TY - JOUR
T1 - Traveling Waves for Spatially Discrete Systems of FitzHugh--Nagumo Type with Periodic Coefficients
AU - Schouten-Straatman, W.M.
AU - Hupkes, Hermen Jan
PY - 2019
Y1 - 2019
N2 - We establish the existence and nonlinear stability of traveling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh--Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable traveling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multicomponent functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen, and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies.
AB - We establish the existence and nonlinear stability of traveling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh--Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable traveling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multicomponent functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen, and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies.
UR - http://www.scopus.com/inward/record.url?scp=85075496464&partnerID=8YFLogxK
U2 - 10.1137/18M1204942
DO - 10.1137/18M1204942
M3 - Article
SN - 0036-1410
VL - 51
SP - 3492
EP - 3532
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 4
ER -