Stability of stationary fronts in a non-linear wave equation with spatial inhomogeneity (Q695070)

Let the spatial real axis be divided in the intervals \(I_0,I_1,\dots ,I_{N+1}\) such that \(\mathbb{R}=\overline{\cup I_i}\). The authors study stability and changes of stability of stationary fronts or solitary waves of the equation \[ u_{tt}=u_{xx}+\frac{\partial}{\partial u}V(u,x)-\alpha u_t, \] where \(\alpha \geq 0\), \(V(u,x)=V_i(u)\) for \(x\in I_i\), \(i=0,1,\dots ,N+1\). The first part of the paper gives an overview of the results for existence of families of stationary fronts or solitary waves, especially the relation between the lengths of the intervals \(I_i\) and the Hamiltonian parameters \(h_i\). Then the relation between potential changes of linear stability (i.e., the existence of eigenvalue zero of the linearisation operator) and critical points of the interval functions parametrised by the Hamiltonian parameters. The results are illustrated with the examples of a long Josephson junction and \(0-\pi \) Josephson junction with a microresistor defect.