Generic q-PII solutions with q close to 1 and a=1 are asymptotic to Jacobi elliptic functions, with the slow modulation described by complete elliptic integrals.
Sur les ´ equations diff´ erentielles du second ordre et du premier degr´ e dont l’int´ egrale g´ en´ erale est ` a points critiques fixes
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Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents
Generic q-PII solutions with q close to 1 and a=1 are asymptotic to Jacobi elliptic functions, with the slow modulation described by complete elliptic integrals.