Deformations of Integrable Systems

DMS-9877151摘要Gieseker教授将研究无限维可积系统的变形，如KdV族。KdV族是一个相互交换的单变量微分算子序列。该补助金的目的是产生变形的这和类似的层次保持完整的可积性。 Gieseker教授已经给出了KdV族的这样一种变形，并将寻求KdV族的更多变形，并寻找其他无限维完全可积系统的变形。非线性科学中的某些重要方程是完全可积的，这大致意味着这些方程存在大量的守恒量。 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教授提出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，使得在计算机上计算出的解与原始方程具有相同类型的守恒量。 Gieseker教授将继续研究KdV的离散化，并开发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.