Continuous limit of the difference second Painlevé equation and its asymptotic solutions

Abstract

The discrete second Painlevé equation dP_II is mapped to the second Painlevé equation P_II by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dP_II also reduces to that of P_II. In this paper, regarding dP_II as a difference equation, we present a certain asymptotic solution that reduces to a triply-truncated solution of P_II in this continuous limit. In a special case our solution corresponds to a rational one of dP_II. Furthermore we show the existence of families of solutions having sequential limits to truncated solutions of P_II.