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

Abstract

We study the long-time asymptotic behavior of the Cauchy problem for the modified Korteweg - de Vries equation with an initial function of the step type. This function rapidly tends to zero as $x\to+\infty$ and to some positive constant c as $x\to-\infty$. In 1989 E. Khruslov and V. Kotlyarov have found that for a large time the solution breaks up into a train of asymptotic solitons located in the domain $4c^2t-C_N \ln t<x\leq4c^2t$ ($C_N$ is a constant). The number N of these solitons grows unboundedly as $t\to\infty$. In 2010 V. Kotlyarov and A. Minakov have studied temporary asymptotics of the solution of the Cauchy problem on the whole line and have found that in the domain $-6c^2 t<x<4c^2t$ this solution is described by a modulated elliptic wave. 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 also breaks up into solitons, which are similar to the asymptotic solitons, but differ from them in phase. It means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain $4c^2t-C_N \ln t<x<4c^2t$. The correct asymptotic behavior of the solution is given by the train 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 can watch precisely a pair of solitons with numbers N. One of them is the asymptotic 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 are also close to each other. This result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well-defined. We have a new and earlier unknown mechanism of matching of the asymptotics of the solution in the adjacent regions.