Long-time behavior of the solution to the mKdV equation with step-like initial data

Abstract

We consider the modified Korteweg–de Vries equation on the line. The initial data is the pure step function, i.e. q(x, 0) = cr for x ⩾ 0 and q(x, 0) = cl for x < 0, where cl > cr > 0 are arbitrary real numbers. Long-time behavior of the solution to the mKdV equation in the case cl > cr = 0 and t → ∞ was studied recently in Kotlyarov and Minakov (2010 J. Math. Phys. 51 093506) where the structure of the compression wave was obtained in the form of a modulated elliptic wave. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t → −∞, i.e. we 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 a different asymptotic behavior in different regions of the xt-plane. In the regions x < 6c2lt and x > 6c2rt the main term of asymptotics of the solution is equal to cl and cr, respectively. In the region the asymptotics of the solution tends to . An influence of the dispersion is also studied: the second term of the asymptotics is obtained in the region x > −6c2rt, where the background constant cr is perturbed by the self-similar vanishing wave.