TY - JOUR

T1 - Continuous limit of the difference second Painlevé equation and its asymptotic solutions

AU - Shimomura, Shun

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Asymptotic solution

KW - Continuous limit

KW - Difference second Painlevé equation

KW - Second Painlevé equation

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U2 - 10.2969/jmsj/06430733

DO - 10.2969/jmsj/06430733

M3 - Article

AN - SCOPUS:84866934261

VL - 64

SP - 733

EP - 781

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -