Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the connection formulas of the third Painlevé transcendent

Pages: 541 - 560, Volume 23, Issue 1/2, January/February 2009      doi:10.3934/dcds.2009.23.541

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Roderick S. C. Wong - Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong (email)
H. Y. Zhang - Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China (email)

Abstract: We consider the connection problem for the sine-Gordon PIII equation $u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general third Painlevé transcendents. The connection formulas are derived by the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. Anal., 1998).

Keywords:  Connection formulas, uniform asymptotics, the third PainlevĂ© transcendent (PIII), sine-Gordon PIII equation, parabolic cylinder functions, Bessel functions
Mathematics Subject Classification:  Primary: 33E17, 34M55; Secondary: 35Q53

Received: November 2007;      Revised: February 2008;      Available Online: September 2008.