论文标题
Bogoyavlensky Lattices和广义加泰罗尼亚
Bogoyavlensky lattices and generalized Catalan numbers
论文作者
论文摘要
我们以Bogoyavlensky Lattices的单位步骤的形式研究了初始数据衰减的问题。与Gurevich-pitaevskii的问题相反，KDV方程的初始不连续性衰减问题，事实证明这是完全可解决的，因为由于半线终止，动力学是可线化的。答案是根据广义超几何函数编写的，该功能是通用加泰罗尼亚数字的指数生成函数。这可以通过以下事实证明，这些数字的普遍汉克尔决定因素等于1，这是组合学的众所周知的结果。另一种方法是基于与动力学一致的非自主对称性还原。它将晶格方程降低到有限维系统，并可以解决更通用的有限参数初始数据家族的问题。
We study the problem of the decay of initial data in the form of a unit step for the Bogoyavlensky lattices. In contrast to the Gurevich--Pitaevskii problem of the decay of initial discontinuity for the KdV equation, it turns out to be exactly solvable, since the dynamics is linearizable due to termination on the half-line. The answer is written in terms of generalized hypergeometric functions, which serve as exponential generating functions for generalized Catalan numbers. This can be proved by the fact that the generalized Hankel determinants for these numbers are equal to 1, which is a well-known result in combinatorics. Another method is based on a non-autonomous symmetry reduction consistent with the dynamics. It reduces the lattice equation to a finite-dimensional system and makes it possible to solve the problem for a more general finite-parameter family of initial data.
