Step-initial function to the mKdV equation: hyper-elliptic long-time asymptotics of the solution

The modified Korteveg–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 $x\geq0$ 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 $x<-6c_l^2t+12c_r^2t$ and $x>4c_l^2t+2c_r^2t$ 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.