  The short pulse equation was introduced by Schaefer--Wayne (2004) for
modeling the propagation of ultrashort optical pulses. While it can describe a
wide range of solutions, its ultrashort pulse solutions with a few cycles,
which the conventional nonlinear Schroedinger equation does not possess, have
drawn much attention. In such a region, existing numerical methods turn out to
require very fine numerical mesh, and accordingly are computationally
expensive. In this paper, we establish a new efficient numerical method by
combining the idea of the hodograph transformation and the structure-preserving
numerical methods. The resulting scheme is a self-adaptive moving mesh scheme
that can successfully capture not only the ultrashort pulses but also exotic
solutions such as loop soliton solutions.
