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 nonautonomous 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. DOI 10.1134/S106192084010011

V.E. Adler

Russian Journal of Mathematical Physics

俄罗斯数学物理杂志是一份同行评审期刊，涉及该学科所包含的所有主题，是当代科学的基础。因此，除了数学物理学本身之外，期刊的报道还包括但不限于泛函分析、线性和非线性偏微分方程、代数、量化、量子场论、现代微分和代数几何和拓扑、LIE 的表示群、变分法、渐近方法、随机过程理论、动力系统和控制理论。

The Russian Journal of Mathematical Physics is a peer-reviewed journal that covers all topics within the field of mathematical physics and serves as a fundamental basis for contemporary science. In addition to mathematical physics itself, the journal also reports on research findings in related areas such as functional analysis, linear and nonlinear partial differential equations, algebra, quantization, quantum field theory, modern differential and algebraic geometry, topology, representations of Lie groups, variational methods, asymptotic methods, stochastic process theory, dynamical systems, and control theory.

《Russian Journal of Mathematical Physics》是一本经过同行评审的期刊，涵盖了数学物理学领域的所有主题，是当代科学的重要基础。除了数学物理学本身，期刊还报道了泛函分析、线性和非线性偏微分方程、代数、量子化、量子场论、现代微分几何和代数几何、拓扑、李群表示、变分法、渐近方法、随机过程理论、动力系统和控制理论等相关领域的研究成果。

Category	Quartile
物理与天体物理	3
物理与天体物理, 物理数学物理	4

Bogoyavlensky Lattices and Generalized Catalan Numbers

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Journal Info

Category	Quartile
PHYSICS, MATHEMATICAL	2