Applied Maths Seminar: Morrison -- Asymptotics of Higher-Order Painlevé Equations

Tegan Morrison, School of Mathematics and Statistics, University of Sydney 

I will discuss results of an asymptotic study of a second Painlevé hierarchy in the
limit as the independent variable approaches infinity.  The hierarchy is defined by an
infinite sequence of non-linear ordinary differential equations, indexed by order, with
the classical second Painlevé equation as the first member.  Particular attention will
be given to the fourth-order analogue of the classical second Painlevé equation.  In
this case, the general asymptotic behaviour is given to leading-order by two related
genus-2 hyperelliptic functions.  The fourth-order equation also admits two classes of
special asymptotic behaviours which are described by algebraic formal power series.
This work builds upon the foundations of established asymptotic results for the
classical second Painlevé equation and so I will begin with a survey of these results.