A Plethora of Integrable Bi-Hamiltonian Equations
A Plethora of Integrable Bi-Hamiltonian Equations
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DOI：
10.1007/978-1-4612-2434-1_5
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
A. Fokas;P. Olver;P. Rosenau
中科院分区：
文献类型：
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作者：
A. Fokas;P. Olver;P. Rosenau
本文讨论了利用多重Hamilton结构构造可积发展方程的几种算法方法。可积孤子方程，如Korteweg-de弗里斯（KdV）和非线性薛定谔（NLS）方程，可以用双哈密顿方法构造的认识可以追溯到20世纪70年代后期。第一作者和Fuchssteiner在20世纪80年代初提出了该方法的扩展，并用于导出KdV和修改的KdV的可积推广。然而，直到这些模型重新出现在物理问题中，并且发现了它们的新解决方案，如双曲子和峰子，该方法才获得认可。在本文中，我们描述的基本方法来构建各种各样的可积双哈密顿方程。除了通常的孤子方程，这些新的层次包括非线性色散方程，支持新类型的孤子解决方案。让我们从简单的标量演化方程开始
This paper discusses several algorithmic ways of constructing integrable evolution equations based on the use of multi-Hamiltonian structures. The recognition that integrable soliton equations, such as the Korteweg-de Vries (KdV) and nonlinear Schrodinger (NLS) equations, can be constructed using a biHamiltonian method dates back to the late 1970's. An extension of the method was proposed by the first author and Fuchssteiner in the early 1980's and was used to derive integrable generalizations of the KdV and of the modified KdV. However it was not until these models reappeared in physical problems, and their novel solutions such as compactons and peakons were discovered, that the method achieved recognition. In this paper, we describe the basic approach to constructing a wide variety of integrable bi-Hamiltonian equations. In addition to usual soliton equations, these new hierarchies include equations with nonlinear dispersion which support novel types of solitonic solutions. Let us start with the simple case of a scalar evolution equation