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

Abstract

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 $4c^{2}t-\epsilon t<x<4c^{2}t-\beta t^{\sigma }\ln t$ for $\epsilon >0,$ $\sigma \in (0,1),$ $\beta >0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\epsilon t}{\ln t}.$ Also we show how to obtain Khruslov--Kotlyarov's asymptotics in the domain $4c^{2}t-\rho \ln t<x<4c^{2}t$ with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.

38 pages, 6 figures