Applications of Transformation Theory for Nonlinear Integrable Systems to Linear Prediction Problems and Isospectral Deformations

Abstract

In 1981, M. Sato introduced the Kadomtsev–Petviashvili (KP) equation hierarchy and completely characterized the solution space of the KP equation being of great interest in terms of an infinite-dimensional Grassmann manifold. The transformation group for the KP hierarchy is the automorphism group GL (∞) of the Grassmann manifold and many other nonlinear integrable systems are derived from the KP hierarchy by reductions. The method of Riemann–Hilbert (RH) transformations was developed in the study of symmetries and solutions to nonlinear integrable systems. The RH transformations were successfully applied to the stationary Einstein equation, which seem to be outside the KP hierarchy. An infinite system was introduced on an infinite-dimensional Grassmann manifold and formulated an algebraic RH transformation for the self-dual Yang Mills (SDYM) equation. The RH transformation theory has become a powerful tool for nonlinear integrable systems. The chapter discusses the applications of the RH transformation theory in mathematical physics to linear prediction problems of stochastic processes and isospectral deformations of linear operators. A discrete isospectral deformation of matrices is also defined as a special member of transformations.