In this paper we study Cellular Nonlinear Network (CNN) computing of double Sine-Gordon equation (SG). SG arises in studying fluxons in Josephson junctions and it has fluxon and antifluxon solutions. We study double SG equation with physical applications. Analysis of the solutions of these types of SG is presented by means of classical physical theory. We construct CNN architecture for the two types of equations. Simulations of the interactions of fluxons, kinks and breathons are provided.