Step-initial function to the MKdV equation: Hyper-elliptic long-time asymptotics of the solution

The modified Korteweg-de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. $q(x,0) = c_r$ for $x > 0$ and $q(x,0) = c_l$ for $x < 0$, where $c_l, c_r$ are real numbers which satisfy $c_l > c_r> 0.$ The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as $t\to+\infty.$ Using the steepest descent method we deform the original oscillatory matrix Riemann-Hilbert problem to 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 $x < -6c_l^2t + 12c_r^2 t$ and $x > 4c_l^2 t + 2c_r^2 t$ the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $(-6c_l^2+ 12c_r^2)t < x < (4c_l^2+ 2c_r^2)t$ the asymptotics of the solution takes the form of a modulated hyper-elliptic wave generated by an algebraic curve of genus 2. V. Kotlyarov and A. Minakov, 2012.