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1 
On the connection formulas of Painlevé transcendents /Zhang, Haiyu. January 2009 (has links) (PDF)
Thesis (Ph.D.)City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [95]100)

2 
Bac̈klund transformations, the Painleve ̓property and some of their applications /Wong, Wingtak, January 1987 (has links)
Thesis (Ph. D.)University of Hong Kong, 1988.

3 
Asymptotics of higherorder Painlevé equationsMorrison, Tegan Ann January 2009 (has links)
Doctor of Philosophy (PhD) / We undertake an asymptotic study of a second Painlevé hierarchy based on the JimboMiwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of nonlinear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higherorder second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourthorder analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leadingorder by two related genus2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leadingorder equations, and we investigate how these parameters change with respect to this variable. We also show that the fourthorder equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higherorder Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reactiondiffusion equations with quadratic and cubic source terms, with a spatiotemporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents.

4 
Asymptotics of higherorder Painlevé equationsMorrison, Tegan Ann January 2009 (has links)
Doctor of Philosophy (PhD) / We undertake an asymptotic study of a second Painlevé hierarchy based on the JimboMiwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of nonlinear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higherorder second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourthorder analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leadingorder by two related genus2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leadingorder equations, and we investigate how these parameters change with respect to this variable. We also show that the fourthorder equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higherorder Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reactiondiffusion equations with quadratic and cubic source terms, with a spatiotemporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents.

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