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.
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