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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Knight, Christopher J. K.¹  Derks, Gianne¹  Doelman, Arjen²  Susanto, Hadi³ 

[1] Univ Surrey, Dept Math, Guildford GU2 7XH, Surrey, England

[2] Leiden Univ, Math Inst, NL-2300 RA Leiden, Netherlands

[3] Univ Nottingham, Sch Math Sci, Nottingham NG7 2RD, England

关键词: Nonlinear wave equations;    Inhomogeneities;    Fronts;    Stability;   

DOI  :  10.1016/j.jde.2012.08.007

来源: Elsevier

【 摘 要 】

We consider inhomogeneous non-linear wave equations of the type u(tt) = u(xx) + V'(u, x) - alpha u(t) (alpha >= 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) = Vi(u) for x is an element of 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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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