Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data

We consider the compressive wave for the modified Korteweg--de Vries equation with background constants $c>0$ for $x\to-\infty$ and $0$ for $x\to+\infty.$ We study the asymptotics of solutions in the transition zone $4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma}\ln t$ for $\varepsilon>0,$ $\sigma\in(0,1),$ $\beta>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\varepsilon t}{\ln t}.$ Also we show how to obtain Khruslov--Kotlyarov's asymptotics in the domain $4c^2t-\rho\ln t<x<4c^2t$ with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.