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Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations
Journal of Nonlinear Mathematical Physics ( IF 1.4 ) Pub Date : 2019-07-09 , DOI: 10.1080/14029251.2019.1640460
Francesco Calogero 1, 2 , Farrin Payandeh 3
Affiliation  

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations, which are algebraically solvable. Here l is the “discrete-time” independent variable taking integer values (l = 0, 1, 2, . . .), xn ≡ xn(l) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F(n)(x1, x2), n = 1, 2, were defined as follows: F(n)(x1, x2) = P2 (xn, xn+1), n = 1, 2 mod[2], with P2(x1, x2) a specific second-degree homogeneous polynomial in the 2 (indistinguishable!) dependent variables x1(l) and x2(l). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary (integer) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x1(l) and x2(l).

中文翻译:

两个特殊类别的可解系统,具有 2 个因变量通过 2 个非线性耦合的一阶递归关系在离散时间演化

在本文中,我们确定了 2 个离散时间演化方程的某些特殊系统,它们是代数可解的。这里 l 是取整数值(l = 0, 1, 2, . .)的“离散时间”自变量,xn ≡ xn(l) 是 2 个因变量,是对应的 2 个更新变量。在之前的一篇论文中,2 个函数 F(n)(x1, x2), n = 1, 2 定义如下: F(n)(x1, x2) = P2 (xn, xn+1), n = 1 , 2 mod[2],其中 P2(x1, x2) 是 2 个(无法区分!)因变量 x1(l) 和 x2(l) 中的特定二阶齐次多项式。在本文中,我们进一步阐明了该模型的某些方面,并且我们将其扩展到以下情况:在 2 个因变量 x1 中具有任意(整数)度 k(因此当 k > 0 时为 k 度的多项式)的特定齐次函数(l) 和 x2(l)。
更新日期:2019-07-09
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