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Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

  • Margaret Beck
  • Anastasia Doikou
  • Simon J. A. Malham
  • Ioannis Stylianidis
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

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.

Notes

Acknowledgements

We are very grateful to the referee for their detailed report and suggestions that helped significantly improve the original manuscript. We would like to thank Percy Deift, Kurusch Ebrahimi–Fard and Anke Wiese for their extremely helpful comments and suggestions. The work of M.B. was partially supported by US National Science Foundation grant DMS-1411460.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Margaret Beck
    • 1
  • Anastasia Doikou
    • 2
  • Simon J. A. Malham
    • 2
  • Ioannis Stylianidis
    • 2
  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Maxwell Institute for Mathematical Sciences, and School of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK

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