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Patterns and Waves Generated by a Subcritical Instability inSystems with a Conservation Law under the Action of a Global FeedbackControl

Published online by Cambridge University Press:  09 June 2010

Y. Kanevsky  and
A.A. Nepomnyashchy
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Abstract

A global feedback control of a system that exhibits a subcritical monotonic instabilityat a non-zero wavenumber (short-wave, or Turing instability) in the presence of a zeromode is investigated using a Ginzburg-Landau equation coupled to an equation for the zeromode. The method based on a variational principle is applied for the derivation of alow-dimensional evolution model. In the framework of this model the investigation of thesystem’s dynamics and the linear and nonlinear stability analysis are carried out. Theobtained results are compared with the results of direct numerical simulations of theoriginal problem.

Keywords

feedback controlpattern formationvariational approach

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References

Casini, D., D’Alessandro, G., Politi, A.. Soft turbulence in multimode lasers . Phys. Rev. A, 55 (1997), 751–760.CrossRefGoogle Scholar
Chávez Cerda, S., Cavalcanti, S.B., Hickmann, J.M.. A variational approach of nonlinear dissipative pulse propagation . Eur. Phys. J. D, 1 (1998), 313–316.CrossRefGoogle Scholar
Coullet, P., Fauve, S.. Propagative phase dynamics for systems with Galilean invariance . Phys. Rev. Lett., 55 (1985), 2857–2859.CrossRefGoogle ScholarPubMed
Coullet, P., Iooss, G.. Instabilities of one-dimensional cellular patterns . Phys. Rev. Lett., 64 (1990), 866–869.CrossRefGoogle ScholarPubMed
Cox, S.M., Matthews, P.C.. Instability and localisation of patterns due to a conserved quantity . Physica D, 175 (2003), 196–219.CrossRefGoogle Scholar
Golovin, A.A., Davis, S.H., Voorhees, P.W.. Self-organization of quantum dots in epitaxially strained solid films . Phys. Rev. E, 68 (2003), 056203.CrossRefGoogle ScholarPubMed
Golovin, A.A., Kanevsky, Y., Nepomnyashchy, A.A.. Feedback control of subcritical Turing instability with zero mode . Phys. Rev. E, 79 (2009), 046218.CrossRefGoogle ScholarPubMed
Golovin, A.A., Nepomnyashchy, A.A.. Feedback control of subcritical oscillatory instabilities . Phys. Rev. E, 73 (2006), 046212.CrossRefGoogle ScholarPubMed
Golovin, A.A., Nepomnyashchy, A.A., Pismen, L.M.. Interaction between short-scale Marangoni convection and long-scale deformational instability . Phys. Fluids, 6 (1994), 34–47.CrossRefGoogle Scholar
Golovin, A.A., Nepomnyashchy, A.A., Pismen, L.M.. Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid-gas system with deformable interface . J. Fluid Mech., 341 (1997), 317–341.CrossRefGoogle Scholar
Kanevsky, Y., Nepomnyashchy, A.A.. Stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control . Phys. Rev. E, 76 (2007), 066305.CrossRefGoogle ScholarPubMed
Y. Kanevsky, A.A. Nepomnyashchy. Dynamics of solitary waves generated by subcritical instabiity under the action of delayed feedback control. Physica D, (2009), DOI: 10.1016/j.physd.2009.10.007. CrossRef
Komarova, N., Newell, A.C.. Nonlinear dynamics of sandbanks and sandwaves . J. Fluid Mech., 415 (2000), 285–321.CrossRefGoogle Scholar
Malomed, B.A.. Variational methods in nonlinear fiber optics and related fields . Progress in Optics, 43 (2002), 69–191.Google Scholar
Matthews, P.C., Cox, S.M.. One-dimensional pattern formation with Galilean invariance near a stationary bifurcation . Phys. Rev. E, 62 (2000), R1473–R1476.CrossRefGoogle Scholar
Matthews, P.C., Cox, S.M.. Pattern formation with a conservation law . Nonlinearity, 13 (2000), 1293–1320.CrossRefGoogle Scholar
Nepomnyashchy, A.A., Golovin, A.A., Gubareva, V., Panfilov, V.. Global feedback control of a long-wave morphological instability . Physica D, 199 (2004), 61–81.CrossRefGoogle Scholar
Newell, A.C., Whitehead, J.A.. Finite amplitude convection . J. Fluid Mech., 38 (1969), 279–303.CrossRefGoogle Scholar
Rubinstein, B.Y., Nepomnyashchy, A.A., Golovin, A.A.. Stability of localized solutions in a subcritically unstable pattern-forming system under a global delayed control . Phys. Rev. E, 75 (2007), 046213.CrossRefGoogle Scholar
Schöpf, W., Kramer, L.. Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation . Phys. Rev. Lett., 66 (1991), 2316–2319.CrossRefGoogle ScholarPubMed
Sheintuch, M., Nekhamkina, O.. Analysis of front interaction and control in stationary patterns of reaction-diffusion systems . Phys. Rev. E, 63 (2001), 056120.CrossRefGoogle ScholarPubMed
Skarka, V., Aleksić, N.B.. Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations . Phys. Rev. Lett., 96 (2006), 013903.CrossRefGoogle ScholarPubMed
Stanton, L.G., Golovin, A.A.. Global feedback control for pattern-forming systems . Phys. Rev. E, 76 (2007), 036210.CrossRefGoogle ScholarPubMed
Tsoy, E.N., Ankiewicz, A., Akhmediev, N.. Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation . Phys. Rev. E, 73 (2006), 036621.CrossRefGoogle ScholarPubMed