Large dispersion, averaging and attractors: three 1D paradigms

Abstract

The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.2000 Mathematics Subject Classification. 35B40, 35B45.

Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation

“…By this reason, it looks natural to utilize the rapid in time oscillations caused by this dispersive term. To this end, following, say, [10] (see also references therein), we do the change of variables Zn(t) = e iωnt zn(t).…”

“…Thus, equation (222) preserves all good properties of equation 2.21, so it is sufficient to prove the unique solvability of (2.22) in the space L 2 (R − , H I ). This equation has an essential advantage since it contains an explicit rapidly oscillating term with zero mean, so by the classical averaging theory (see [10] and references therein), we expect that this term will be averaged to zero and the solvability of (2.22) for a general β(t) should follow from the particular case β = 0 (where it is obvious). Let us justify this idea.…”

“…By this reason, it looks natural to use the rapid in time oscillations caused by this dispersive term. To this end, following, say, [27] (see also references therein), we do the change of variables Znfalse(tfalse)=eiωntznfalse(tfalse). Then, we get ∂tZn+false(λn−θfalse)Zn+e2iωntβfalse(tfalse)Z¯n=eiωntitalicΦnfalse(falsefalse{e−iωktZkfalsefalse}false)+eiωntgn:=italicΦnεfalse(Zfalse)+Gn. Since our transform is an isometry in H I , we have right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278emfalse|false|Zfalse||L2(double-struckR−,HI…”

“…The proposed method is a combination of the spatial averaging principle of Sell and Mallet-Paret with the classical temporal averaging for equations with large dispersion (in the spirit of [10]). Roughly speaking, at the first step we use spatial averaging in order to get rid of the dependence on the spatial variable x, in the equation of variations, and replace f (•)v by f v. However, the matrix f is not a scalar matrix, so this is not enough to get the result.…”

Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions

“…a large amount of dispersion) can regularize chaotic dynamics for any system length L , as solutions are observed to be attracted to nonlinear travelling waves. Surprisingly, third-order dispersion acts as a destabilizing mechanism for this equation, competing with the stabilizing nonlinear term—it hinders the transfer of energy from low to high wavenumbers, and consequently analyticity of solutions reduces as δ is increased (see [ 61 ] for a discussion of this for the 1D case). Turning to the 2D problem, Toh et al [ 41 ] and Indireshkumar & Frenkel [ 42 ] observed pulse solutions of ( 1.3 ) for large values of dispersion on large periodic domains—the usual streamwise cellular structures are found to be unstable and give way to the 2D pulses.…”