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We study the long-time asymptotic behavior of the Cauchy problem for themodified Korteweg - de Vries equation with an initial function of the steptype. This function rapidly tends to zero as $x\to+\infty$ and to some positiveconstant c as $x\to-\infty$. In 1989 E. Khruslov and V. Kotlyarov have foundthat for a large time the solution breaks up into a train of asymptoticsolitons located in the domain $4c^2t-C_N \ln t<x\leq4c^2t$ ($C_N$ is aconstant). The number N of these solitons grows unboundedly as $t\to\infty$. In2010 V. Kotlyarov and A. Minakov have studied temporary asymptotics of thesolution of the Cauchy problem on the whole line and have found that in thedomain $-6c^2 t<x<4c^2t$ this solution is described by a modulated ellipticwave. We considere here the modulated elliptic wave in the domain $4c^2t-C_N\ln t<x<4c^2t$. Our main result shows that the modulated elliptic wave alsobreaks up into solitons, which are similar to the asymptotic solitons, butdiffer from them in phase. It means that the modulated elliptic wave does notrepresent the asymptotics of the solution in the domain $4c^2t-C_N \lnt<x<4c^2t$. The correct asymptotic behavior of the solution is given by thetrain of asymptotic solitons. However, in the asymptotic regime as $t\to\infty$in the region $4c^2t-\frac{N+1/4}{c}\ln t<x<4c^2t-\frac{N-3/4}{c}\ln t$ we canwatch precisely a pair of solitons with numbers N. One of them is theasymptotic soliton while the other soliton is generated from the elliptic wave.Their phases become closer to each other for a large N, i.e. these solitons arealso close to each other. This result gives the answer on a very importantquestion about matching of the asymptotic formulas in the mentioned regionwhere the both formulas are well-defined. We have a new and earlier unknownmechanism of matching of the asymptotics of the solution in the adjacentregions.

Modified Korteweg-de Vries equation: modulated elliptic wave and a train of asymptotic solitons