We consider the Cahn-Hilliard (CH) equation with a Burgers-type convective term that is used as a model of coarsening dynamics in laterally driven phase-separating systems. In the absence of driving, it is known that solutions to the standard CH equation are characterized by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or drops/holes or islands are obtained) followed by the coarsening process, where the average size of the structures grows in time and their number decreases. Moreover, two main coarsening modes have been identified in the literature, namely, coarsening due to volume transfer and due to translation. In the opposite limit of strong driving, the well-known Kuramoto-Sivashinsky (KS) equation is recovered, which may produce complicated chaotic spatio-temporal oscillations. The primary aim of the present work is to perform a detailed and systematic investigation of the transitions in the solutions of the convective CH (cCH) equation for a wide range of parameter values, and, in particular, to understand in detail how the coarsening dynamics is affected by an increase of the strength of the lateral driving force. Considering symmetric two-drop states, we find that one of the coarsening modes is stabilized at relatively weak driving, and the type of the remaining mode may change as driving increases. Furthermore, there exist intervals in the driving strength where coarsening is completely stabilized. In the intervals where the symmetric two-drop states are unstable they can evolve, for example, into one-drop states, two-drop states of broken symmetry or even time-periodic two-drop states that consist of two traveling drops that periodically exchange mass. We present detailed stability diagrams for symmetric two-drop states in various parameter planes and corroborate our findings by selected time simulations.

中文翻译：

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驱动对相分离系统粗化动力学的影响

我们考虑带有 Burgers 型对流项的 Cahn-Hilliard (CH) 方程，该方程用作横向驱动相分离系统中的粗化动力学模型。在没有驱动的情况下，已知标准 CH 方程的解的特征在于初始阶段的相分离为一相被另一相包围的区域（即，获得簇或液滴/孔或岛），然后是粗化过程，其中结构的平均尺寸随时间增长而其数量减少。此外，文献中已经确定了两种主要的粗化模式，即由于体积转移和由于平移而导致的粗化。在强驱动的相反极限下，恢复了著名的 Kuramoto-Sivashinsky (KS) 方程，这可能会产生复杂的混沌时空振荡。本工作的主要目的是对大范围参数值的对流 CH (cCH) 方程解的转变进行详细和系统的研究，特别是详细了解粗化动力学如何受横向驱动力强度增加的影响。考虑对称二降态，我们发现粗化模式之一在相对较弱的驱动下稳定，其余模式的类型可能随着驱动的增加而变化。此外，在驱动强度中存在粗化完全稳定的区间。在对称两滴状态不稳定的区间中，它们可以演化，例如，一滴状态，对称性破缺的两滴状态，甚至是由两个周期性交换质量的移动滴组成的时间周期两滴状态。我们提供了各种参数平面中对称二降状态的详细稳定性图，并通过选定的时间模拟证实了我们的发现。