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arXiv 1904.02853nlin.SI
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一族由簇代数产生的可积和不可积差分方程

A family of integrable and non-integrable difference equations arising from cluster algebras

Atsushi Nobe, Junta Matsukidaira

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17 pages, 1 figure
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摘要：

研究了由$$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{β-1}+x_{n+1} \qquad(β\in\mathbb{N}) $$给出的单参数二阶非线性差分方程族。由于上面的方程是由秩为2的簇代数的种子突变产生的，所以它的解仅在$β\leq3$。为了用$β\geq4$计算动力学，研究了与差分方程等价的双有理映射的代数熵，当$β=4$为零，而当$β\geq5$为正.这一事实表明，具有$β\leq4$的差分方程是可积的，而具有$β\geq5$的差分方程是不可积的。此外，还表明，差分方程与$β\geq4$的奇异约束测试失败。这一事实与$β=4$方程的线性化一致，并加强了$β\geq5$方程的不可积性。

英文摘要：

The one-parameter family of second order nonlinear difference equations each of which is given by $$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{β-1}+x_{n+1} \qquad(β\in\mathbb{N}) $$ is explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when $β\leq3$. In order to evaluate the dynamics with $β\geq4$, algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when $β=4$ but is positive when $β\geq5$. This fact suggests that the difference equation with $β\leq4$ is integrable but that with $β\geq5$ is not. It is moreover shown that the difference equation with $β\geq4$ fails the singularity confinement test. This fact is consistent with linearizability of the equation with $β=4$ and reinforces non-integrability of the equation with $β\geq5$.

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