An initial-value problem for the defocusing modified Korteweg-de Vries equation

In this paper we address an initial-value problem for the defocusing modified Korteweg-de Vries (mKdV(-)) equation. The normalized modified Korteweg-de Vries equation considered is given by u(tau) - gamma u(2)u(x) + u(xxx) = 0, -infinity < x < infinity, tau > 0, where x and tau represent dimensionless distance and time respectively and gamma (> 0) is a constant. We consider the case when the initial data has a discontinuous step, where u(x, 0) = u(0) (> 0) for x >= 0 and u(x, 0) = -u(0) for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the complete large-tau asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave (kink) solution propagating in the -x direction with speed -u(0)(2)gamma/3 and connecting u = u(0) to u = -u(0), while the solution is oscillatory in x < -gamma u(0)(2)tau as tau ->infinity on (oscillating about u = -u(0)), with the oscillatory envelope being of O(tau(-1/2)) as tau ->infinity. The asymptotic correction to the propagation speed of the travelling wave solution is given by 1/2u(0) root 3/2 gamma 1/tau as tau -> infinity, and the rate of convergence of the solution of the initial-value problem to the travelling wave solution is found to be algebraic in tau, as tau -> infinity, being of O(1/tau). A brief discussion of the structure of the large-time solution to the mKdV(-) equation when the initial data is given by the general discontinuous step, u(x, 0) = u(+) for x >= 0 and u(x, 0) = u(-) (not equal u(+)) for x < 0, is also given. (C) 2011 Elsevier Inc. All rights reserved.