Viscous Shocks in the Destabilized Kuramoto-Sivashinsky Equation

[+] Author and Article Information
Jens D. M. Rademacher1

 Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany

Ralf W. Wittenberg

Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canadaralf@sfu.ca


Current address: C.W.I. Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.

J. Comput. Nonlinear Dynam 1(4), 336-347 (Mar 29, 2006) (12 pages) doi:10.1115/1.2338656 History: Received December 11, 2005; Revised March 29, 2006

We study stationary periodic solutions of the Kuramoto-Sivashinsky (KS) model for complex spatio-temporal dynamics in the presence of an additional linear destabilizing term. In particular, we show the phase space origins of the previously observed stationary “viscous shocks” and related solutions. These arise in a reversible four-dimensional dynamical system as perturbed heteroclinic connections whose tails are joined through a reinjection mechanism due to the linear term. We present numerical evidence that the transition to the KS limit contains a rich bifurcation structure even within the class of stationary reversible solutions.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

(a) An odd bubble solution for the KS equation (α=0) with period L=128; (b) the (u,ux) projection of another bubble solution with more oscillations

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Figure 2

Odd viscous shock of period L=60 for α=0.4

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Figure 3

(a) Schematic depiction of direction of transport of localized perturbations along the tail of a viscous shock. (b) Time evolution of the dKS equation for L=60, α=0.5; initial condition is a viscous shock with a small Gaussian perturbation near x=4. Note the accelerated transport near the interface.

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Figure 4

Phase portraits for stationary solutions of the Burgers-Sivashinsky system 9 for (a)α=−1, (b)α=0, and (c)α=1

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Figure 5

Profile of a “viscous shock” of the Burgers-Sivashinsky Eq. 8 with α=1

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Figure 6

Sketch of the phase space projected into the (v,v1) plane for δ>0; note that the v1 axis lies in SR. The dashed line is the heteroclinic Hc for δ=0.

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Figure 7

Phase space projections of the viscous shock from Fig. 2 onto the planes (a)(u,u1) and (b)(u,u2)

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Figure 8

(a) Intersections of viscous shocks with SR for fixed α≈0.3435 parametrized by L∊(40,62). (b) Projections to the (u,uxx) plane of trajectories of Eq. 5 with α=0.4 and initial conditions u=uxx=uxxx=0, and A: ux=0.3999999991 B: ux=0.39999999915 C: ux=0.3999999992.

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Figure 9

(a) Paths of reversible periodic orbits of period L=60, continuing in α; labeled solutions are shown in Figs.  101112. (b) A section through several sheets of shocks for L=60. The upper branch is one of viscous shocks, while the shocks on branches with lower amplitude have a longer flat region.

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Figure 10

Viscous shocks: Labeled solutions (a) A and (b) B from Fig. 9

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Figure 11

Flat and roll shocks: (a) A flat shock: solution C from Fig. 9, remaining near ℓα for a while; (b) a roll shock: solution D from Fig. 9

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Figure 12

Roll (cellular) solutions: (a) Solution E from Fig. 9; (b) a family of rolls for α between 2 and 7

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Figure 13

Folds on different sheets of solutions in the (α,L) plane. The horizontal dotted line marks the slice shown in Fig. 1, and the box the range of Fig. 1. The sufficient existence criterion from Sec. 4 is sketched by the thick dotted line, but viscous shocks continue until the relevant fold curves to the left and below this curve.

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Figure 14

Creation of a rip in the surface of viscous shocks at a fold point of folds: two sheets of solutions touch, and a pair of folds is created. (a)L=62.20: the upper branch A-D is a branch of viscous shocks, terminating in a fold to the left of D with smaller α; (b)L=62.22: two new folds created after the sheets of solutions touch and tear apart; the fold of viscous shocks is now the rightmost one connecting A and B; (c)L=62.6: cross section near the cusp point, at which two folds of flat shocks annihilate each other.

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Figure 15

Part of the essential spectrum of a viscous shock with (α,L)=(0.15,60) (only the spectrum in the upper half plane is needed, due to symmetry under complex conjugation). Bullets are used to enlarge the four unstable, extremely small closed isolated curves, each of which contains an eigenvalue of Lper(u).

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Figure 16

(a) The onset of Hopf bifurcation in the (α,L) plane; only the thick line is relevant for viscous shocks. (b) The relevant part of (a) plotted together with fold curves from Fig. 1. The bullet denotes the location of the viscous shock used in Fig. 1.




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