Airy and Painlevé asymptotics for the mKdV equation

We consider the higher order asymptotics for the modified Korteweg-de Vries equation in the Painlevé 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 ( 3 t ) − 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 Painlevé II equation vanishes. We show that the leading asymptotics are 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.