Browsing by Author "Sakka, Ayman"
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Item Open Access The solvability of PVI equation and second-order second-degree Painleve type equations(Bilkent University, 1998) Sakka, AymanA rigorous method was introduced by Fokas and Zhou for studying the Riernaiin-Hilhert problem associated with the Painleve II and IV. The same methodology has been applied to Painleve I, III and V. In this thesis, we applied the same methodology to the Painleve VI equation. VVe showed that The Cauchy problem for the Painleve VI equation admits in general global meromorphic solution in /. Furthermore, a. special solution which can lie written in terms of hypergeometric function is obtained via sob’ing the special case of the Riemann-Hilbert problem. Moreover, an algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painleve equations and a generalization of it are used to obtain one-to-one correspondence between the Painleve equations and the second-ord(U‘ seconddegree equations of Painleve type.Item Open Access Transformation properties of Painleve VI equation(Bilkent University, 1995) Sakka, AymanIn this thesis, we studied the Schlesinger transformations of Painleve VI equation. We showed that Painleve VI equation admits Schlesinger transformations which relate a given solution of Paileve VI to solution of Painleve VI but with different values of the parameters. Using these transformations we obtained the corresponding Bäcklund transformations for Painleve VI. Also, we showed that the Schlesinger transformations and the corresponding Bäcklund transformations break down if and only if Painleve VI has certain one-parameter family of solutions.