Absolute and Convective Instability for Evolution PDEs on the Half-Line
We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let (k) be the associated symbol, i.e., let exp[ikx−(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 (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.
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Document Type: Research Article
Affiliations: 1: University of Cambridge 2: New Jersey Institute of Technology
Publication date: January 1, 2005