Turing pattern formation in the Brusselator system with nonlinear diffusion

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In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in theBrusselator system. Through linear stability analysis of the basic solution we determine the Turing and theoscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusionfavors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis wederive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations showsthe occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patternswith multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves:When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors topatterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern whichinvades the domain. We show the emergence of radially symmetric target patterns, and, through a matchingprocedure, we construct the outer amplitude equation and the inner core solution.
Original languageEnglish
Pages (from-to)042925-
Number of pages12
Publication statusPublished - 2013

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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