Absolute and convective instability for evolution PDEs on the half-line

Document Type

Article

Publication Date

1-1-2005

Abstract

We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let ω(k) be the associated symbol, i.e., let exp[ikx ω(k)t] be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition q0(x), where q0(x) decays as [x] → ∞. By making use of a certain transformation in the complex k-plane, which leaves ω(k) invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half-line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial-boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second-order equation is a linearized Ginzburg-Landau equation arising in Rayleigh-Bénard convection and in the stability of plane Poiseuille flow, while the fourth-order equation is a linearized Kuramoto-Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.

Identifier

12344254215 (Scopus)

Publication Title

Studies in Applied Mathematics

External Full Text Location

https://doi.org/10.1111/j.0022-2526.2005.01541.x

ISSN

00222526

First Page

95

Last Page

114

Issue

1

Volume

114

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