On camel-like traveling wave solutions in cellular neural networks. (Q1428264)

The authors are interested in the existence of traveling wave solutions of cellular neutral networks distributed in the one-dimensional integer lattice \(\mathbb{Z}^1\). 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 \[ -c\phi'(s)= -\phi(s)+ \sum^n_{j=0} \alpha_j f(\phi(s- j)),\tag{1} \] where \(f\) (called the output function) is defined by \[ f(x)= \textstyle{{1\over 2}}(| x+ 1|-| x-1|),\qquad x\in\mathbb{R}. \] 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 \(\phi_1(s)= \sum^m_{j=0} \alpha_j\), \(s\in\mathbb{R}\), are solutions of (1). Assume that \(m\) is even, \(\alpha_0\geq 1\) and \(\sum^m_{j=0} \alpha_j> 1\). In the paper, it is proved for example that (i) if \(\alpha_j> 0\) ,\(1\leq j\leq m\), then, for every \(c< 0\), there exists a monotone increasing solution \(\phi\) of (1) such that \[ \lim_{s\to-\infty}\, \phi(s)= 0,\qquad \lim_{s\to\infty}\, \phi(s)= \sum^m_{j=0} \alpha_j;\tag{2} \] (ii) if the signs of \(\{\alpha_j\}^m_{j=1}\) are alternating with \(|\alpha_j|\geq |\alpha_{j+ 1}|\) for \(1\leq j\leq m-1\), then there exists \(c_*< 0\) with \(| c_*|\) sufficiently large such that (1) possesses for \(c< c_*\) a monotone increasing solution satisfying (2) and if, in addition, \(\alpha_1> 0\) \((1- \alpha_0\leq\alpha_1< 0)\) 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 \(\alpha_j< 0\), \(1\leq j\leq m\), and \(\alpha_j\geq \alpha_{j+1}\), \(1\leq j\leq m-1\), or \(\alpha_j=(- 1)^j\alpha\), \(1\leq j\leq m\), and \(\alpha\neq 0\) are discussed, too. Also more general output functions are considered and some numerical results are given.