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Register a new userDiffusive stability of Turing patterns via normal forms
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We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a normal form coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in $L^1cap L^infty$, with sharp rates.
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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 sub-critical 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.
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We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Greens functions estimates for functional differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.
This paper is concerned with the derivative nonlinear Schr{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.
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