### 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.

Original language | English |
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Title of host publication | Computation and Combinatorics in Dynamics, Stochastics and Control |

Subtitle of host publication | Abelsymposium 2016 |

Editors | Giulia Di Nunno, Elena Celledoni, Kurusch Ebrahimi-Fard, Hans Zanna Munthe-Kaas |

Publisher | Springer |

Pages | 71-98 |

Number of pages | 28 |

ISBN (Electronic) | 9783030015930 |

ISBN (Print) | 9783030015923 |

DOIs | |

Publication status | Published - 14 Jan 2019 |

### Publication series

Name | Abel Symposia |
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Publisher | Springer |

Volume | 13 |

ISSN (Print) | 2193-2808 |

ISSN (Electronic) | 2197-8549 |

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Grassmannian flows and applications to nonlinear partial differential equations'. Together they form a unique fingerprint.

## Profiles

## Simon John A. Malham

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Mathematics - Associate Professor

Person: Academic (Research & Teaching)

## Cite this

Beck, M., Doikou, A., Malham, S. J. A., & Stylianidis, I. (2019). Grassmannian flows and applications to nonlinear partial differential equations. In G. Di Nunno, E. Celledoni, K. Ebrahimi-Fard, & H. Z. Munthe-Kaas (Eds.),

*Computation and Combinatorics in Dynamics, Stochastics and Control: Abelsymposium 2016*(pp. 71-98). (Abel Symposia; Vol. 13). Springer. https://doi.org/10.1007/978-3-030-01593-0_3