标题： 最小非互惠Cahn-Hilliard系统中的尖锐界面动力学 显示英文标题

标题： Sharp Interface Dynamics in a Minimal Non-Reciprocal Cahn-Hilliard System

摘要： 非互惠耦合系统引起了兴趣，最近Brauns和Marchetti在2024年arXiv:2306.08868中引入了一个最小的非互惠耦合Cahn-Hilliard（CH）模型，我们称之为Brauns-Marchetti（BM）模型。 这个模型可以看作是空间扩展的FitzHugh-Nagumo模型的保守对应物。 由于缺乏梯度结构，BM模型被观察到在某些参数范围内表现出有趣的动态，包括行进的周期波列和其他相干结构，以及时空混沌。 在本文中，我们在尖锐界面极限下，推导了BM模型在$\mathbb{R}^2$中的解的界面动力学的有效方程。 所得的方程组是对经典Mullins-Sekerka（MS）方程的推广，我们称之为修改的MS方程。 我们证明了修改的MS方程与经典对应方程有一些相似的性质，但重要的是，它通常不是长度最小化流。 为了说明尖锐界面极限中这种渐近约简的实用性，我们对静态和周期波列进行了详细分析，系统地推导了波列速度和稳定性阈值的表达式。 这里使用的方法应适用于其他非互惠耦合的CH模型，因此为它们的更详细分析提供了另一条途径。 显示英文摘要

摘要： Interest in non-reciprocally coupled systems recently led to the introduction of a minimal non-reciprocally coupled Cahn-Hilliard (CH) model by Brauns and Marchetti in 2024 arXiv:2306.08868, which we refer to as the Brauns-Marchetti (BM) model. This model can be seen as a conservative counterpart to the spatially extended FitzHugh-Nagumo model. Lacking a gradient structure, the BM model was observed to exhibit interesting dynamics including traveling periodic wave-trains and other coherent structures, as well as spatiotemporal chaos in certain parameter regimes. In this paper, we derive an effective equation for the interface dynamics of solutions to the BM model in $\mathbb{R}^2$ in the sharp-interface limit. The resulting system of equations is a generalization of the classical Mullins-Sekerka (MS) equations, which we refer to as the modified MS equations. We show that the modified MS equation shares some properties with its classical counterpart, but importantly, it is not in general a length minimizing flow. To illustrate the utility of this asymptotic reduction in the sharp interface limit, we perform a detailed analysis of stationary and periodic wave-trains, systematically deriving expressions for wave-train speeds and stability thresholds. The methods used here should be applicable to other non-reciprocally coupled CH models and therefore provide another avenue for their more detailed analysis.