Grassmannian flows and applications to non-commutative non-local and local integrable systems

Abstract

We present a method for linearising classes of matrix-valued nonlinear partial differential equations with local and nonlocal nonlinearities. Indeed we generalise a linearisation procedure originally developed by Pöppe based on solving the corresponding underlying linear partial differential equation to generate an evolutionary Hankel operator for the ‘scattering data’, and then solving a linear Fredholm equation akin to the Marchenko equation to generate the evolutionary solution to the nonlinear partial differential system. Our generalisation involves inflating the underlying linear partial differential system for the scattering data to incorporate corresponding adjoint, reverse time or reverse space–time data, and it also allows for Hankel operators with matrix-valued kernels. With this approach we show how to linearise the matrix nonlinear Schrödinger and modified Korteweg de Vries equations as well as nonlocal reverse time and/or reverse space–time versions of these systems. Further, we formulate a unified linearisation procedure that incorporates all these systems as special cases. Further still, we demonstrate all such systems are example Fredholm Grassmannian flows.