Speaker: 

Peter Miller

Institution: 

University of Michigan

Time: 

Tuesday, March 5, 2019 - 1:00pm

Location: 

NS II 1201

This talk will survey two recent studies of solutions of the Painlev\'e-III equation.  First, we describe work with T. Bothner and Y. Sheng on the asymptotic behavior of rational solutions of the Painlev\'e-III equation in the limit where the degree is large.  Then we turn to a class of rational solutions of a different equation, namely the focusing nonlinear Schr\"odinger equation, which are believed to model rogue waves.  In work with D. Bilman and L. Ling we studied the fundamental rogue wave solutions in the limit of large order, and found a new transcendental solution of the focusing nonlinear Schr\"odinger equation that we call the rogue wave of infinite order.  This solution turns out to also satisfy ordinary differential equations in the Painlev\'e-III hierarchy.