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Airy and Painlevé asymptotics for the mKdV equation
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 …