# Applications of Grassmannian flows to integrable systems

## TL;DR

This paper explores how many nonlinear partial differential systems, including integrable ones, can be represented as Fredholm Grassmannian flows, enabling solutions to be constructed from linear systems, thus unifying and extending solution methods.

## Contribution

It demonstrates that a broad class of nonlinear systems, including non-commutative KdV and NLS equations, can be linearized via Grassmannian flows, extending previous work and providing new applications.

## Key findings

- Nonlinear systems can be realized as Fredholm Grassmannian flows.
- Solution flows of non-commutative KdV and NLS are examples of Grassmannian flows.
- The approach simplifies solving complex nonlinear PDEs by linear methods.

## Abstract

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.