  In this paper, we investigate the long-time asymptotic behavior of the
solution to the initial value problem for the modified Camassa-Holm (mCH)
equation with cubic nonlinearity. The equation is known to be integrable, which
we mean it admits an Lax pair. We formulate the initial value problem as an
associate vector Riemann-Hilbert problem, which allows us to give a parametric
representation of the solution to the initial value problem in terms of the
solution of the Riemann-Hilbert problem. And then by adopting the nonlinear
steepest descent method, we can get the explicit leading order asymptotic of
the solution as time goes to infinity.
