Deformations of Integrable Systems

Gieseker教授将研究无限维可积系统的变形，例如KdV层次。KdV层次是在一个变量中相互交换的微分算子的序列。本授权的目的是产生这种和类似层次的变形，并保持完全可积性。Gieseker教授已经产生了KdV层次结构的这种变形，并将寻求产生更多的KdV变形，并找到其他无限维完全可积系统的变形。非线性科学中某些重要的方程是完全可积的，这大致意味着这些方程有很多守恒量。这些方程适用于各种各样的实际情况，例如在理解激光光学。gieseker教授建议应用代数几何的方法来寻找这类方程的离散化，使离散化是可积的。这意味着找到在计算机上求解这些方程的算法，以便在计算机上计算的解具有与原始方程相同类型的守恒量。Gieseker教授将继续研究他的KdV离散化，并发展其他无限维可积系统的新离散化。

DMS-9877151ABSTRACTProfessor Gieseker will investigate deformations of infinite dimensionalintegrable systems such as the KdV hierarchy. The KdV hierarchy is asequence of mutually commuting differential operators in one variable. Theobject of this grant is to produce deformations of this and similarhierarchies which preserve complete integrability. Professor Gieseker hasproduced such a deformation of the KdV hierarchy and will seek to producemany more deformations of KdV and also to find deformations of otherinfinite dimensional completely integrable systems. Certain important equations of nonlinear science are completely integrable,which roughly means that there are a lot of conserved quantities of theseequations. These equations are applicable in a wide variety of practicalsituations, for instance in understanding laser optics. Professor Giesekerproposes to apply the methods of algebraic geometry to find discretizationsof this type of equation so that the discretizations are integrable. Thismeans finding algorithms to solve these equations on the computer so thatthe solutions computed on the computer have the same type of conservedquantities as the original equations. Professor Gieseker will continue toinvestigate his discretization of KdV, as well as develop newdiscretizations of other infinite dimensional integrable systems.