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Linear Stability Theory for Fronts with Algebraically Decaying Tails

Leonid Brevdo
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 460, No. 2050 (Oct. 8, 2004), pp. 3013-3035
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/4143086
Page Count: 23
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Linear Stability Theory for Fronts with Algebraically Decaying Tails
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Abstract

Based on our recently developed theory of absolute and convective instabilities of spatially varying unbounded and semi-bounded flows and media with algebraically decaying tails, we develop in this paper a linear stability theory for continuous fronts whose tails have similar decay asymptotics at infinity. It is assumed that the base state of the front, W(x), tends to constant states, $W^R and W^L \noy = W^R, when x rightarrow \propto and x \rightarrow - \propto, respectively, and the tails, R^R(x) and R^L(x)$, in the governing equation linearized about W(x) decay as $\midx\mid^{-\alpha}$, when $x rightarrow \pm\propto, respectively$, where α is sufficiently large. No restrictions on the rate of variability of the tails in the finite domain are imposed. The Laplace-transformed problem, $Z_x(x,w) = A(x,w)Z(x,w) + G(x,w), x \in R$, governing the perturbation dynamics of the front is treated by using the decompositions of the fundamental matrix of the system obtained by us previously, $\Phi(x,w) = B^R(x,w) e^A^R(w)x[B^R(0,w)}^{-1}$, with the asymptotics $B^R(x,w) = I + O(x^\in), \in > 0, when x rightarrow \propto, and \Phi(x,w) = B^L(x,w)e^A^L_(w)[B^L(0,w)]^{-1}$, with the asymptotics $B^L(x,w) = I + O(\midx\mid^{-\in}), when x rightarrow-\propto$, where I is the identity matrix. Here, Z(x,w) denotes the Laplace-transformed perturbation, x ∈ R is the spatial coordinate, $\omega \in C$ is a frequency (and a Laplace transform parameter), G(x, w) is the source function, $A^R(\omega) = lim_x rightarrow \propto A(x,\omega) and A^L(\omega) = lim_x rightarrow A(x,\omega) \not= A^R(\omega)$. The principal part of the analysis is the formulation of conditions equivalent to the boundary conditions of decay for $Z(x,\omega), when x rightarrow \pm \propto$, derived by applying a result due to Kato (1980 Perturbation theory for linear operators). The boundary-value problem for Z(x,ω) is solved formally. Its solution has the form similar to that obtained in Brevdo (2003 Proc. R. Soc. Lond. A 459, 1403-1425). The absolute and convective instabilities of, and signalling in, the front are studied by applying to the solution the treatments in the above papers, whereas new elements present due to the different limits in $\pm \propto$ of the matrix $A(x,\omega)$ are taken account of. We express the stability results in terms of the dispersion relation functions, Dn(ω), for the global normal modes, for the corresponding regions, $R_n \subset C, n \geq 1$, the dispersion relation functions of the associated uniform states, $D_0^R(k,\omega) = det[ikI - A^R(\omega)] and D_0^L(k,\omega) = det[ikI - A^L(\omega)]$, and the singularities of the matrices $B^R(x,\omega) and B^L(x,\omega)$, and of the projectors $P^R+(\omega) and P^L-(\omega)$ related to the operators $A^R(\omega) and A^L(\omega)$, respectively. Since all of the above objects controlling the instabilities are essentially global properties of the front, it is argued that the concept of local stability cannot be consistently defined for the fronts treated. A procedure for computing the instabilities is outlined.

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