Front Dynamics with Time Delays in a Bistable System of the Reaction-Diffusion Type: Role of the Symmetry of the Rate Function

• Authors: R. Bakanas , F. Ivanauskas , V. Jasaitis
• Languages of publication: EN

Abstracts

EN

The retardation effects in dynamics of the ac driven "bistable" fronts joining two states of the different stability in a bistable system of the reaction-diffusion type are investigated by use of the macroscopic kinetic equation of the reaction kinetics. We approximate the rate (reaction) function in the governing equation of "bistable" fronts by the piecewise linear dependence of the flexible symmetry, encompassing both cases of the symmetrical and asymmetrical rate functions. By numerically simulating the drift motion of the ac driven front being subjected to the time-dependent step-like (rectangular) forcing we investigate the lag time between the ac force and the instantaneous velocity of the ac driven front. We find that the time lags derivable by the symmetrical and asymmetrical rate functions notably differ, namely, we show that (a) the lag time is a function of the outer slope coefficients of the rate function and is not sensitive to the inner, (b) it has only weak dependence on the strength of the applied forcing, (c) the retardation effects (time lags) in the front dynamics are describable adequately enough by use of the perturbation theory. Another aspect of the front dynamics discussed in this report is the influence of the retardation effects on the ratchet-like transport of the ac driven fronts being described by the asymmetrical rate functions of the "low" symmetry. By considering the response of "bistable" front to the single-harmonic ac force we find that the occurrence of the time lags in the oscillatory motion of the ac driven front shrink the spurious drift of the front; the spurious drift practically disappears if the frequency of the oscillatory force significantly exceeds the characteristic relaxation rate of the system. Furthermore, the occurrence of the time lags in the front dynamics leads to the vanishing of the reversals in the directed net motion of the ac driven fronts, being always inherent in the case of the slow (quasi-stationary) ac drive, i.e., the possibilities of controlling the directed net motion of the self-ordered fronts by the low- and high-frequency zero-mean ac forces radically differ.

Keywords

EN

05.45.-a; 02.30.Jr; 82.40.Ck

Discipline

• 05.45.-a: Nonlinear dynamics and chaos(see also section 45 Classical mechanics of discrete systems; for chaos in fluid dynamics, see 47.52.+j; for chaos in superconductivity, see 74.40.De)
• 02.30.Jr: Partial differential equations
• 82.40.Ck: Pattern formation in reactions with diffusion, flow and heat transfer(see also 47.54.-r Pattern selection; pattern formation and 47.32.C- Vortex dynamics in fluid dynamics)

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2011
• Volume: 119
• Issue: 3
• Pages: 282-297

Dates

published

2011-03

received

2010-01-28

(unknown)

2010-04-28

(unknown)

2010-10-24

Contributors

author

R. Bakanas

• Semiconductor Physics Institute, A. Goštauto 11, 2600 Vilnius, Lithuania

author

F. Ivanauskas

• Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
• Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania

author

V. Jasaitis

• Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

References

• 1. F. Julicher, A. Adjari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997); D. Abbott, Chaos 11, 526 (2001); P. Reimann, Phys. Rep. 361, 57 (2002); R. Mallik, P. Gross, Current Biol. 14, R971 (2004); P. Hanggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)
• 2. A V. Savin, G.P. Tsironis, Y. Zolotariuk, Phys. Rev. E 56, 2457 (1997); N. Quintero, A. Sanchez, Eur. Phys. J. B 6, 133 (1998); G. Costantini, F. Marchesoni, M. Borromeo, Phys. Rev. E 65, 051103 (2002); M. Salerno, Y. Zolotaryuk, Phys. Rev E 65, 056603 (2002); S. Flach, Y. Zolotaryuk, A.E. Morishnichenko, M.V. Fistul, Phys. Rev. Lett. 88, 184101 (2002); L. Morales-Molina, N.R. Quintero, F.G. Mertens, A. Sanchez, Phys. Rev. Lett. 91, 234102 (2003); N.R. Quintero, B. Sanchez-Rey, M. Salerno, Phys. Rev. E 72, 016610 (2005); L. Morales-Molina, N.R. Quintero, A. Sanchez, F.G. Mertens, Chaos 16, 013117 (2006).
• 3. J. Armero, J.M. Sancho, J. Casademunt, A.M. Lacasta, L. Ramirez-Piscina, F. Sagues, Phys. Rev. Lett. 76, 3045 (1996); J. Armero, J. Casademunt, L. Ramirez-Piscina, J.M. Sancho, Phys. Rev. E 58, 5494 (1998); M.A. Santos, J.M. Sancho, Phys. Rev. E 64, 016129 (2001); Ch.R. Doering, C. Muller, P. Smereka, Physica A 325, 243 (2003)
• 4. J.M. Sancho, A. Sanchez, Eur. Phys. J. B 16, 127 (2000); N.R. Quintero, A. Sanchez, F.G. Mertens, Phys. Rev. E 60, 222 (1999)
• 5. A.S. Mikhailov, L. Schimansky-Geier, W. Ebeling, Phys. Lett. A 96, 453 (1983); L. Schimansky-Geier, Ch. Zulicke, Z. Phys. B 82, 157 (1991)
• 6. F. Marchesoni, Phys. Rev. Lett. 77, 2364 (1996); M.G. Clerc, C. Falconi, E. Tirapegui, Phys. Rev. Lett. 94, 148302 (2005)
• 7. F.G. Bass, R. Bakanas, Europhys. Lett. 53, 444 (2001)
• 8. R. Bakanas, Phys. Rev. E 69, 016103 (2004)
• 9. R. Bakanas, Phys. Rev. E 71, 026201 (2005)
• 10. R. Bakanas, Phys. Rev. E 78, 046202 (2008)
• 11. A. Raguotis, F. Ivanauskas, R. Bakanas, Phys. Scr. 74, 629 (2006)
• 12. R. Bakanas, A. Raguotis, F. Ivanauskas, Phys. Scr. 77, 055003 (2008)
• 13. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993); J. Fort, V. Mendez, Rep. Prog. Phys. 65, 985 (2002); Wim van Saarloos, Phys. Rep. 386, 29 (2003); J. Fort, T. Pujol, Rep. Prog. Phys. 71, 086001 (2008)
• 14. G. Abramson, A.R. Bishop, V.M. Kenkre, Phys. Rev. E 64, 066615 (2001); E.P. Zemskov, K. Kassner, S.C. Muller, Eur. Phys. J. B 34, 285 (2003); E.P. Zemskov, K. Kassner, Eur. J. Phys. 25, 361 (2004); E.P. Zemskov, Phys. Rev. E 69, 036208 (2004); V. Mendez, V. Ortega-Cejas, E.P. Zemskov, J. Casas-Vazquez, Physica A 375, 50 (2007)
• 15. R. Bakanas, Nonlinearity 16, 313 (2003)
• 16. A. Kurganov, P. Rosenau, Nonlinearity 19, 171 (2006)
• 17. F. Schlogl, C. Escher, R.S. Berry, Phys. Rev. A 27, 2698 (1983); F.G. Bass, R. Bakanas, Phys. Lett. A 214, 301 (1996); F.G. Bass, R. Bakanas, Waves Random Media 10, 217 (2000)
• 18. R. Jackiw, Rev. Mod. Phys. 49, 681 (1977); A.R. Bishop, J.A. Krumhansl, S.E. Trullinger, Physica D 1, 1 (1980)
• 19. V. Jasaitis, F. Ivanauskas, R. Bakanas, Nonlinear Anal. Model. Control 13, 433 (2008)

Identifiers

DOI

10.12693/APhysPolA.119.282