Geometry of Quadrics and Spectral Theory
Geometry of Quadrics and Spectral Theory
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DOI：
10.1007/978-1-4613-8109-9_7
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发表时间：
1980
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影响因子：
--
通讯作者：
J. Moser
中科院分区：
文献类型：
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作者：
J. Moser
本文研究可积哈密顿系统。这个概念可以追溯到上个世纪的经典分析动力学。简而言之，这些是由哈密顿函数描述的非线性常微分方程组，具有足够多的积分(或守恒量)，使得它们或多或少可以通过求积显式求解。因此，在世纪之交发展出更多的微分方程定性方法之前，这些系统在上个世纪起到了至关重要的作用。随后，人们对这些系统的兴趣减少了，部分原因是意识到只有例外的哈密顿系统才能建立整体积分的存在性。在过去的15年里，随着一些偏微分方程的发现，可积哈密顿系统重新引起了人们的极大兴趣，这些偏微分方程可以被视为具有无限自由度的系统。在这种情况下，积分形成了守恒泛函的无限序列。最著名的例子是Korteweg-DeVries方程：UT+uUx+uxxx=O。对该方程的广泛研究导致了与散射理论、谱理论、超椭圆曲线及其0-函数的复分析和微分几何之间惊人的联系。
In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary differential equations described by a Hamiltonian function and possessing sufficiently many integrals (or conserved quantities) so that they are more or less explicitly solvable by quadrature. Therefore these systems played a crucial role in the last century before more qualitative methods for differential equations were developed at the turn of the century. Subsequently interest in these systems decreased, partly due to the realization that the existence of global integrals can be established only for exceptional Hamiltonian systems.In the last 15 years the subject of integrable Hamiltonian system has regained considerable interest with the discovery of some partial differential equations which can be viewed as such systems with infinite degrees of freedom. In this case the integrals form an infinite sequence of conserved functionals. The most celebrated example is the Korteweg-deVries equation: ut+ uUx+ uxxx= O. Extensive investigations of this equation have led to surprising links with scattering theory, spectral theory, complex analysis of hyperelliptic curves and their 0-functions, and differential geometry.