Hurwitz numbers and integrable hierarchy of Volterra type
Hurwitz numbers and integrable hierarchy of Volterra type
复制标题
DOI：
10.1088/1751-8121/aae10b
复制
发表时间：
2018-06
期刊：
影响因子：
--
通讯作者：
K. Takasaki
中科院分区：
文献类型：
--
作者：
K. Takasaki
黎曼球面的单个Hurwitz数的母函数是格子Kp族的tau函数。与之相关的Lax算子L被表示为，其中是形式的差-微分算子。满足一组松弛方程，形成Bogoyavlensky-Itoh(又名饥饿Lotka-Volterra)层次结构的连续版本。对于二维Toda族的Lax算子和Orlov-Schulman算子，用广义弦方程的语言进一步解释了这种潜在的可积结构的出现。这就产生了对数弦方程，并借助于运算符的因式分解问题得到了证实。
A generating function of the single Hurwitz numbers of the Riemann sphere is a tau function of the lattice KP hierarchy. The associated Lax operator L turns out to be expressed as , where is a difference-differential operator of the form . satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky–Itoh (aka hungry Lotka–Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov–Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.