The direct method in soliton theory
The direct method in soliton theory
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DOI：
10.1017/cbo9780511543043
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
広田 良吾;永井 敦;J. Nimmo;C. Gilson
中科院分区：
文献类型：
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作者：
広田 良吾;永井 敦;J. Nimmo;C. Gilson
双线性或Hirota的直接方法是在20世纪70年代早期发明的，作为构造孤子解的基本方法，它避免了使用反散射变换的繁重机器，并成功地用于构造许多新方程的多孤子解。在20世纪80年代，这种方法中使用的工具——Hirota导数和双线性形式——的深层意义开始被理解为佐藤理论和与仿射李代数的联系的关键成分。这本书的主要部分是关于以行列式和定式形式表示解的方法的更现代的版本。在保持使用相对简单的数学的原始哲学的同时，它受到了京都学派工作中产生的更深层次理解的影响。这本书对所有研究孤子理论的人都是必不可少的。
The bilinear, or Hirota's direct, method was invented in the early 1970s as an elementary means of constructing soliton solutions that avoided the use of the heavy machinery of the inverse scattering transform and was successfully used to construct the multisoliton solutions of many new equations. In the 1980s the deeper significance of the tools used in this method - Hirota derivatives and the bilinear form - came to be understood as a key ingredient in Sato's theory and the connections with affine Lie algebras. The main part of this book concerns the more modern version of the method in which solutions are expressed in the form of determinants and pfaffians. While maintaining the original philosophy of using relatively simple mathematics, it has, nevertheless, been influenced by the deeper understanding that came out of the work of the Kyoto school. The book will be essential for all those working in soliton theory.