Airy and Painlev\'e asymptotics for the mKdV equation

TL;DR

This paper derives higher order asymptotics for the mKdV equation in the Painlevé sector, revealing a transition from Painlevé II to Airy function behavior when the reflection coefficient vanishes at the origin.

Contribution

It provides a uniform expansion of solutions to the mKdV equation to all orders in powers of t^{-1/3} and characterizes the asymptotics in the special case where the reflection coefficient vanishes at zero.

Findings

Solution admits a uniform expansion in powers of t^{-1/3}

Leading asymptotics described by the derivative of the Airy function when reflection coefficient vanishes

Explicit expression for the subleading term in terms of the Airy function

Abstract

We consider the higher order asymptotics for the mKdV equation in the Painlev\'e sector. We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of $x(3t)^{-1/3}$. We then consider the special case when the reflection coefficient vanishes at the origin. In this case, the leading coefficient which satisfies the Painlev\'e II equation vanishes. We show that the leading asymptotics is instead described by the derivative of the Airy function. We are also able to express the subleading term explicitly in terms of the Airy function.