Stable stationary solutions in reaction-diffusion systems consisting of a 1-D array of bistable cells (Q2754833)

This paper deals with the reaction diffusion lattice of identical cells NEWLINE\[NEWLINE\dot v(n)=d(v(n+1)+v(n-1)-2v(n))-f(v(n)),\tag{1}NEWLINE\]NEWLINE where \(d\) is the cell coupling constant and \(f(v)\) is the cubic function \(v(v^2-1)\) or the cubic-like piece-wise linear function: \(f=v+1\) for \(v<0\), \(f(0)=0\), \(f=v-1\) for \(v>0\).NEWLINENEWLINENEWLINELet \(v(n)\) be a stable stationary solution to (1). The function \(\hat v:\mathbb Z \mapsto \mathbb R\) defined by \(\hat v(n)=\text{sign} v(n)\) is called the skeleton of \(v(n)\). For \(d<0.06\) in the case of cubic nonlinearity it is shown that any function \(\hat v:\mathbb Z \mapsto \{-1,1\}\) is the skeleton of a stable stationary solution. NEWLINENEWLINENEWLINEBy using the fact that the function \(v(x)=\tanh (x-x_0)\) is a kink for the equation \(v''+2v(1-v^2)=0\), an approximate analytical expression for kink-type stationary solutions are obtained.NEWLINENEWLINENEWLINEIn the case of piece-wise cubic-like nonlinearity it is shown that each stable stationary solution \(v(n)\) of (1) is uniquely determined by its skeleton for arbitrary coupling constant. Sufficient condition for a given function \(\hat v:\mathbb Z \mapsto \{-1,1\}\) to be the skeleton of exactly one solution \(v(n)\) is obtained. This proves that the spatial chaos in system (1) exists for arbitrary \(d>0\).NEWLINENEWLINENEWLINENext, the value of the spatial entropy is found. Lastly, the effects of self-organization and pattern formation are analyzed.