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- 저자 : Zumbrun, Kevin (검색결과 2 건)
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Barker, Blake,Jung, Soyeun,Zumbrun, Kevin Elsevier 2018 Physica. D, Nonlinear phenomena Vol.367 No.-
<P><B>Abstract</B></P> <P>Turing patterns on unbounded domains have been widely studied in systems of reaction–diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of Oh–Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram – specifically, determination of rigorous Eckhaus-type stability conditions – remains an interesting open problem.</P> <P><B>Highlights</B></P> <P> <UL> <LI> Conditions for Turing instability in conservation laws are derived. </LI> <LI> There exist no Turing-type instabilities in conservation laws for 2 × 2 systems. </LI> <LI> Stable periodic waves in conservation laws are numerically observed. </LI> </UL> </P>
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Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model
Sukhtayev, Alim,Zumbrun, Kevin,Jung, Soyeun,Venkatraman, Raghavendra Springer-Verlag 2018 Communications in mathematical physics Vol.358 No.1
<P>Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction-diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg-Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.</P>
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