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

Abstract

We consider inhomogeneous non-linear wave equations of the type u =u +V (u, x)-αu (α≥0). The spatial real axis is divided in intervals I , i=0,..., N+1 and on each individual interval the potential is homogeneous, i.e., V(u, x)=V (u) for x∈I . 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 V . 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. © 2012 Elsevier Inc

Stability of Fronts in Inhomogeneous Wave Equations

“…More examples with N = 1 and N = 2 can be found in [6,22]. This example is similar to the example in [23] and can be related to long Josephson junctions with defects.…”

“…For more details, see [22]. The condition that (u 2 , p 2 ) is not a fixed point of the V mi -dynamics, i = 1, 2, can be rephrased as the condition that the derivative of the front u x (x; g, h) does not have a non-simple zero in the middle intervals.…”

“…In this paper, we will present an introductory overview of the systematic analysis of the existence and stability of stationary waves in inhomogeneous wave equations with finite length spatial inhomogeneities. The overview is based mainly on the work in [22,23] and will highlight the main steps and key ideas in the analysis. It aims to put the reader in a good position to use the ideas and techniques in various applications.…”

“…This section is based on [22], in which full details can be found. Multiplying (5) through by u x , writing p = u x , and integrating with respect to x gives the following Hamiltonian description for the stationary fronts (recall that R = I ℓ ∪ I m1 ∪ I m2 ∪ I r ):…”

“…In the lemmas below, the continuation of the matching points as function of h 1 and/or h 2 is discussed. More background and proofs of those lemmas can be found in [22]. To avoid too many technical details, we will assume that the following non-degeneracy condition holds on the potential functions.…”