Long Time Asymptotic Behavior for the Nonlocal mKdV Equation in Solitonic Space-Time Regions

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions q(t )(x , t) - 6 sigma q(x , t)q(-x, -t)q(x)(x , t) + q(xxx)(x , t) = 0, q(x , 0) = q(0)(x), (x ->+/-infinity)lim q(0 )(x) = q +/-,where |q +/-| = 1 and q(+) = delta q- , sigma delta = -1. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region -6 < xi < 6 with xi = (t)/(x) . In this paper, we give the asymptotic expansion of the solution q(x , t) for other solitonic regions xi < -6 and xi > 6. Based on the Riemann-Hilbert formulation of the Cauchy problem, further using the & part;(& macr;) steepest descent method, we derive different long time asymptotic expansions of the solution q(x , t) in above two different space-time solitonic regions. In the region xi < -6, phase function theta(z) has four stationary phase points on the R. Correspondingly, q(x , t) can be characterized with an N(lambda)-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function Im nu (zeta(i) ). In the region xi > 6, phase function theta(z) has four stationary phase points on iR , the corresponding asymptotic approximations can be characterized with an N(lambda)-soliton with diverse residual error order O(t(-1)).