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| Abstract |
Using the projection operator formalism we explore the decay form of the time correlation function U_n(t)≡⟨û_n(t)û^∗_n(0)⟩ of the state variable û_n(t) in the chaotic Kuramoto-Sivashinsky equation. Th...e decay form turns out to be the algebraic decay 1/[1+(γ_{na}t)^2] in the initial regime t<1/γ_{ne} and the exponential decay exp(-γ_{ne}t) in the final regime t>1/γ_{ne} . The memory function Γ_n(t) that represents the chaos-induced transport is found to obey the Gaussian decay exp[-(β_{ng}t)^2] in the case of large wave numbers, but the 3/2 power decay exp[-(β_{n3}t)^{3/2}] in the case of small wave numbers. The power spectrum of û_n(t) is given by the real part U'_n(ω) of the Fourier-Laplace transform of U_n(t) and has a dominant peak at ω=0 . This peak within the linewidth γ¯_{ne}(≈γ_{ne}) is given by the Lorentzian spectrum γ¯^2_{ne}/(ω^2+γ¯^2_{ne}) . However, the wings of the peak outside the width γ¯_{ne} turn out to take the exponential spectrum exp(-ω/γ_{na}) . Thus it is found that the exponential decay exp(-γ_{ne}t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1/[1+(γ_{na}t)^2] arises to bring about the exponential wing.show more
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PhysRevE.76.061104