  We show how many classes of partial differential systems with local and
nonlocal nonlinearities are linearisable in the sense that they are realisable
as Fredholm Grassmannian flows. In other words, time-evolutionary solutions to
such systems can be constructed from solutions to the corresponding underlying
linear partial differential system, by solving a linear Fredholm equation. For
example, it is well-known that solutions to classical integrable partial
differential systems can be generated by solving a corresponding linear partial
differential system for the scattering data and then solving the linear
Fredholm (or Volterra) integral equation known as the
Gel'fand-Levitan-Marchenko equation. In this paper and in a companion paper,
Doikou et al. [DMSW:graphflows], we both, survey the classes of nonlinear
systems that are realisable as Fredholm Grassmannian flows, and present new
example applications of such flows. We also demonstrate the usefulness of such
a representation. Herein we extend the work of Poppe and demonstrate how
solution flows of the non-commutative potential Korteweg de Vries and nonlinear
Schrodinger systems are examples of such Grassmannian flows. In the companion
paper we use this Grassmannian flow approach as well as an extension to
nonlinear graph flows, to solve Smoluchowski coagulation and related equations.
