Airy and Painlevé asymptotics for the mKdV equation

Abstract

We consider the higher order asymptotics for the mKdV 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 $x(3t{)}^{-1/3}$. We then consider the special case when the reflection coefficient vanishes at the origin. In this case, the leading coefficient which satisfies the Painlevé 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.

arXiv comment32 pages, 12 figures

References in corpus (3)

• Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Long-Time Asymptotics for the Korteweg-de Vries Equation with Steplike Initial Data
• Rarefaction Waves of the Korteweg-de Vries Equation via Nonlinear Steepest Descent

Cited by in corpus (4)

• Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
• Transient asymptotics of the modified Camassa-Holm equation
• Painleve-type asymptotics for the defocusing Hirota equation in transition region
• Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions