Mathematical Sciences: Scattering and Stability of NonlinearWaves

这个数学项目的工作分为三个主要领域：(A)无限维哈密顿系统的非线性散射和轨道渐近稳定性，(B)向不稳定性的转变和(C)向量Zakharov方程和非线性薛定谔方程。第一类问题是关于非线性有界态通道(模拟非衰变行为)和色散辐射场的非线性色散系统解的长时间行为。控制方程具有耦合到无限维色散辐射场的有限维动力系统的结构。第二部分，将一类本征值问题的新理论应用于色散介质中长波传播的非线性方程。目前正在开发一种独立于动力系统框架的高级公式。我们将寻求在流体和等离子体研究中出现的非局部方程的应用。第三个目标是研究向量Zakharov方程和薛定谔方程的非线性束缚态的存在性和性质。这些系统是对无碰撞等离子体中波的物理更好的近似。与这些方程相关的是标量系统中不存在的现象，如各向同性基态，它们可能参与奇点形成的动力学。在自然科学中，微分方程式构成了数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是描述关系，而是从它们中提取定性和定量的意义。

Work on this mathematics project is grouped into three main areas: (a) nonlinear scattering and orbital asymptotic stability in infinite dimensional Hamiltonian systems, (b) transitions to instability and (c) vector Zakharov and nonlinear Schrodinger equations. The first class of problems is concerned with the long-time behavior of solutions of nonlinear dispersive systems in terms of a nonlinear bound state channel (modeling nondecaying behavior) and a dispersive radiation field. The governing equations have the structure of a finite dimensional dynamical system which is coupled to the infinite dimensional dispersive radiation field. In the second, a new theory for a class of eigenvalue problems will be applied to nonlinear equations modeling long wave propagation in dispersive media. An advanced formulation is being developed which is independent of the dynamical systems framework. Applications to nonlocal equations arising in the study of fluids and plasmas will be sought. The third goal is to study the existence and properties of nonlinear bound states of the vector Zakharov and Schrodinger equations. These systems are a better approximation of the physics of waves in a collisionless plasma. Associated with these equations are believed to be phenomena not present in scalar systems, such as non-isotropic ground states, which may participate in the dynamics of singularity formation. Differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.