|The short pulse (SP) equation was derived recently as a model equation for the propagation of ultra-short optical pulses in nonlinear media.
The SP equation admits various interesting exact solutions such as loop soliton solutions and breather solutions.
Discrete integrable systems have received much attention recently because of many applications to numerical algorithms, computer visualization,
mathematical physics, etc. In the study of discrete integrable systems, finding integrable discretizations of continuous integrable systems
such as soliton equations is one of keys to explore the world of discrete integrable systems.
Although integrable discretizations for many of soliton equations such as the KdV, mKdV, NLS equations were found, integrable discretizations
of some classes of soliton equations having solutions with singularities (e.g. the SP equation) are still missing.
In this talk, we propose new integrable semi-discretization and fully discretization of the SP equation by using the method proposed by us
recently. These discrete equations have various exact solutions which correspond to exact solutions of the continuous SP equation.
Using the semi-discrete analogue of the SP equation, we perform numerical computations of loop solitons.