Abstract

In recent decades, much interest has been shown in nonlinear lattice vibrations because crystalline materials are subjected to high-amplitude impacts in many fields of human activity. One of the effects of nonlinearity in discrete periodic structures is the possibility of existence of spatially localized high-amplitude vibrations, referred to as discrete breathers (DBs), or intrinsic localized modes. The problem of searching for DBs in nonlinear chains (i.e., one-dimensional crystals) can be solved in a fairly simple way, because the variety of possible DBs is small in this case. However, no general approaches to the search for DBs have been developed for high-dimension crystal lattices. Such an approach was derived based on the works by Chechin, Sakhnenko et al., who developed the theory of bushes of nonlinear normal modes, which (as applied to crystals) were later referred to as delocalized nonlinear vibrational modes (DNVMs). It has recently been noted that all known DBs can be obtained by superimposing localizing functions on DNVMs with a frequency beyond the phonon spectrum of the lattice. Since the Chechin and Sakhnenko theory makes it possible to find all possible DNVMs by considering the lattice symmetry, it has become possible to formulate the problem of determining all possible DBs in a given lattice. This approach has recently been applied with success to the search for DBs in a two-dimensional triangular lattice. The purpose of this study is to analyze and describe DBs in a two-dimensional square lattice obtained using a localizing function. As a result, new types of DBs of a square lattice are obtained, including one-dimensional DBs (i.e., those localized only in one of two orthogonal directions) and zero-dimensional DBs (i.e., those localized in two directions).

中文翻译：

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基于离域模式的方格子中的离散呼吸器

摘要

近几十年来，人们对非线性晶格振动表现出了极大的兴趣，因为晶体材料在人类活动的许多领域都会受到高振幅的影响。离散周期结构中非线性的影响之一是可能存在空间局部高振幅振动，称为离散呼吸器 (DB) 或固有局部模式。在非线性链（即一维晶体）中搜索数据库的问题可以通过相当简单的方式解决，因为在这种情况下可能的数据库的种类很少。然而，尚未开发出针对高维晶格寻找 DB 的通用方法。这种方法是基于 Chechin、Sakhnenko 等人的工作推导出来的，他们发展了非线性简正模式的灌木理论，该理论（应用于晶体）后来被称为离域非线性振动模式 (DNVM)。最近人们注意到，所有已知的 DB 都可以通过在 DNVM 上叠加频率超出晶格声子谱的定位函数来获得。由于 Chechin 和 Sakhnenko 理论使得通过考虑晶格对称性来找到所有可能的 DNVM 成为可能，因此可以公式化确定给定晶格中所有可能的 DB 的问题。这种方法最近已成功应用于二维三角晶格中数据库的搜索。本研究的目的是分析和描述使用定位函数获得的二维方格中的数据库。结果，获得了方晶格的新型DB，包括一维DB（即，仅集中在两个正交方向之一上的DB）和零维DB（即，集中在两个方向上的DB）。