# Grassmannian flows and applications to nonlinear partial differential equations

## TL;DR

This paper introduces a novel method to generate solutions for a broad class of nonlinear PDEs with nonlocal Riccati-type nonlinearities by extending finite-dimensional Grassmannian flow techniques to infinite dimensions, supported by numerical simulations.

## Contribution

It extends the Grassmannian flow approach to infinite-dimensional PDEs with nonlocal nonlinearities, providing a new integral equation framework for solution generation.

## Key findings

- Solutions to scalar PDEs with nonlocal quadratic nonlinearities can be explicitly generated.
- Numerical simulations demonstrate the method's effectiveness on Fisher--Kolmogorov--Petrovskii--Piskunov equations.
- The approach suggests potential extensions to more general nonlinear PDE systems.

## Abstract

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.