Amplitude equations for activator-inhibitor system with superdiffusion

Prytula Z.
AttachmentSize
PDF icon 2016_3_2_191_198.pdf175.89 KB
Abstract: 
The generalized activator-inhibitor model with cubic nonlinearity, in which the classical Laplacian is replaced by fractional operator has been studied. The fractional operator reflects the nonlocal behavior of superdiffusion. A spatially homogeneous, time independent solution has been found and its linear stability was studied. We have also performed a weakly nonlinear analysis and obtained a system of amplitude equations that are the basis for analysing pattern formation as well as parameter regimes for which various steady-state patterns would exist.
References: 
  1. Henry B. I., Wearne S. L. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62, n. 3, 870–887 (2002).
  2. Datsko B., Luchko Y., Gafiychuk V. Pattern formation in fractional reaction-diffusion systems with multiple homogeneous states. Int. J. Bifurcation Chaos. 22, 1250087 (2012).
  3. Datsko B., Gafiychuk V., Podlubny I. Solitary travelling auto-waves in fractional reaction–diffusion systems. Communications in Nonlinear Science and Numerical Simulation. 23 (1), 378–387 (2015).
  4. Nec Y., Ward M. J. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. Math. Model. Nat. Phenom. 8 (5), 206–232 (2013).
  5. Fomin S., Chugunov V., Hashida T. Mathematical modeling of anomalous diffusion in porous media. Fract. Different. Calc. 1, n. 1, 1–28 (2011).
  6. Farago J., Meyer H., Semenov A. N. Anomalous Diffusion of a Polymer Chain in an Unentangled Melt. Phys. Rev. Lett. 107 (17), 178301 (2011).
  7. Carcione J. M., Sanchez-Sesma F. J., Luzon F., Gavilan J. J. P. Theory and simulation of time-fractional fluid diffusion in porous media. J. Phys. A: Math. Theor. 46, 345501 (2013).
  8. Aarão Reis F. D. A., di Caprio D. Crossover from anomalous to normal diffusion in porous media. Phys. Rev. E. 89, 062126 (2014).
  9. Garra R. Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks. Phys. Rev. E. 84, 036605 (2011).
  10. Roubinet D., de Dreuzy J. R., Tartakovsky D. M. Particle-tracking simulations of anomalous transport in hierarchically fractured rocks. Computers & Geosciences. 50, 52–58 (2013).
  11. Carreras B. A., Lynch V. E., Zaslavsky G. M. Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas. 8 (12), 5096–5103 (2001).
  12. Priego M., Garcia O. E., Naulin V., Rasmussen J. J. Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence. Phys. Plasmas. 12 (6), 062312 (2005).
  13. Krivolapov Y., Levi L., Fishman Sh., Segev M., Wilkinson M. Super-diffusion in optical realizations of Anderson localization. New J. Phys. 14, 043047 (2012).
  14. Barkai E., Jung Y., Silbey R. Theory of single-molucule spectroskopy: beyond the ensemble average. Annu. Rev. Phys. Chem. 55, 457–507 (2004).
  15. Golovin A. A., Matkowsky B. J., Volpert V. A. Turing pattern formation in the Brusselator model with superdiffusion. J. Appl. Math. 69, n. 1, 251–272 (2008).
  16. Zhang L., Tian C. Turing pattern dynamics in an activator-inhibitor system with superdiffusion. Phys. Rev. E. 90, 062915 (2014).
  17. Dufiet V., Boissonade J. Dynamics of Turing pattern monolayers close to onset. Phys. Rev. E. 53, 4883 (1996).
  18. Prytula Z. Mathematical modelling of nonlinear dynamics in activator-inhibitor systems with superdiffusion. The Bulletin of Lviv Polytechnic National University titled “Computer Sciences and Information Technologies”. 826, 230–237 (2015).
  19. Samko S. G., Kilbas A. A., Marichev O. I. Fractional integrals and derivatives, theory and applications. Gordon and Breach, Amsterdam (1993).
  20. Uchaikin V. Method of fractional derivatives. Artishok-Press (2008), (in Russian).
  21. Petráš I. Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Springer (2011).
  22. Podlubny I. Fractional Differential Equations. San Diego: Acad. Press (1999).
  23. Walgraef D. Spatio-Temporal Pattern Formation. Springer, New York (1997).
Bibliography: 
Math. Model. Comput. Vol.3, No.2, pp.191-198 (2016)