Discrete Painlevé transcendent solutions to the multiplicative type discrete KdV equations

Hirota's discrete KdV equation is an integrable partial difference equation on $\mathbb{Z}^2$, which approaches the Korteweg-de Vries (KdV) equation in a continuum limit. In this paper, we show that its multiplicative-discrete versions have the special solutions given by the solutions of $q$-Painlevé equations of types $A_J^{(1)}$ $(J=3,4,5,6)$.

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