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

Shun Shimomura

Research output: Contribution to journalArticle

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)733-781
Number of pages49
JournalJournal of the Mathematical Society of Japan
Issue number3
Publication statusPublished - 2012 Oct 5



  • Asymptotic solution
  • Continuous limit
  • Difference second Painlevé equation
  • Second Painlevé equation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this