Dynamics of Nonlinear Wave Equations

主要研究者：Grozdena Todorova，田纳西大学，诺克斯维尔DMS-0245578该项目旨在研究当前非线性波动方程的问题，涉及解的渐近行为，奇点的发展和不稳定性。我们打算研究在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Schroedinger（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。与双曲型问题相关联的定常方程的变分结构和过程的演化性质之间的相互作用使得这类问题在非线性分析中成为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，研究了各种重要的双曲型方程的小振幅解的性质，目的是得到爆破和整体解存在的尖锐判据。进一步的问题是：由于阻尼项的存在，双曲型方程的渐近抛物结构，阻尼对解的主支集的影响，阻尼对能量衰减率的影响，阻尼对解的主支集的影响，阻尼对解的主支集的影响，阻尼对解的主支集的影响，阻尼对解的主支集的影响。以及外部区域解的渐近性态，其中边界上的几何条件和耗散之间的微妙相互作用必须考虑。所有提出的问题，影响解决方案的各种力量之间的潜在平衡，都将得到考虑。重要的是要补充的是，解决提案中的许多问题需要开发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Schroedinger方程广泛存在于流体力学、经典力学、量子力学、等离子体物理和非线性光学等领域，尤其是非线性光学是通信技术的主要工具，其主要基础是光纤的大量使用。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至关重要。由于非线性薛定谔方程是设计实际光纤通信系统的有用工具，因此理解其解的稳定性/不稳定性现象是很重要的。本文中所概述的问题的解将有助于我们理解与上述基本方程相关的物理现象。考虑到这些方程在许多技术领域中的关键作用，理解它们的原理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.