Symplectic forms in the theory of solitons
Symplectic forms in the theory of solitons
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孤子理论中的辛形式
DOI：
10.4310/sdg.1998.v4.n1.a6
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
D. H. Phong
中科院分区：
文献类型：
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作者：
I. Krichever;D. H. Phong
本文发展了二维孤子方程的哈密顿理论。特别是，我们确定了空间的双周期算子上，可以引入一个完整的层次的交换流，并表明，这些流量是哈密顿相对于一个普遍的辛形式$\omega={1\over 2}\r_{\infty} \d k$。我们还构造了其他高阶辛形式，并与一维孤子的情形进行了比较。限制在有限间隙孤子的空间中，泛辛形式与最近出现在非线性WKB理论、拓扑场论和Seiberg-Witten理论中的辛形式一致。我们借此机会调查在这些领域的辛形式发挥了重要作用的一些发展。
We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $\omega={1\over 2}\r_{\infty} \d k$. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.