Using the projection operator formalism we explore the decay form of the time correlation function Un (t) un (t) un (0) of the state variable u n (t) in the chaotic Kuramoto-Sivashinsky equation. The decay form turns out to be the algebraic decay 1 [1+ (na t) 2] in the initial regime t ne and the exponential decay exp (a ne t) in the final regime t>1â• I ne. The memory function Î"n (t) that represents the chaos-induced transport is found to obey the Gaussian decay exp [a (ng t) 2] in the case of large wave numbers, but the 3/2 power decay exp [a (I n3 t) 3a 2] in the case of small wave numbers. The power spectrum of u n (t) is given by the real part Una (Ï‰) of the Fourier-Laplace transform of Un (t) and has a dominant peak at Ï‰=0. This peak within the linewidth ne (ane) is given by the Lorentzian spectrum ne 2 a (Ï‰2 + ne 2). However, the wings of the peak outside the width ne turn out to take the exponential spectrum exp (na). Thus it is found that the exponential decay exp (ne t) appears to lead to the universal Lorentzian peak, while the algebraic decay 1 [1+ (na t) 2] arises to bring about the exponential wing.
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|Publication status||Published - Dec 5 2007|
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics