Abstract
We show how solutions to a large class of partial differential equations with nonlocal Riccatitype 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 

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 EbrahimiFard, Hans Zanna MuntheKaas 
Publisher  Springer 
Pages  7198 
Number of pages  28 
ISBN (Electronic)  9783030015930 
ISBN (Print)  9783030015923 
DOIs  
Publication status  Published  14 Jan 2019 
Publication series
Name  Abel Symposia 

Publisher  Springer 
Volume  13 
ISSN (Print)  21932808 
ISSN (Electronic)  21978549 
ASJC Scopus subject areas
 Mathematics(all)
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Profiles

Simon John A. Malham
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)