Abstract
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction-diffusion systems with nonlocal quadratic nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic nonlinearity. In each case we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended.
Original language | English |
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Article number | 195 |
Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |
Volume | 376 |
Issue number | 2117 |
Early online date | 5 Mar 2018 |
DOIs | |
Publication status | Published - 13 Apr 2018 |
Keywords
- math.AP
- math-ph
- math.MP
- nlin.SI
- quant-ph
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Dive into the research topics of 'Partial differential systems with nonlocal nonlinearities: Generation and solutions'. Together they form a unique fingerprint.Profiles
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Simon John A. Malham
- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Mathematics - Associate Professor
Person: Academic (Research & Teaching)