Painlev\'e transcendents in the defocusing mKdV equation with non-zero boundary conditions

Abstract

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlev\'{e} transcendent.