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

Knight, CJK and Derks, G and 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). 408 - 468. ISSN 0022-0396

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Abstract

We consider inhomogeneous non-linear wave equations of the type u tt=u xx+V '(u, x)-αu t (α≥0). The spatial real axis is divided in intervals I i, i=0,..., N+1 and on each individual interval the potential is homogeneous, i.e., V(u, x)=V i(u) for x∈I 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 i. 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.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 12 Nov 2014 19:51
Last Modified: 30 Jan 2019 16:17
URI: http://repository.essex.ac.uk/id/eprint/11557

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