Stability of stationary fronts in a non-linear wave equation with spatial inhomogeneity

We consider inhomogeneous non-linear wave equations of the type utt=uxx+V′(u,x)−αut (α⩾0). The spatial real axis is divided in intervals Ii, i=0,…,N+1 and on each individual interval the potential is homogeneous, i.e., V(u,x)=Vi(u) for x∈Ii. By varying the lengths of the middle intervals, typically one can obtain large families of stationary front or solitary wave solutions. In these families, the lengths are functions of the energies associated with the potentials Vi. In this paper we show that the existence of an eigenvalue zero of the linearisation operator about such a front or stationary wave is related to zeroes of the determinant of a Jacobian associated to the length functions. Furthermore, the methods by which the result is obtained is fully constructive and can subsequently be used to deduce the stability and instability of stationary fronts or solitary waves, as will be illustrated in examples.