Partial differential systems with nonlocal nonlinearities: Generation and solutions

Margaret Beck, Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis

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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 languageEnglish
Article number195
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume376
Issue number2117
Early online date5 Mar 2018
DOIs
Publication statusPublished - 13 Apr 2018

Keywords

  • math.AP
  • math-ph
  • math.MP
  • nlin.SI
  • quant-ph

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