Structure of a class of traveling waves in delayed cellular neural networks

This work investigates the structure of a class of traveling wave solutions of delayed cellular neural networks distributed in the one-dimensional integer lattice $\mathbb Z^1$. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with its two left neighbors in which one is instantaneous and the other is distributively delayed due to, for example, finite switching speed and finite velocity of signal transmission. Applying the method of step with the aid of positive roots of the corresponding characteristic function of the profile equation, we can directly figure out the solution in explicit form. We then partition the parameter space $(\alpha, \beta)$-plane into four regions such that the qualitative properties of traveling waves can be completely determined for each region. In addition to the existence of monotonic traveling wave solutions, we also find that, for certain parameters, there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points.