TY - JOUR
T1 - Effects of boundaries on one-dimensional reaction-diffusion equations near threshold
AU - Hohenberg, P. C.
AU - Kramer, Lorenz
AU - Riecke, Hermann
N1 - Funding Information:
edge the Physics of the University of California at Barbara, benefitted of us Tesauro. National PHY 77-27084. Travel grants by the Deutsche Forschungsgemeinschaft mission (L.K.) (P.C.H.)
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 1985
Y1 - 1985
N2 - 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. Wavevector selection is studied analytically for parameters varying slowly in space from supercritical to subcritical conditions. The nonuniversality of the selection process is demonstrated explicitly for 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 II), 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.
AB - 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. Wavevector selection is studied analytically for parameters varying slowly in space from supercritical to subcritical conditions. The nonuniversality of the selection process is demonstrated explicitly for 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 II), 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.
UR - http://www.scopus.com/inward/record.url?scp=0022051003&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0022051003&partnerID=8YFLogxK
U2 - 10.1016/S0167-2789(85)80007-5
DO - 10.1016/S0167-2789(85)80007-5
M3 - Article
AN - SCOPUS:0022051003
VL - 15
SP - 402
EP - 420
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 3
ER -