1 Introduction Magnetic domain walls (DWs) have attracted a lot of attention as important elements of new magnetoelectronic devices [1 -3]. In particular, it was demonstrated recently that they can be used in a new type of memory device (the racetrack memory) effectively controlled by an electric current [4,5]. On the other hand, the DW can be viewed as a kind of local imperfection in an ordered magnetic system, like an impurity or defect. The substantial difference is that DWs can move and therefore they can be relatively easily put into motion by an external field or electric current, and also by an interaction between them.A great amount of theoretical and experimental work was dedicated to studying the resistance of DWs, the current-induced spin torque, the dynamics of DW motion, and other effects related to a single DW strongly coupled to the electron system [3,6,7]. When the density of DWs increases, it is important to include into these considerations the effects of their interaction. It has already been demonstrated [8 -11] that an electric current in a magnetic wire with DWs influences the DW interaction, so that by using a current the DW coupling can be controlled, and the dynamics of strongly coupled DWs can be affected.In this work we consider in detail the indirect exchange coupling between the DWs in equilibrium. Essentially, the analysis of such interaction is the first necessary step to understand the basic mechanisms of DW interactions.