Stability of stationary fronts in a non-linear wave equation with spatial inhomogeneity
Knight, CJK, Derks, G, Doelman, A and Susanto, H (2013) Stability of stationary fronts in a non-linear wave equation with spatial inhomogeneity Journal of Differential Equations, 254 (2). pp. 408-468.
Available under License : See the attached licence file.
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.
|Divisions :||Faculty of Engineering and Physical Sciences > Mathematics|
|Date :||15 January 2013|
|Identification Number :||https://doi.org/10.1016/j.jde.2012.08.007|
|Additional Information :||NOTICE: this is the author’s version of a work that was accepted for publication in the Journal of Differential Equations. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Differential Equations, 254(2), January 2013, DOI 10.1016/j.jde.2012.08.007.|
|Depositing User :||Symplectic Elements|
|Date Deposited :||14 Aug 2013 08:42|
|Last Modified :||23 Sep 2013 20:13|
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