Bogoyavlensky 格与广义 Catalan 数
Bogoyavlensky lattices and generalized Catalan numbers
AI总结：
本文研究 Bogoyavlensky 格单位阶跃初始数据衰减问题，利用半直线终止的可线性化性质、广义 Catalan 数及其 Hankel 行列式，以及非自治对称性约化，给出精确解并推广到有限参数初值族。
摘要：
我们研究 Bogoyavlensky 格的单位阶跃形式初始数据的衰减问题。与 KdV 方程初始间断衰减的 Gurevich–Pitaevskii 问题不同，该问题被证明是精确可解的，因为在半直线上终止后，其动力学可线性化。答案用广义超几何函数表示，这些函数充当广义 Catalan 数的指数生成函数。这一点可由这些数的广义 Hankel 行列式均等于 1 这一组合数学中的著名结果来证明。另一种方法基于与动力学相容的非自治对称性约化。它将格方程约化为有限维系统，并使得能够求解更一般的有限参数初始数据族的问题。
英文摘要：
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.
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