Painlevé-Type Asymptotics for the Camassa–Holm Equation

Abstract

We consider the initial value problem for the Camassa–Holm equation on the line with decaying initial data. We apply the nonlinear steepest descent approach to compute the long-time asymptotics of the solution in two transition regions. In both regions the asymptotics is expressed in terms of second Painlevé transcendents.