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.
|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|
|Number of pages||28|
|Publication status||Published - 14 Jan 2019|
ASJC Scopus subject areas
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