Mathematical Sciences: Classical And Quantum Integrable Systems

研究二阶能量相关薛定谔算符的等谱流。相关的逆散射问题不能用Gel’fand- levitan - marchenko方法解决；而是必须通过在合适的黎曼曲面上求解黎曼-希尔伯特问题来处理。流程本身分为两种情况。只有当初始数据被限制为合适的不变子流形时，这两个不同的流才具有全局解。我们将利用Deift和Zhou最近开发的Riemann-Hilbert问题解的长时间渐近性分析方法来研究解的渐近性。将调查不适当数据的解决方案分解的性质。三波相互作用的哈密顿量包含一个三次相互作用项和一个动能项，允许无界的正能量和负能量，就像在量子电动力学中一样。因此，该模型包含了许多有趣的特征，包括创造和湮灭过程。该模型可作为基本粒子物理学中这类过程的范例。分析非线性波的现代技术将应用于具有物理意义的模型。第一个模型由近似流体中的波动的方程组成。这些模型允许奇点的发展，类似于海洋中的波浪破碎。我们将尝试分析奇点的结构。第二个模型由非线性光学中出现的经典方程的量子版本组成。该系统展示了类似于基本粒子相互作用的过程，在这些过程中，粒子被创造和毁灭。

DMS-9501233 Sattinger The isospectral flows of a second order energy dependent Schrodinger operator will be investigated. The associated inverse scattering problem cannot be resolved by the Gel'fand-Levitan-Marchenko method; but instead must be treated by solving a Riemann-Hilbert problem on an appropriate Riemann surface. The flows themselves divide into two cases. The two distinct flows have global solutions only when the initial data is restricted to suitable invariant submanifolds. The asymptotic behavior of the solutions will be investigated using the recent methods developed by Deift and Zhou for analyzing long time asymptotics of solutions to Riemann-Hilbert problems. The nature of the break-down of the solutions for inapprmpriate data will be investigated. The Hamiltonian for the three wave interaction contains a cubic interaction term and a kinetic energy term that allows unbounded positive and negative energies, as in quantum electrodynamics. The model thus contains a number of interesting features, including creation and annihilation processes. The model serves as a paradigm for such processes in elementary particle physics. Modern techniques for analyzing nonlinear waves will be applied to models of physical interest. The first model consists of equations approximating waves in fluids. These models allow for singularities to develop, similar to waves breaking in the ocean. An attempt will be made to analyze the structure of the singularities. The second model consists of quantum versions of classical equations that arise in nonlinear optics. The system exhibits processes similar to those in elementary particle interactions, in which particles are created and destroyed.