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Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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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.

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Notes

  1. 1.

    We quote from the referee: “numerical integration in time will usually become inaccurate for large time t, but the nature of the exact solution gives you a precise answer for arbitrary t, and maybe allows access to information about long time behaviour which is inaccessible via standard numerical schemes.”

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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Margaret Beck

  2. Maxwell Institute for Mathematical Sciences, and School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK

    Anastasia Doikou, Simon J. A. Malham & Ioannis Stylianidis

Authors
  1. Margaret Beck
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  2. Anastasia Doikou
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  3. Simon J. A. Malham
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  4. Ioannis Stylianidis
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Corresponding author

Correspondence to Simon J. A. Malham .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Beck, M., Doikou, A., Malham, S.J.A., Stylianidis, I. (2018). Grassmannian Flows and Applications to Nonlinear Partial Differential Equations. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_3

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