Deformations of Integrable Systems

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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.