A WKB-derived reduced wave-number equation captures unstable states, self-similar singularities, and traveling shocks in the real and complex Ginzburg-Landau equations, with exact profiles and DNS agreement in the nearly-real limit.
Proof: Convergencek→k ∞ asξ→ ±∞requires thatf(k ∞) =f ′(k∞) = 0 and, by assumption, f ′′(k∞)>0
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Reduced wave number dynamics in the real and complex Ginzburg-Landau equations
A WKB-derived reduced wave-number equation captures unstable states, self-similar singularities, and traveling shocks in the real and complex Ginzburg-Landau equations, with exact profiles and DNS agreement in the nearly-real limit.