Homoclinic solutions to an integral equation: Existence and stability (Q1293258)

The authors study (stationary) evolution equation in the form of the following nonlinear convolution type integral equation \[ (J\ast u)(x)- u(x)-f[u(x)]=0,\tag{1} \] with the condition \(u(\pm \infty)=0.\) It is assumed, that \(J(z)\geq 0\), \(\int J(z) dz =1\) and \(f\) is bistable. They construct homoclinic (even) solutions to the equation (1).