Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity

TL;DR

This paper investigates the existence of spatially localized solutions to a parameterized Hammerstein equation with sigmoid nonlinearity, showing how solutions for large steepness parameters can be approximated by a discontinuous limit case, with applications in biological models.

Contribution

The paper proves the existence of localized solutions for large sigmoid steepness and introduces an iterative method for better approximation, applicable to neural and reaction-diffusion models.

Findings

Localized solutions exist for large sigmoid steepness.

Solutions for finite steepness can be approximated by the discontinuous limit case.

The iterative method improves approximation and can construct homoclinic orbits.

Abstract

We study the existence of fixed points to a parameterized Hammertstain operator $\cH_\beta,$ $\beta\in (0,\infty],$ with sigmoid type of nonlinearity. The parameter $\beta<\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $\beta=\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large $\beta$ exist and can be approximated by the fixed points of $\cH_\infty.$ These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltionian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh-Nagumo reaction-diffusion equation and a neural field model.