Stability of Fronts in Inhomogeneous Wave Equations

Abstract

This paper presents an introduction to the existence and stability of stationary fronts in wave equations with finite length spatial inhomogeneities. The main focus will be on wave equations with one or two inhomogeneities. It will be shown that the fronts come in families. The front solutions provide a parameterisation of the length of the inhomogeneities in terms of the local energy of the potential in the inhomogeneity. The stability of the fronts is determined by analysing (constrained) critical points of those length functions. Amongst others, it will shown that inhomogeneities can stabilise non-monotonic fronts. Furthermore it is demonstrated that bi-stability can occur in such systems.

Pinned Solutions in a Heterogeneous Three-Component FitzHugh–Nagumo Model

“…Hence, the (15) and (17) are O(ε 2 ) in the slow regions. Consequently, the leading order terms of the action functionals in the slow region I s are actual O(ε).…”

“…We now take a closer look to the individual terms of L (16) inside the action functionals (15) and (17). As discussed above, to construct the profiles of the defect front and pulse solutions we split the spatial domain into slow regions I s -that are away from the defect and away from the interfaces-and fast regions I f -near the defect and interfaces.…”

“…x -terms in L of (15) and (17) are O(ε 4 ) in the slow regions. For the antiderivative F(u), we have…”

“…Before we compute the higher order correction terms of Z f ,ld , we first look at the contribution of the leading order profile to the action functional J f (17). That is, we compute…”

“…We are now in the position to compute the higher order correction term of the action functional J f (17) for a front solution Z f ,ld pinned to the left of the defect. We start by computing the higher order contributions to the action functional coming from the first slow region I 1 s .…”