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On camel-like traveling wave solutions in cellular neural networks. (English) Zbl 1057.34068
The authors are interested in the existence of traveling wave solutions of cellular neutral networks distributed in the one-dimensional integer lattice $\bbfZ^1$. The dynamics of each given cell depends on itself and its nearest $m$ left neighbor cells with instantaneous feedback. It is described by the functional differential equation of delay-type $$-c\phi'(s)= -\phi(s)+ \sum^n_{j=0} \alpha_j f(\phi(s- j)),\tag1$$ where $f$ (called the output function) is defined by $$f(x)= \textstyle{{1\over 2}}(\vert x+ 1\vert-\vert x-1\vert),\qquad x\in\bbfR.$$ Let $\phi$ be a solution (called a traveling wave solution) of (1). We say that $\phi$ is a solution of $n$-type if it has exactly $n$ critical points with strictly local extreme values. Nonmonotone solutions are called camel-like traveling wave solutions. If $\sum^m_{j=0} \alpha_j> 1$, then the constant functions $\phi_0(s)= 0$ and $\phi_1(s)= \sum^m_{j=0} \alpha_j$, $s\in\bbfR$, are solutions of (1). Assume that $m$ is even, $\alpha_0\ge 1$ and $\sum^m_{j=0} \alpha_j> 1$. In the paper, it is proved for example that (i) if $\alpha_j> 0$ ,$1\le j\le m$, then, for every $c< 0$, there exists a monotone increasing solution $\phi$ of (1) such that $$\lim_{s\to-\infty}\, \phi(s)= 0,\qquad \lim_{s\to\infty}\, \phi(s)= \sum^m_{j=0} \alpha_j;\tag2$$ (ii) if the signs of $\{\alpha_j\}^m_{j=1}$ are alternating with $\vert\alpha_j\vert\ge \vert\alpha_{j+ 1}\vert$ for $1\le j\le m-1$, then there exists $c_*< 0$ with $\vert c_*\vert$ sufficiently large such that (1) possesses for $c< c_*$ a monotone increasing solution satisfying (2) and if, in addition, $\alpha_1> 0$ $(1- \alpha_0\le\alpha_1< 0)$ then there exists $\widetilde c< 0$ such that for $\widetilde c< c< 0$, (1) has a solution of $(m-1)$-type ($m$-type) $\phi$ satisfying (2). Solutions of (1) with $\alpha_j< 0$, $1\le j\le m$, and $\alpha_j\ge \alpha_{j+1}$, $1\le j\le m-1$, or $\alpha_j=(- 1)^j\alpha$, $1\le j\le m$, and $\alpha\ne 0$ are discussed, too. Also more general output functions are considered and some numerical results are given.

34K10Boundary value problems for functional-differential equations
37L60Lattice dynamics (infinite-dimensional dissipative systems)
34B45Boundary value problems for ODE on graphs and networks
Full Text: DOI
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