，研究数学物理基本方程：非线性Klein-Gordon （NLKG）、非线性薛定谔（NLS）、非线性阻尼波（NLDW）的束缚态和束缚态驻波的稳定性和不稳定性问题。与双曲型问题相关的平稳方程的变分结构与过程的演化性质之间的相互作用使这类问题成为非线性分析中一个非常有趣的领域。我们还打算研究平凡平衡的稳定性/不稳定性，即各种重要双曲方程的小振幅解的行为，目的是得到爆炸和整体存在的尖锐判据。进一步感兴趣的问题是：由于阻尼项的存在，双曲方程的渐近抛物线结构；阻尼对解的主支承的影响阻尼对能量衰减率的影响；在考虑边界几何条件与耗散之间微妙相互作用的外域上，解的渐近行为。在所有提出的问题中，将考虑影响解决方案的各种力量之间的潜在平衡。重要的是要补充指出，解决提案中的许多问题需要发展新技术以及改进更经典的方法。非线性Klein-Gordon方程、非线性阻尼波方程和非线性薛定谔方程出现在流体力学、经典力学、量子力学、等离子体物理和非线性光学中。特别是非线性光学是电信技术的主要工具，主要是基于光纤的大量使用。这里没有必要强调光纤在大容量连接网络中的相关性，这在高速互联网应用中是至关重要的。由于非线性薛定谔方程是实际光纤通信系统的一个有用的设计工具，因此了解其解的稳定/不稳定现象是很重要的。提案中概述的问题的解答将加深我们对与上述基本方程有关的物理现象的理解。考虑到这些方程在许多技术领域的关键作用，理解支配它们的原则对于建立在技术基础上的社会至关重要。

PI: Grozdena Todorova, University of Tennessee, KnoxvilleDMS-0245578The proposed project intends to investigate current issues in nonlinear wave equations involving the topics of asymptotic behavior of solutions, development of singularities, andinstability. We intend to study problems that deal with the dynamics in the vicinity of the least energy standing waves, investigate the question of stability and instability of bound states and bound states standing waves for fundamental equations of Mathematical Physics: Nonlinear Klein-Gordon (NLKG), Nonlinear Schroedinger (NLS), Nonlinear Damped Wave (NLDW). The interaction between the variational structure of stationary equations associated with hyperbolic problems and the evolutionary nature of the process makes this type of problems a very interesting area in nonlinear analysis. We also intend to study the stability/instability of the trivial equilibrium, i.e., the behavior of the small amplitude solutions of various important hyperbolic equations, with the purpose to derive sharp criteria for blow--up and global existence. Further questions of interest are: the asymptotically parabolic structure of the hyperbolic equations due to the presence of damping terms; the influence of damping on the principal support of solutions; the influence of damping on the decay rate of the energy; and the asymptotic behavior of solutions in external domains where the delicate interaction between the geometricalcondition on the boundary and the dissipation has to be taken into account.In all of the proposed problems, the underlying balance between different kinds of forces, which affect the solution, will be considered. It is important to add that the solution of many of the problems in the proposal requires the development of new techniques as well as the refinement of more classical methods. Nonlinear Klein-Gordon, Nonlinear Damped Wave and Nonlinear Schroedinger equations arise in fluid dynamics, classical mechanics, quantum mechanics, plasma physics and nonlinear optics.In particular nonlinear optics is a main tool in telecommunication technology, mainly based on the large use of fiber optics. There is no need here to emphasize the relevance of the fiber optics in large capacity connection nets, which are crucial in high-speed Internet applications. Since the Nonlinear Schroedinger equation is a useful design tool for realistic fiber communication systems, it is important to understand the stability/instability phenomena of its solutions.The solutions of the problems outlined in the proposal will develop our understanding of the physical phenomena related with the above fundamental equations. Given the crucial role of these equations in many areas of technology, understanding the principles governing them is of basic importance for a society built on technology.