Partial differential systems with nonlocal nonlinearities: Generation and solutions

TL;DR

This paper introduces a novel method for generating solutions to a broad class of evolutionary PDE systems with nonlocal nonlinearities, using a Fredholm integral equation approach inspired by integrable systems, demonstrated through examples and numerical simulations.

Contribution

The paper presents a new solution-generating technique for nonlocal PDE systems based on a Fredholm integral equation, extending methods from integrable systems.

Findings

Method successfully generates solutions for reaction-diffusion systems with nonlocal quadratic nonlinearities.

Approach is applicable to the nonlinear Schrödinger equation with nonlocal cubic nonlinearity.

Numerical simulations confirm the effectiveness of the proposed method.

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.