rsta20170195_si_005.m (2.45 kB)

Matlab file for Riccati generalized NLS solution from Partial differential systems with non-local nonlinearities: generation and solutions

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posted on 29.01.2018 by Margaret Beck, Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis
We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local 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 non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local 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. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.