Exponential Asymptotics and Generalized Solitary Waves
Exponential Asymptotics and Generalized Solitary Waves
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指数渐进和广义孤立波
DOI：
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发表时间：
2010
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通讯作者：
R. Grimshaw
中科院分区：
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作者：
R. Grimshaw
流体力学中的许多问题都涉及适当小参数的幂级数形式的渐近展开。这种展开必然无法找到相对于该参数呈指数小的项。这些缺失的术语虽然很小，但往往具有重要的物理意义。本章将描述如何使用复平面中的匹配渐近展开式和 Borel 求和作为主要工具来找到这种指数小项。这些技术是在主要与弱非局域孤立波（也称为广义孤立波）理论相关的模型问题的背景下开发的，这些问题出现在重力毛细波、内波和其他几种物理背景的研究中。这些波具有有限振幅的中心核心，但伴随着振幅呈指数级小的共同传播振荡尾部。特别令人感兴趣的是对于某些参数值，振荡尾部的幅度可能为零的可能性，从而产生了嵌入孤立波的重要概念。
Many problems in fluid mechanics involve asymptotic expansions in the form of a power series for a suitable small parameter. Such expansions necessarily fail to find terms which are exponentially small with respect to this parameter. Although small these missing terms are often of physical importance. This chapter will describe how to find such exponentially small terms, using as the main tool matched asymptotic expansions in the complex plane and Borel summation. The techniques are developed in the context of model problems related primarily to the theory of weakly nonlocal solitary waves (also called generalized solitary waves) which arise in the study of gravity-capillary waves, internal waves and in several other physical contexts. These waves have a central core of finite amplitude, but are accompanied by co-propagating oscillatory tails whose amplitude is exponentially small. Special interest lies in the possibility that for certain parameter values, the amplitude of the oscillatory tails may be zero, leading to the important concept of embedded solitary waves.