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→ −∞ and 0 for x→ +∞. We study the asymptotics of solutions in the transition zone 4ct− εt < x < 4ct − βt ln t for ε > 0, σ ∈ (0, 1), β > 0. In this region we have a bulk of nonvanishing oscillations, the number of which grows as εt ln t . Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain 4ct − ρ ln t < x < 4ct with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.