Asymptotics of rarefaction wave solution to the mKdV equation

The modified Korteweg-de Vries equation $q_t+ 6q^2q_x+ q_{xxx}= 0$ on the line is considered. The initial data is the pure step function, i.e. $q(x, 0) = c_r$ for $x>0$ and $q(x, 0) = c_l$ for $x < 0$, where $c_l > c_r> 0$ are arbitrary real numbers. The goal of this paper is to study the asymptotic behavior of the solution of initial-value problem as $t\to-infty$, i.e. to study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann-Hilbert problem to the explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the xt-plane. In the regions $x < 6c_l^2 t$ and $x > 6c_r^2 t$, the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $6c_l^2 t < x < 6c_r^2 t$, the asymptotics of the solution tends to $\sqrt{\frac{x}{6t}}$. A. Minakov, 2011.