Absolute and Convective Instability for Evolution PDEs on the Half‐Line

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.