  In this paper, we consider the initial value problem for the complex short
pulse equation with a Wadati-Konno-Ichikawa type Lax pair.
  We show that the solution to the initial value problem has a parametric
expression in terms of the solution of $2\times 2$-matrix Riemann-Hilbert
problem, from which an implicit one-soliton solution is obtained on the
discrete spectrum. While on the continuous spectrum we further establish the
explicit long-time asymptotic behavior of the non-soliton solution by using
Deift-Zhou nonlinear steepest descent method.
