Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension
复制标题
一维反应扩散系统阴影极限附近台面图案的稳定性转变和动力学
DOI：
10.3934/dcdsb.2012.17.191
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发表时间：
2011
影响因子：
1.2
通讯作者：
T. Kolokolnikov
中科院分区：
文献类型：
--
作者：
Rebecca McKay;T. Kolokolnikov
We consider a class of one-dimensional reaction-diffusion systems,
\[
\left\{
\begin{array}
[
u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\
\tau w_{t}=Dw_{xx}+g(u,w)
\end{array}
\right.
\]
with homogeneous Neumann boundary conditions on a one dimensional interval.
Under some generic conditions on the nonlinearities $f,g$ and in the singular
limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which
$u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$
and for sufficiently small $\tau$, such a steady state is known to be stable
in time. On the other hand, it is also known that in the so-called shadow
limit $D\rightarrow\infty,$ patterns having more than one interface are
unstable. In this paper we analyse in detail the transition between the stable
patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We
show that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg
D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if
$D D_{K}$. We also study the dynamics of the
interfaces when $D$ is exponentially large; this allows us to describe in
detail the mechanism leading to the instability. Direct numerical computations
of stability and dynamics are performed, and these results are in excellent
agreement with corresponding results as predicted by the asymptotic theory.
We consider a class of one-dimensional reaction-diffusion systems,
\[
\left\{
\begin{array}
[
u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\
\tau w_{t}=Dw_{xx}+g(u,w)
\end{array}
\right.
\]
with homogeneous Neumann boundary conditions on a one dimensional interval.
Under some generic conditions on the nonlinearities $f,g$ and in the singular
limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which
$u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$
and for sufficiently small $\tau$, such a steady state is known to be stable
in time. On the other hand, it is also known that in the so-called shadow
limit $D\rightarrow\infty,$ patterns having more than one interface are
unstable. In this paper we analyse in detail the transition between the stable
patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We
show that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg
D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if
$D D_{K}$. We also study the dynamics of the
interfaces when $D$ is exponentially large; this allows us to describe in
detail the mechanism leading to the instability. Direct numerical computations
of stability and dynamics are performed, and these results are in excellent
agreement with corresponding results as predicted by the asymptotic theory.