Step-Initial Function to the MKdV Equation: Hyper-Elliptic Long-Time Asymptotics of the Solution

摘要

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 ≥ 0 and q ( x , 0) = c_l for x 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 → ∞. 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 l ² t + 12 c _r ² and x > 4 c _l ² t + 2 c _r ² t the main term of asymptotics of the solution is equal to c_l and c_r , respectively. In the region (-6 c _l ² + 12 c _r ² ) t l ² + 2 c _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.