\(p\)-adic difference-difference Lotka-Volterra equation and ultra-discrete limit (Q5957218)

The equation mentioned in the title is NEWLINE\[NEWLINE \frac{c_n^{m+1}}{c_n^m}=\frac{1+\delta_pc_{n- 1}^m}{1+\delta_pc_{n+1}^{m+1}},\quad n,m\in \mathbb Z,\tag{1} NEWLINE\]NEWLINE where \(\delta_p\in p\mathbb Z_p\). The author proves the existence of a solution different from the trivial one, for which \(c_n^m=\text{const}\) for all \(n,m\). NEWLINENEWLINENEWLINEThe equation (1) is a \(p\)-adic analog of the well-known partial difference equation of the KdV type. If one sets \(f_n^m=-\text{ord}_p(c_n^m)\), the equation for \(f_n^m\) obtained from (1) coincides with the ultra-discrete Lotka-Volterra equation used in the theory of cellular automata. This equation is also interpreted in terms of a valuation on a certain function ring.