从守恒的Kuramoto-Sivashinsky方程到凝聚粒子模型
From the conserved Kuramoto-Sivashinsky equation to a coalescing particles model
摘要：
最近在晶体生长的背景下导出了守恒的Kuramoto-Sivashinsky（CKS）方程u_t = -（u+u_xx+u_x^2）_xx，并且它也与出现的类似方程严格相关，例如，在沙纹动力学中。我们表明，这个方程可以映射到一个质点系的运动与吸引力的相互作用，衰减为他们的距离的倒数。粒子代表曲率发散的消失区域，由单个抛物线的弧连接，并在遇到时合并。聚结粒子模型比原CKS方程更容易模拟。增长的粒子间距离\ell表示系统的粗化，并且我们能够牢固地建立缩放\ell（t）\sim \sqrt{t}。我们得到它的概率分布函数，g（\ell），数值，并研究分析假设不相关的间隔内，发现在大的距离高估。最后，我们介绍了一种基于聚结波的方法，这可能是有用的，以获得更好的分析见解的模型。
英文摘要：
The conserved Kuramoto-Sivashinsky (CKS) equation, u_t = -(u+u_xx+u_x^2)_xx, has recently been derived in the context of crystal growth, and it is also strictly related to a similar equation appearing, e.g., in sand-ripple dynamics. We show that this equation can be mapped into the motion of a system of particles with attractive interactions, decaying as the inverse of their distance. Particles represent vanishing regions of diverging curvature, joined by arcs of a single parabola, and coalesce upon encounter. The coalescing particles model is easier to simulate than the original CKS equation. The growing interparticle distance \ell represents coarsening of the system, and we are able to establish firmly the scaling \ell(t) \sim \sqrt{t}. We obtain its probability distribution function, g(\ell), numerically, and study it analytically within the hypothesis of uncorrelated intervals, finding an overestimate at large distances. Finally, we introduce a method based on coalescence waves which might be useful to gain better analytical insights into the model.
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