Minakov O. Riemann - Hilbert problems and the modified Korteweg - de Vries equation: asymptotic analysis of the solutions with the step-like initial data

Purpose of the work is constructing of step-like solutions of the modified Korteweg - de Vries equation and the development of analytic method for large time asymptotic analysis as Riemann - Hilbert problems, as solutions of the modified Korteweg - de Vries equation. The object of research is the modified Korteweg - de Vries equation. The results obtained are new. The main results are the following. We propose the algorithm of constructing of the set of matrix Riemann - Hilbert problems, which lead to step-like solutions of the modified Korteweg - de Vries equation. We prove theorems of existence and uniqueness of solutions of the Riemann - Hilbert problems, which are in the set. We prove smoothness of solutions in external parameters (space -time). We study large time asymptotic behavior of the solution of the matrix Riemann - Hilbert problem. In order to do this we find the corresponding factorizations of the jump matrix, construct the sequence of transformations of the Riemann - Hilbert problem, construct the corresponding phase functions. Due to this the problem is reduced to the model problems, which are solved explicitly in elementary and special functions. We obtain the explicit formulas for the solutions of the model problems in elliptic, hyperelliptic and parabolic cylinder functions. Due to this we obtain explicit asymptotic formulas for solutions of the modified Korteweg - de Vries equation.