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

Abstract

The one-parameter family of second order nonlinear difference equations each of which is given by $$x_{n-1}{x}_n{x}_{n+1}=x_{n-1}+(x_n{)}^{\beta -1}+x_{n+1}(\beta \in \mathbb{N})$$ is explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when $\beta \le 3$. In order to evaluate the dynamics with $\beta \ge 4$, algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when $\beta =4$ but is positive when $\beta \ge 5$. This fact suggests that the difference equation with $\beta \le 4$ is integrable but that with $\beta \ge 5$ is not. It is moreover shown that the difference equation with $\beta \ge 4$ fails the singularity confinement test. This fact is consistent with linearizability of the equation with $\beta =4$ and reinforces non-integrability of the equation with $\beta \ge 5$.

17 pages, 1 figure

References in corpus (2)

• The solution to the initial value problem for the ultradiscrete Somos-4 and 5 equations
• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations

Cited by in corpus (1)

• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$