Collective Coordinates and Length-Scale Competition in Spatially Inhomogeneous Soliton-Bearing Equations
Collective Coordinates and Length-Scale Competition in Spatially Inhomogeneous Soliton-Bearing Equations
复制标题
空间非齐次孤子方程中的集体坐标和长度尺度竞争
DOI：
10.1137/s0036144597317418
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
A. Bishop
中科院分区：
文献类型：
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作者：
A. Sánchez;A. Bishop
扰动的、一维的、可积的（即，孤子承载）方程出现在许多应用环境中，当试图根据纯可积方程改进感兴趣的问题的（通常高度理想化的）描述时。特别是，当考虑到不可避免的杂质或缺陷而放弃完美均匀性的假设时，必须将取决于空间坐标的扰动添加到原始方程中。在这篇综述中，我们使用的一维sine-Gordon（SG）方程扰动的空间周期项作为一个通用的范例，讨论主要的扰动技术可用于研究这类问题。为了将工作的背景下，我们总结了迄今为止开发的方法，并集中在集体协调的方法作为最有用的工具之一。我们介绍了几个版本的微扰方法，并将它们与更复杂的程序。我们详细分析了它在sG方程中的应用，但过程是很一般的.为了说明这一点的其他例子的集体坐标的应用程序进行了简要回顾。在我们的案例研究中，这种方法可以帮助我们识别微扰和非微扰制度，产生一个非常简单的图片。除了微扰计算，同样的例子，非齐次SG方程允许我们引入一种现象，称为长度尺度竞争，我们表明这是一个相当普遍的机制，在扰动可积系统中出现复杂的时空行为，在审查中讨论的其他实例显示。这种非微扰的结果是通过数值模拟的充分扰动的问题，数值线性稳定性分析也被用来澄清起源于这种竞争的不稳定性。为了补充我们在这些研究中所采用的技术的描述，我们的数值模拟的计算细节也包括在内。最后，本文结束了上述想法的讨论，并对有关非线性与无序的相互作用的一般问题进行了推测性的展望。
Perturbed, one-dimensional, integrable (i.e., soliton-bearing) equations arise in many applied contexts when trying to improve the (usually highly idealized) description of problems of interest in terms of the purely integrable equations. In particular, when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects, perturbations depending on the spatial coordinate must be added to the original equation. In this review, we use the one-dimensional sine-Gordon (sG) equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems. To place the work in context, we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools. We introduce several versions of this perturbative method and relate them to more involved procedures. We analyze in detail the application to the sG equation, but the procedure is very general. To illustrate this other examples of the application of collective coordinates are briefly revisited. In our case study, this approach helps us identify perturbative and nonperturbative regimes, yielding a very simple picture of the former. Beyond perturbative calculations, the same example of the inhomogeneous sG equation allows us to introduce a phenomenon, termed length scale competition, which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems, as other instances discussed in the review show. Such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem; numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition. To complement our description of the techniques employed in these studies, computational details of our numerical simulations are also included. Finally, the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder.