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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Oleynik, Anna¹  Ponosov, Arcady¹  Kostrykin, Vadim²  Sobolev, Alexander V.³ 

[1] Norwegian Univ Life Sci, Dept Math Sci & Technol, Postboks 5003 NMBU, N-1432 As, Norway

[2] Johannes Gutenberg Univ Mainz, FE Inst Math 08, Staudinger Weg 9, D-55099 Mainz, Germany

[3] UCL, Dept Math, Gower St, London WC1E 6ET, England

关键词: Nonlinear integral equations;    Sigmoid type nonlinearities;    Neural field model;    FitzHugh-Nagumo model;    Bumps;   

DOI  :  10.1016/j.jde.2016.08.026

来源: Elsevier

【 摘 要 】

We study the existence of fixed points to a parameterized Hammerstein operator H-beta, beta is an element of (0, infinity], with sigmoid type of nonlinearity. The parameter beta < infinity indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case beta = infinity 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 H-infinity. 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 Hamiltonian 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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