Effects of boundaries on one-dimensional reaction-diffusion equations near threshold

✍ Scribed by P.C. Hohenberg; Lorenz Kramer; Hermann Riecke

• Book ID: 104297067
• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 951 KB
• Volume: 15
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/s0167-2789(85)80007-5

✦ Synopsis

A simple set of two reaction-diffusion equations is analyzed near the threshold for the appearance of spatially periodic solutions in one dimension. Exact amplitude equations are derived to second order in the deviation from the threshold. Wavcvcctor selection is studied analytically for parameters varying slowly in space from supercriticalto subcritical conditions. The nonuniversality of the selection process is demonstrated explicitly {or nonpotential cases. The effect of boundary conditions on restricting the available band of wavevectors is studied for solutions which fall below their bulk value near the boundary (type I). The existence of another class of static solutions (type 11), remaining finite at the boundary even near threshold, is demonstrated. A linear stability analysis of the uniform state reveals that type-I solutions are often unstable immediately above threshold. Then one has type-II states or oscillatory solutions. Some numerical results are presented. which confirm the analytic calculations.