Applications of Grassmannian and graph flows to nonlinear systems
Abstract
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear partial differential system together with a linear Fredholm integral equation. The flows of such nonlinear systems are examples of linear flows on Fredholm Stiefel manifolds that can be projected onto Fredholm Grassmann manifolds or of further projections onto natural subspaces thereof. Detailed expositions of such flows are provided for the Korteweg de Vries and nonlinear Schrodinger equations, as well as Smoluchowski's coagulation and related equations in the constant frequency kernel case. We then consider Smoluchowski's equation in the additive and multiplicative frequency kernel cases which correspond to the inviscid Burgers equation. The solution flow in these cases prompts us to introduce a new class of flows we call ``graph flows''. These generalise flows on a Grassmann manifold from sets of graphs of linear maps to sets of graphs of nonlinear maps. We include a detailed discussion of directions in which these flows can be generalised to include many other partial differential systems with local and nonlocal nonlinearities.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.05035
 Bibcode:
 2019arXiv190505035D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Quantum Physics
 EPrint:
 44 pages, 3 figures