Grassmannian flows and applications to nonlinear partial differential equations

Margaret Beck, Anastasia Doikou, Simon John A. Malham, Ioannis Stylianidis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)
21 Downloads (Pure)

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 languageEnglish
Title of host publicationComputation and Combinatorics in Dynamics, Stochastics and Control
Subtitle of host publicationAbelsymposium 2016
EditorsGiulia Di Nunno, Elena Celledoni, Kurusch Ebrahimi-Fard, Hans Zanna Munthe-Kaas
PublisherSpringer
Pages71-98
Number of pages28
ISBN (Electronic)9783030015930
ISBN (Print)9783030015923
DOIs
Publication statusPublished - 14 Jan 2019

Publication series

NameAbel Symposia
PublisherSpringer
Volume13
ISSN (Print)2193-2808
ISSN (Electronic)2197-8549

ASJC Scopus subject areas

  • Mathematics(all)

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    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