arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1802.02737math.APmath.DS
收藏

具有时变参数的奇摄动反应扩散系统中消失脉冲的动力学

The dynamics of disappearing pulses in a singularly perturbed reaction-diffusion system with parameters that vary in time and space

Robbin Bastiaansen, Arjen doelman

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们考虑在一个扩展的Klausmeier方程的参数，在时间和/或空间变化的多脉冲模式的演变。我们正式表明，$N$脉冲配置的完整PDE动力学可以减少到一个$N$维动力系统描述的$N$维流形$\mathcal{M}_N$上的动力学。接下来，我们通过与不断发展的$N$ -脉冲模式相关联的准稳态谱来确定$\mathcal{M}_N$的局部稳定性，这提供了关于边界$\partial\mathcal{M}_N$的明确信息。在$\mathcal{M}_N$上的动力学之后，$N$ -脉冲模式可以移动通过$\partial\mathcal{M}_N$并且“脱落”$\mathcal{M}_N$。我们的线性分析的直接非线性外推预测随后的快速PDE动力学的模式“跳”到另一个不变流形$\mathcal{M}_M$，并具体预测的脉冲消失的数量$N-M$。将渐近分析与各种不变流形上的动力学的数值模拟相结合，产生了描述以$N$脉冲模式开始并且通常以没有脉冲的平凡均匀状态结束的全过程的混合渐近-数值方法。我们广泛地测试这种方法对PDE模拟和推导出的脉冲相互作用与消失的脉冲的性质的一般说明。我们特别考虑不规则和规则模式的演化之间的差异。在前一种情况下，消失的过程是渐进的：不规则的模式一个接一个地失去它们的脉冲。相比之下，规则的，空间周期性的，图案经历灾难性的转变，其中一半或所有脉冲消失。然而，在这两个截然不同的过程之间进行精确区分是相当微妙的，因为不规则的$N$ -不穿过$\partial\mathcal{M}_N$的脉冲模式通常朝着规则性发展。

英文摘要：

We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a $N$-pulse configuration can be reduced to a $N$-dimensional dynamical system describing the dynamics on a $N$-dimensional manifold $\mathcal{M}_N$. Next, we determine the local stability of $\mathcal{M}_N$ via the quasi-steady spectrum associated to evolving $N$-pulse patterns, which provides provides explicit information on the boundary $\partial\mathcal{M}_N$. Following the dynamics on $\mathcal{M}_N$, a $N$-pulse pattern may move through $\partial\mathcal{M}_N$ and `fall off' $\mathcal{M}_N$. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern `jumps' to another invariant manifold $\mathcal{M}_M$, and specifically predicts the number $N-M$ of pulses that disappear. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic-numerical method describing the full process that starts with a $N$-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns loose their pulses one by one. In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear. However, making a precise distinction between these two drastically different processes is quite subtle, since irregular $N$-pulse patterns that do not cross $\partial\mathcal{M}_N$ typically evolve towards regularity.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑